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+{"text":"\\section{Introduction}\n\nRecent investigations seem to show that soft $\\gamma$-ray repeaters and some\nanomalous X-ray pulsars are neutron stars which may have surface magnetic fields larger that\n$10^{15}$ G~\\cite{duncan,usov,pacz}, the so called magnetars.  Until recently, the strongest \nestimated magnetic field is of the order of $B=2\\times 10^{15}$ G and was detected \nin a quite young star, SGR 1806-20~\\cite{sgr}. According to~\\cite{kouve} a fraction as high as \n$10\\%$ of the neutron star population could be magnetars. \n\nThe effect of the strong magnetic fields on the equation of state (EOS) of\nstellar matter in neutron stars has been\nstudied both at low densities below neutron drip, of interest for the study of \nthe outer crust of neutron stars~\\cite{low}, and at  high densities, of interest for the\nstudy of the interior of compact stars~\\cite{chakrabarty96,broderick}. In\nthis last case field-theoretical descriptions based on the non-linear \nWalecka model (NLWM)~\\cite{bb}  were used and several\nparametrizations compared. It was shown that they have an overall similar\nbehavior.\nIt was recently shown in~\\cite{aziz08} that the EOS at subsaturation densities, including densities\nof the order of the densities at the\ninner edge of the crust of a compact star, was particularly affected by fields of\nthe order of 10$^{18}$ G. \n\nAn important characteristic of the nuclear matter is the appearance of a\nliquid-gas phase transition at subsaturation densities. The role of the\nisospin is of particular importance. Indeed, since nuclear matter is composed\nof two different fluids, namely protons and neutrons, the liquid-gas phase\ntransition can lead to an isospin distillation phenomenon~\\cite{chomaz}. \n The region of \ninstability is determined by the spinodal curve. Due to the symmetry energy, the EOS of $\\beta$-equilibrium of magnetic free nuclear matter is thermodynamically stable. The stability of the EOS is determined from the curvature of the free-energy: a positive curvature corresponds to thermodynamically stable matter.\n\nIf we\nconsider stellar matter at very low densities, nuclei in matter are expected\nto form a Coulomb lattice embedded in the neutron-electron sea that\nminimizes the Coulomb interaction energy. With an increase of the density,\nnuclear \"pasta\" structures emerge~\\cite{rpw83}. The existence of pasta phases\nmay modify some important processes by\nchanging the hydrodynamic properties and the neutrino opacity in supernova\nmatter and in the matter of newly born neutron stars~\\cite{hor04}. Also, the\npasta phases may influence neutron star quakes and pulsar glitches via the\nchange of mechanical properties of the crust matter~\\cite{quakes}. It is\ntherefore important to study how the magnetic field could affect the extension\nof the pasta phase and the isospin distillation effect.\n\n\n\n In fact, sufficiently strong magnetic field affect the extension of the unstable region.\nIn order to\nhave a better understanding of the effect of the magnetic field on the\ninstabilities of nuclear matter at subsaturation densities we study in the\npresent work the effect of a strong magnetic field on the thermodynamical\nspinodal instabilities obtained from the free energy curvature matrix~\\cite{umodes06,abmp06}.\nRecently, it was shown that the magnetic field and Joule heating have \nthe important effect of maintaining compact stars warm for a\nlonger time~\\cite{pons08}. This kind of simulations need the EOS of the crust. It is,\ntherefore, important to make a study that shows when should the magnetic field \nbe taken explicitly into account in the EOS of the crust. An unstable\nregion in a wider density range  will correspond to a larger crust and the\nproperties of the star depending on the crust will be affected. \n\nIn the present paper, we will consider two  relativistic effective approaches: a\nNLWM, TM1~\\cite{tm1}, with constant coupling parameters,\nand a density dependent relativistic hadronic (DDRH) model TW~\\cite{tw} \nwith density-dependent coupling parameters. DDRH models  seem to give results closer Skyrme interactions  than NLWM,\n at subsaturation densities~\\cite{camille08}.\n\nIn section II we make a brief review of the models, the EOS under the effect of\na magnetic field and  the stability conditions.  Results are discussed in section III and conclusions are drawn\nin the last section. \n\n\\section{The formalism}\n\n\\subsection{EOS of nuclear matter under a strong magnetic field}\n\nIn the present section we make a short review of\nthe field-theoretical approach used to obtain the EOS of nuclear matter. Within this approach, the baryons\ninteract via the exchange of $\\sigma$, $\\omega$ and $\\rho$ mesons in the presence of a\nuniform magnetic field $B$ along the $z$-axis.  We start from the Lagrangian\ndensity of TW~\\cite{fuchs,tw} model\n\\begin{equation}\n{\\cal L}= \\sum_{b=n, p}{\\cal L}_{b} + {\\cal L}_{m}.\n\\label{lan}\n\\end{equation}\nThe baryon ($b$=$n$, $p$) and meson ($\\sigma$, $\\omega$ and $\\rho$) Lagrangians are given by\n\\begin{widetext}\n\\begin{eqnarray}\n{\\cal L}_{b}&=&\\bar{\\Psi}_{b}\\left(i\\gamma_{\\mu}\\partial^{\\mu}-q_{b}\\gamma_{\\mu}A^{\\mu}- \nm_{b}+\\Gamma_{\\sigma}\\sigma\n-\\Gamma_{\\omega}\\gamma_{\\mu}\\omega^{\\mu}-\\frac{1}{2}\\Gamma_{\\rho}\\tau_{3 b}\\gamma_{\\mu}\\rho^{\\mu}\n-\\frac{1}{2}\\mu_{N}\\kappa_{b}\\sigma_{\\mu \\nu} F^{\\mu \\nu}\\right )\\Psi_{b}, \\cr\n{\\cal L}_{m}&=&\\frac{1}{2}\\partial_{\\mu}\\sigma \\partial^{\\mu}\\sigma\n-\\frac{1}{2}m^{2}_{\\sigma}\\sigma^{2}\n+\\frac{1}{2}m^{2}_{\\omega}\\omega_{\\mu}\\omega^{\\mu}\n-\\frac{1}{4}\\Omega^{\\mu \\nu} \\Omega_{\\mu \\nu}  \\cr\n&-&\\frac{1}{4} F^{\\mu \\nu}F_{\\mu \\nu}\n+\\frac{1}{2}m^{2}_{\\rho}\\rho_{\\mu}\\rho^{\\mu}-\\frac{1}{4}  P^{\\mu \\nu}P_{\\mu \\nu},\n\\label{lagran}\n\\end{eqnarray}\n\\end{widetext}\nrespectively, where $\\Psi_{b}$ are the baryon Dirac fields. The nucleon mass and isospin projection for the protons and neutrons are denoted by $m_{b}$  and $\\tau_{3 b}=\\pm 1$, respectively. The mesonic and electromagnetic field strength tensors are given by their usual expressions: $\\Omega_{\\mu \\nu}=\\partial_{\\mu}\\omega_{\\nu}-\\partial_{\\nu}\\omega_{\\mu}$, $P_{\\mu \n\\nu}=\\partial_{\\mu}\\rho_{\\nu}-\\partial_{\\nu}\\rho_{\\mu}$, and  $F_{\\mu\n\\nu}=\\partial_{\\mu}A_{\\nu}-\\partial_{\\nu}A_{\\mu}$. The nucleon anomalous\nmagnetic moments are introduced via the coupling of the baryons to the\nelectromagnetic field tensor with $\\sigma_{\\mu\n  \\nu}=\\frac{i}{2}\\left[\\gamma_{\\mu},  \\gamma_{\\nu}\\right] $ and strength\n$\\kappa_{b}$ with $\\kappa_{n}=g_n\/2=-1.91315$ for the neutron and\n$\\kappa_{p}=(g_p\/2-1)=1.79285$ for the proton, respectively, and where $g_i$\nare the Land\\'e $g$ factors for the particle $i$ ($g_p=5.5856912$ and\n$g_n=-3.8260837$), and $\\mu_N$ is the nuclear magneton. The electromagnetic field is assumed to be externally generated (and thus has no associated field equation), and only frozen-field configurations will be considered.  The electromagnetic couplings are denoted by $q$.\nThe parameters of the model are the nucleon mass $m_b=939$~MeV, the\nmasses of mesons $m_\\sigma$, $m_\\omega$ and $m_\\rho$ and the density\ndependent coupling parameters $\\Gamma$ which are adjusted in order to\nreproduce some of the nuclear matter bulk properties and  DBHF (Dirac\nBrueckner Hartree-Fock) calculations~\\cite{dbhf}, using the following parametrization\n\\begin{equation}\n\\Gamma_{i}(\\rho)=\\Gamma_{i}(\\rho_{sat})f_{i}(x),\\quad i=\\sigma, \\omega, \\rho\n\\label{gam1}\n\\end{equation}\nwhere $x={\\rho}\/{\\rho_{sat}}$, with\n\\begin{equation}\nf_{i}(x)=a_{i}\\frac{1+b_{i}\\left(x+d_{i}\\right)^{2}}{1+c_{i}\\left(x+d_{i}\\right)^{2}},\n\\quad i=\\sigma, \\omega,\n\\label{gam2}\n\\end{equation}\nand,\n\\begin{equation}\nf_{\\rho}(x)=\\exp\\left[ -a_{\\rho}(x-1)\\right], \n\\end{equation}\nwith the values of the parameters $m_i$, $\\Gamma_{i}$, $a_{i}$, $b_{i}$,\n$c_{i}$ and $d_{i}$, $i=\\sigma, \\omega, \\rho$ \ngiven in Table~\\ref{table1} for TW model~\\cite{tw}. Other possibilities for these parameters are also found in the\nliterature~\\cite{hbt}. \n\nThe symmetry energy and its first and second derivatives are important to understand the instability \nregion.\nNLWM models become very stiff above saturation densities while DDRH models have \na softer behavior. On the other hand, at subsaturation densities DDRH models\nhave larger symmetry energies and  a larger extension of the\ninstability region for very asymmetric matter. In Fig. \\ref{esym} we compare\nthe symmetry energy of the models we will consider in the present study: TM1\nand TW. As expected TM1 has a smaller symmetry energy at subsaturation\ndensities and a larger one above the saturation density.\n\n\\begin{table}[Htb]\n\\begin{tabular}{ccccccc}\n\\hline\n\\hline\ni &$m_{i}$(MeV)& $\\Gamma_{i}$ & $a_{i}$ & $b_{i}$ & $c_{i}$ & $d_{i}$  \\\\\n\\hline\n$\\sigma$   & 550 &10.72854 &1.365469 &0.226061 &0.409704&0.901995\\\\\n$\\omega$ &783 &13.29015 & 1.402488&0.172577 &0.344293&0.983955 \\\\\n$\\rho$       &763 & 7.32196 & 0.515 & & & \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Parameters of the TW model.}\n\\label{table1}\n\\end{table} \n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure1.eps}\n\\caption{Symmetry energy for all models used in the present work. }\n\\label{esym}\n\\end{figure}\n\nNotice that in the DDRHM  the nonlinear meson terms are not present, in contrast\nwith the usual NLWM. For TM1 model we add\nto the Langrangian density, Eq. (\\ref{lagran}) , with $g_i=\\Gamma_i$,\n$${\\cal L}=\\frac{1}{3}bm_n(g_{\\sigma}\\sigma)^3+\\frac{1}{4}c(g_{\\sigma}\\sigma)^4+\\frac{1}{4!}\\xi g^4_\\omega(\\omega_{\\mu}\\omega^{\\mu} )^2. $$\nThe coupling parameters are constant and given in~\\cite{tm1}.\n\n\nFrom now we take the standard mean-field theory (MFT) approach and display\nonly some of the equations. A complete set of equations and description of the\nmethod can be found in the literature (\\textit{e.g.}, Ref.~\\cite{yuan,\n  broderick}). For  the description of the system, we need the baryonic\ndensity, the energy density of nuclear  matter, and the pressure. The energy density of nuclear\nmatter is given by \n\\begin{equation}\n\\varepsilon=\\sum_{b=p,n} \\varepsilon_{b}+\\frac{1}\n{2}m^{2}_{\\sigma}\\sigma^{2}+\\frac{1}{2}m^{2}_{\\omega}\\omega^{2}_{0}+\\frac{1}{2}m^{2}_{\\rho}\\rho^{2}_{0}\n\\label{ener}\n\\end{equation} \nwhere the energy densities of nucleons have the following forms\n\\begin{eqnarray}\n\\varepsilon_{p}&=&\\frac{q_{p}B}{4\\pi^ {2}}\\sum_{\\nu, s}\\left[k^{p}_{F,\\nu,s}E^{p}_{F}\n+\\left(\\sqrt{m^{* 2}+2\\nu q_{p}B}-s\\mu_{N}\\kappa_{p}B\\right) ^{2} \n\\ln\\left|\\frac{k^{p}_{F,\\nu,s}+E^{p}_{F}}{\\sqrt{m^{* 2}+2\\nu q_{p}B}-s\\mu_{N}\\kappa_{p}B} \\right|\\right] , \\cr\n\\varepsilon_{n}&=&\\frac{1}{4\\pi^ {2}}\\sum_{s}\\bigg[\\frac{1}{2}k^{n}_{F, s}E^{n 3}_{F}-\\frac{2}\n{3}s\\mu_{N}\\kappa_{n} B E^{n 3}_{F}\\left(\\arcsin\\left(\\frac{\\bar{m}_{n}}{E^{n}_{F}} \\right)-\\frac{\\pi}\n{2}\\right)-\\left(\\frac{1}{3}s\\mu_{N}\\kappa_{n} B +\\frac{1}{4}\\bar{m}_{n}\\right) \\cr\n&&\\left(\\bar{m}_{n}k^{n}_{F, s}E^{n}_{F}+\\bar{m}^{3}_{n}\\ln\\left|\\frac{k^{n}_{F,s}+E^{n}_{F}}{\\bar{m}_{n}} \n\\right|\\right) \\bigg].\n\\end{eqnarray}\nFor the neutrons we have introduced\n\\begin{equation}\n\\bar{m}_{n}=m^{*}-s\\mu_{N}\\kappa_{n}B,\n\\label{barm}\n\\end{equation}\nwhere the effective baryon masses are given by \n\\begin{equation}\nm^{*}=m-\\Gamma_{\\sigma}\\sigma.\n\\end{equation} \n\nThe pressure of the system is obtained from the expression\n\\begin{equation}\nP_{m}=\\sum_{b=n,p}\\mu_{b}\\rho_{b}-\\varepsilon.\n\\label{press}\n\\end{equation}\n  \nThe energy spectra for protons are neutrons are given by\n\\begin{eqnarray}\nE^{p}_{\\nu, s}&=& \\sqrt{k^{2}_{z}+\\left(\\sqrt{m^{* 2}+2\\nu q_{p}B}-s\\mu_{N}\\kappa_{p}B \\right) \n^{2}}+\\Gamma_{\\omega} \\omega^{0}+\\frac{1}{2}\\Gamma_{\\rho}\\rho^{0}+\\Sigma^{R}_{0}, \\label{enspc1}\\\\\nE^{n}_{s}&=& \\sqrt{k^{2}_{z}+\\left(\\sqrt{m^{* 2}+k^{2}_{x}+k^{2}_{y}}-s\\mu_{N}\\kappa_{n}B \n\\right)^{2}}+\\Gamma_{\\omega} \\omega^{0}-\\frac{1}{2}\\Gamma_{\\rho}\\rho^{0}+\\Sigma^{R}_{0},\n\\label{enspc2} \n\\end{eqnarray}\nrespectively, where $\\nu=n+\\frac{1}{2}-sign(q)\\frac{s}{2}=0, 1, 2, \\ldots$ enumerates the Landau levels of the fermions with electric charge $q$, the quantum number $s$ is $+1$ for spin up and $-1$ for spin down\ncases, and the rearrangement term is given by \n\\begin{equation}\n\\Sigma^{R}_{0}=\\frac{\\partial \\Gamma_{\\omega}}{\\partial \\rho}\\rho_b\\omega_{0}+\\frac{\\partial \\Gamma_{\\rho}}{\\partial \n\\rho}\\rho_{3}\\frac{\\rho_{0}}{2}-\\frac{\\partial \\Gamma_{\\sigma}}{\\partial \\rho}\\rho^s\\sigma,\n\\end{equation}\nwhere $\\rho^s=\\rho^s_p+\\rho^s_n$ and $\\rho_b=\\rho_p+\\rho_n$, with the expressions of the scalar and vector densities for protons and neutrons given by~\\cite{broderick}\n\\begin{eqnarray}\n\\rho^{s}_{p}&=&\\frac{q_{p}Bm^{*}}{2\\pi^{2}}\\sum_{\\nu=0}^{\\nu_{\\mbox{\\small max}}}\\sum_{s}\\frac{\\sqrt{m^{* 2}+2\\nu \nq_{p}B}-s\\mu_{N}\\kappa_{p}B}{\\sqrt{m^{* 2}+2\\nu q_{p}B}}\\ln\\left|\\frac{k^{p}_{F,\\nu,s}+E^{p}_{F}}\n{\\sqrt{m^{* 2}+2\\nu q_{p}B}-s\\mu_{N}\\kappa_{p}B} \\right|, \\cr\n\\rho^{s}_{n}&=&\\frac{m^{*}}{4\\pi^{2}}\\sum_{s} \\left[E^ {n}_{F}k^{n}_{F, s}-\\bar{m}^{2}_{n}\\ln\\left|\n\\frac{k^{n}_{F,s}+E^{n}_{F}}{\\bar{m}_{n}} \\right|\\right],  \\cr\n\\rho_{p}&=&\\frac{q_{p}B}{2\\pi^{2}}\\sum_{\\nu, s}k^{p}_{F,\\nu,s},  \\cr\n\\rho_{n}&=&\\frac{1}{2\\pi^{2}}\\sum_{s}\\left[ \\frac{1}{3}\\left(k^{n}_{F, s}\\right) ^{3}-\\frac{1}\n{2}s\\mu_{N}\\kappa_{n}B\\left(\\bar{m}_{n}k^{n}_{F,s}+E^{n 2}_{F}\\left(\\arcsin\\left( \\frac{\\bar{m}_{n}}\n{E^{n}_{F}}\\right) -\\frac{\\pi}{2} \\right)  \\right) \\right] \n\\end{eqnarray}\nwhere $k^{p}_{F, \\nu, s}$ and $ k^{n}_{F, s}$ are the Fermi momenta of protons and neutrons which are related to the Fermi energies $E^{p}_{F}$ and $E^{n}_{F}$ through\n\\begin{eqnarray}\nk^{p 2}_{F,\\nu,s}&=&E^{p 2}_{F}-\\left[\\sqrt{m^{* 2}+2\\nu q_{p}B}-s\\mu_{N}\\kappa_{p}B\\right] ^{2} \\cr\nk^{n 2}_{F,s}&=&E^{n 2}_{F}-\\bar{m}^{2}_{n}.\n\\end{eqnarray}\n\nThe chemical potentials of nucleons within TW are given  by\n\\begin{eqnarray}\n\\mu_{p}&=& E^{p}_{F}+\\Gamma_{\\omega}\\omega^{0}+\\frac{1}{2}\\Gamma_{\\rho}\\rho^{0}\n+\\Sigma^{R}_{0} \\cr\n\\mu_{n}&=& E^{n}_{F}+\\Gamma_{\\omega}\\omega^{0}-\\frac{1}{2}\\Gamma_{\\rho}\\rho^{0}\n+\\Sigma^{R}_{0}.\n\\label{mu}\n\\end{eqnarray}\nFor TM1 we have similar expressions with the last term, the rearrangement\nterm, equal to zero. \n\n\\subsection{Stability conditions for nuclear matter}\n\nAt subsaturation densities nuclear matter has a liquid-gas phase transition\nand homogeneous matter is not stable within a given range of densities. \nThe stability conditions for asymmetric nuclear matter, keeping volume and temperature constant, \nare obtained from the free energy density $\\cal F$, imposing that this\nfunction is a convex function of the densities $\\rho_p$ and $\\rho_n$. For\nstable homogeneous matter, the symmetric matrix with the elements~\\cite{Bar03,  marg03}, \n\\begin{equation}\n{\\cal F}_{ij}=\\left( \\frac{\\partial^{2} {\\cal F}}{\\partial \\rho_{i}\\partial\\rho_{j}}\\right) _{T},\n\\end{equation}\nknown as stability matrix,  must be positive. This corresponds to imposing that the trace and the\ndeterminant of ${\\cal F}_{ij}$ are positive. In terms of the proton and\nneutron chemical potentials $\\mu_i$, the stability matrix is given by\n\\begin{equation}\n{\\cal F}=\n\\begin{pmatrix}\n\\displaystyle\\frac{\\partial\\mu_{n}}{\\partial \\rho_{n}} \n&\\displaystyle\\frac{\\partial\\mu_{n}}{\\partial \\rho_{p}} \\\\\n\\displaystyle\\frac{\\partial\\mu_{p}}{\\partial \\rho_{n}}\n&\\displaystyle\\frac{\\partial\\mu_{p}}{\\partial \\rho_{p}}\n\\label{cur}\n\\end{pmatrix},\n\\end{equation}\nwith $\\mu_{i}=\\frac{\\partial{\\cal F}}{\\partial \\rho_{i}}|_{T,\\rho_{j\\neq i}} $.\n\nFor nuclear matter,\nthe largest eigenvalue of the stability matrix is always positive and the\nother becomes negative at subsaturation densities.\nWe define the thermodynamical spinodal at T=0 as the curve on the  $\\rho_n$,\n$\\rho_p$ plane defined by the points  for which the determinant of $ {\\cal  F}_{ij}$ is zero; \nthat is, the smallest  eigenvalue is zero. Inside the region limited by the thermodynamical\nspinodal  the smallest eigenvalue of  ${\\cal F}_{ij}$ is negative and nuclear matter is \nunstable~\\cite{marg03}. At $T=0$, $\\cal F$ is equal to the energy density defined \nby Eq. (\\ref{ener}). The eigenvalues of the stability matrix are given by\n\\begin{equation}\n\\lambda_{\\pm}=\\frac{1}{2}\\left(\\hbox{Tr}({\\cal F})\\pm\\sqrt{\\hbox{Tr}({\\cal F})^2-4 \\hbox{Det}({\\cal F})}\\right).\n\\label{lambd}\n\\end{equation}\nThe stability condition requires that they are both positive. When one curvature becomes negative the system is thermodynamically unstable and can decrease its free energy by going in the instability direction \\cite{marg03}, defined by the direction of the eigenvector associated to the negative eigenvalue.\nThe eigenvectors $\\delta {\\bf \\rho}_{\\pm}$ of the stability matrix are given by \n\\begin{equation}\n\\frac{\\delta \\rho^{\\pm}_{i}}{\\delta \\rho^{\\pm}_{j}}=\\frac{\\lambda_{\\pm}-{\\cal F}_{jj}}{{\\cal F}_{ji}}, \\: i, j = p, n.\n\\end{equation}\nIn the following we study the direction of instability inside the spinodal\nsection for both  models considered. \n\n\n\\subsection{Stability conditions  for $npe$ matter}\nStellar matter at low densities is formed by protons, neutrons and\nelectrons in equilibrium with respect to weak interaction processes. Until now we have considered the subsaturation instability region\nof neutron-proton ($np$) nuclear matter.  In this section we investigate the\neffect of the inclusion of electrons on the stability conditions of nuclear matter when electrons are\nincluded. In particular, we will calculate the thermodynamical spinodal\nsections for $npe$ (neutron-proton-electron) neutral matter. Since matter is neutral the proton and electron densities must be equal, i.e. $\\rho_p=\\rho_e$.\n\nElectrons are included in a minimal way in the system, and are   described by the following the Lagrangian density \n\\begin{equation}\n{\\cal L}_{e} = \\bar{\\psi}_{e}\\left(i\\gamma_{\\mu}\\partial^{\\mu}-q_{e}\\gamma_{\\mu}A^{\\mu}\n-m_{e}\\right )\\psi_{e},\n\\label{lagran}\n\\end{equation}\nwhere $\\psi_{l} $ are the electron Dirac fields and $m_e=0.511 \\hbox{ MeV}$. \n\nIncluding the electrons, the stability matrix  (\\ref{cur}) becomes\n\\begin{equation}\n{\\cal F}=\n\\begin{pmatrix}\n\\displaystyle\\frac{\\partial\\mu_{n}}{\\partial \\rho_{n}} &\\displaystyle\\frac{\\partial\\mu_{n}}{\\partial \\rho_{p}} \\\\\n\\displaystyle\\frac{\\partial\\mu_{p}}{\\partial \\rho_{n}}\n&\\displaystyle\\frac{\\partial\\left(\\mu_{p}+\\mu_{e}\\right)}{\\partial \\rho_{p}}\n\\end{pmatrix},\n\\label{cur1}\n\\end{equation}\nand the stability conditions are equivalent to the ones indicated in the previous subsection: the trace and the determinant of ${\\cal F}$ must be positive.\n\nThe electron density is given by\n\\begin{equation}\n\\rho_e=\\frac{|q_{e}|B}{2\\pi^{2}}\\sum_{\\nu, s}k^{e}_{F, \\nu, s}\n\\end{equation}\nwhere $k^{e}_{F, \\nu, s}$ is  the electron Fermi momentum related to the Fermi energy $E^{e}_{F}$ by\n\\begin{equation}\nk^{e 2}_{F,\\nu,s}=E^{e 2}_{F}-\\left(m^{2}_{e}+2\\nu |q_{e}| B\\right).\n\\end{equation}\nFor $npe$ neutral matter\n$${\\cal F}=\\epsilon+ \\epsilon_e$$\nwhere $\\epsilon$ was defined in (\\ref{ener}) and\nthe electron contribution is given by\n\\begin{equation}\n\\varepsilon_{e}=\\frac{|q_{e}|B}{4\\pi^ {2}}\\sum_{\\nu, s}\\left[k^{l}_{F,\\nu,s}E^{e}_{F}\n+\\left(m^{2}_{e}+2\\nu |q_{e}|B\\right) \n\\ln\\left|\\frac{k^{e}_{F,\\nu,s}+E^{e}_{F}}{\\sqrt{m^{2}_{e}+2\\nu |q_{e}| B}} \\right|\\right] .\n\\end{equation}\nWe next discuss $np$  nuclear matter and  $npe$ neutral matter. $\\beta$-equilibrium stellar matter is a particular case of  $npe$ neutral matter, with the proton and electron fractions defined by chemical equilibrium conditions, namely\n$$\\mu_p=\\mu_n-\\mu_e,$$\nwith $\\mu_p$ and $\\mu_n$ defined in (\\ref{mu}) and $\\mu_e=E^{e }_{F}$.\n \n\\section{Results and discussion}\n\nIn the present section we first show the dependence of the spinodal section on  the magnetic field, \nboth for neutron-proton  ($np$) and neutron-proton-electron ($npe$) matter. \nFrom the crossing of the  $\\beta$-equilibrium  EOS of stellar matter with the thermodynamical spinodal we make a prevision of the transition density and transition pressure at the inner edge of the crust of a compact star. \n\nFor a zero magnetic field,  the direction of instability of nuclear matter gives rise to a \ndistillation effect, corresponding to the formation of droplets of  dense matter with low isospin asymmetry\nin a background of a neutron gas with a small fraction of protons. This effect has been observed experimentally in heavy-ion reactions \\cite{chomaz}. Therefore, \nwe also discuss the effect of the magnetic field on the direction of instability, namely in which way it affects the distillation effect. \n\n\n\\subsection{Spinodal section $np$ matter}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure2.eps}\n\\caption{(Color online) Spinodal section on the  $\\rho_p$, $\\rho_n$\n  plane for TM1 at $T = 0 \\hbox{ MeV}$ and for severals values of\n  magnetic fields, $B=B^*\\, B_e$, a)  without  and b) with  AMM. For $B^*=10^5$ the spinodal section is formed by two separate regions.}\n\\label{spztm1npn}\n\\end{figure}\n\nWe will first consider $np$ nuclear  matter and determine the instability region limited by the spinodal surface.\nIn Fig.~\\ref{spztm1npn} and Fig.~\\ref{spztwnp} we show the spinodal sections obtained making $\\lambda_-=0$, where $\\lambda_-$ was defined in Eq.  (\\ref{lambd}),  on\nthe ($\\rho_p$, $\\rho_n$) plane for TM1  and TW and  for several values of the\nmagnetic field.\nWe define the magnetic field in units of the critical  field $B^c_e=4.414 \\times 10^{13}$~G, so that  $B=B^* \\, B^c_e$. For a field with the intensity $B^c_e$ the electron cyclotron energy is equal to the electron mass. \n\n We present the numerical results both not including and including the contribution of the anomalous magnetic moment (AMM).  In all figures where both cases are considered we \nshow on the left panel the results without the magnetic field   and on the right panel the results\nincluding  the AMM. \n\nThe magnetic\nfield has a strong effect not only on the size, giving rise to larger\ninstability regions, but also on the shape of the\ninstability zones which is not symmetric with respect to the $\\rho_p=\\rho_n$ line, contrary to $np$ matter for $B=0$. \nThe increase of the instability region is due to \nLandau quantization which softens the EOS. \n For magnetic fields $B^{*}>2\\times10^{5}$, the\nprotons are totally polarized for all the values of the densities\nconsidered and the size of the spinodal zone is larger than\nthe one obtained for $B=0$.\nIncluding AMM decreases the spinodal region with respect to the results\nwithout AMM for all the values of the magnetic field considered.\nThis behavior is explained by the extra stiffness that  \nthe inclusion of AMM brings  into  the system due to neutron spin polarization.\n\\bigskip\n\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure3.eps}\n\\caption{(Color online) Spinodal section in terms of $\\rho_p$ versus $\\rho_n$\nfor TW at $T = 0 \\hbox{ MeV}$ and for severals values of magnetic fields\na) without  and b) with  AMM. For $B^*=10^5$ and $2\\times 10^5$ the spinodal section has respectively three and two separated parts.}\n\\label{spztwnp}\n\\end{figure}\n\nThe effect of the Landau quantization on the spinodal section  is explicitly seen\n in the spinodal for $B^{*}=10^{5}$ for TM1 [dashed line\nFig. \\ref{spztm1npn} a)]. The\nspinodal region consists of two separate zones each one corresponding to one\nLandau level, the one corresponding to the first Landau level extends to\nlarger neutron densities than the one corresponding to the second level.\nIn order to understand this effect,  we plot in Fig. \\ref{chem2} the proton chemical potential for $B^*=0$, \n$B^{*}=10^{5}$ and $B^{*}=3\\times 10^{5}$. We have  identified the\ninstable regions with thick lines. It is seen that for the larger\nfield the proton chemical potential changes smoothly  because for all\nthe densities shown only the first Landau level (LL) is occupied. At\nlow densities the chemical potential decreases with density and only\nabove 0.025 fm$^{-3}$ it starts increasing with density. For\n$B^{*}=10^{5}$ the proton chemical potential shows a cusp\ncorresponding to the end of the first LL and beginning of the second LL. The unstable regions, in this case, correspond to the beginning of each LL when the slope of the chemical potential is smaller. The smaller the magnetic field the larger the number of LL occupied at subsaturation densities and the larger of independent sections which make up the whole spinodal section. For reference we include the chemical potential at $B=0$: it increases smoothly with density with a quite constant slope.\n\n\nIf the AMM is included the instabilities regions are smaller, as discussed above.\nThe structure (bump) appearing at $\\rho_n\\sim 0.05$ fm$^{-3}$ is due to the neutron\npolarization: below that value of the density the neutrons are totally polarized. \n\n\nFor the TW model with $B^{*}=2 \\times 10^{5}$  and  $10^{5}$, Fig. \\ref{spztwnp},  we also get a \nspinodal region formed by several bands, respectively two and three bands. \nAn interesting feature of this model, is that the band with the largest Landau level may\nextend to larger neutron densities than lower levels. This does not occur for NLWM and \nhas to do with the behavior of the symmetry energy which increases in a smoother \nway for DDRHM than for NLWM above $\\rho=0.1$ fm$^{-3}$. As a result the slope of the chemical potentials is not so large.\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure4.eps}\n\\caption{(Color online) The proton chemical potential for $B=0$ and for two magnetic field  intensities, $B^*=10^5$ and  $B^*=2\\times 10^5$, obtained within the TM1\nmodel for the neutron density $\\rho_n=0.05$ fm$^{-3}$ and excluding the\nAMM. The thick lines represent the regions of instability.}\n\\label{chem2}\n\\end{figure}\n \nAnother feature of the spinodal regions with magnetic field and without AMM, that is\npresent in both models we have studied, is the extension of the spinodal for\nzero proton fraction: in the absence of the magnetic field there is no\ninstability but the inclusion of the magnetic field changes this behavior: the instability region \nat $\\rho_p=0$ extends until a finite  $\\rho_n$ value, model dependent, but independent of\n$B$. This value is $\\sim 0.15$ fm$^{-3}$ for TW and  $\\sim 0.186$ fm$^{-3}$ for TM1. \n\nTo understand this behavior seen at zero proton fraction, we consider the TM1 model.  For $\\rho_p=0$, we obtain the\ncorresponding finite value of $\\displaystyle\\rho_n=\\frac{k^{n 3}_ {F}}{3\\pi^2}$ from the Fermi neutron momenta $k^{n}_ {F}$ solution of the equation $\\hbox{Det}({\\cal F})=0$ with $\\rho_p=0$. Explicitly, the latter equation can be written as follows \n\n\\begin{eqnarray}\n\\left({\\cal A}_{+}-\\frac{\\cal C}{m^{* 2}} -{\\cal D} \\right) \\left(\\frac{\\pi^{2}}{E^{n}_{F} k^{n}_{F}}+{\\cal A}_{+}-\\frac{\\cal C}{E^{n 2}_{F}}-{\\cal D} \\right) -\\left({\\cal A}_{-}-\\frac{\\cal C}{m^{*} E^{n }_{F}}-{\\cal D} \\right)^{2} = 0\n\\end{eqnarray}\nwhere $\\displaystyle{\\cal A}_{\\pm}=\\left(\\frac{g_{\\omega}}{m_{\\omega}}\\right)^{2}\n\\pm\\frac{1}{4}\\left(\\frac{g_{\\rho}}{m_{\\rho}}\\right)^{2}$, $E^{n}_{F}=\\sqrt{k^{n 2}_ {F}+m^{* 2}}$ , and $\\displaystyle {\\cal C}=\\left(\\frac{g_\\sigma}{m_{\\sigma}}\\right)^2 \\frac{m^{* 2}}{\\cal K}$ with,\n\\begin{eqnarray}\n{\\cal K}=1+\\left(\\frac{g_{\\sigma}}{m_{\\sigma}}\\right)^{2}\\left[2 b (g_{\\sigma}\\sigma)+3c(g_{\\sigma}\\sigma)^2+\\frac{1}{2 \\pi^ 2 E^{n}_{F}}\\left(k^{n 3}_ {F}+3 m^{* 2} k^{n}_ {F}-3 m^{* 2} E^{n}_{F}\\log \\left|\\frac{k^{n}_ {F}+E^{n}_{F}}{m^{*}}\\right|\\right) \\right] \n\\end{eqnarray}\nand $\\displaystyle{\\cal D}=\\frac{\\frac{1}{2}\\xi\\left( \\frac{g_{\\omega}}{m_{\\omega}}\\right)^{4}\\left(g_{\\omega}\\omega^0 \\right)^2}{1+\\frac{1}{2}\\xi\\left( \\frac{g_{\\omega}}{m_{\\omega}}\\right)^{2}\\left(g_{\\omega}\\omega^0 \\right)^2}.$ \\\\\nThis equation is independent of the magnetic field and, therefore, all the\nspinodal regions for different magnetic fields, without AMM, in\nFigs.~\\ref{spztm1npn} and~\\ref{spztwnp} have\nthe same value of $\\rho_n$ for $\\rho_p=0$. The inclusion of AMM changes this\nfeature: $\\rho_n$ is still finite for $\\rho_p=0$ but its value is $B$ dependent.\n\nFor models with density dependent couplings the spinodal region \nextends to larger  densities for the larger proton densities\nwhen compared with NLWM.\nThis is due to the behavior of the symmetry energy: while for\nNLWM the symmetry energy increases quite steeply for densities above\nsaturation densities, DDRH models have a much smoother behavior and the\nsymmetry energy of these models take much smaller values than NLWM for\ndensities above $\\rho=0.15$ fm$^{-3}$. \n\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure5.eps}\n\\caption{(Color online) Proton and neutron chemical potentials  as function of\nthe proton density for $\\rho_n = 0.02$  $\\hbox{ fm}^{-3}$ and for $B^*=10^5$\na) without and b) with  AMM for TM1 (dashed line)  and TW (full line).\nThe thick segments of each curve represent the regions of instability.}\n\\label{chem}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure6.eps}\n\\caption{(Color online) Spinodal section in the  $\\rho_p$,  $\\rho_n$ plane for\nTM1 at $T = 0 \\hbox{ MeV}$ and for severals values of the magnetic field without\nAMM. For each value of the magnetic field, it also plotted the EOS of\nstellar matter in $\\beta$-equilibrium  (thin lines). The crossing of the EOS\nwith the respective spinodal, large dot in each spinodal, represent the\ntransition density to continuous matter. }\n\\label{spztm1npd}\n\\end{figure}\n\n\n\n\n In Fig. \\ref{chem} we plot $\\mu_p$ and  $\\mu_n$ for $B^*=10^5$ with\nthe models TM1 (dashed lines) and TW (full lines). The thick segments of the curves lines identify\nthe instability regions defined by the two (TM1) and three (TW) bands which form the spinodal.  These curves are obtained for a fixed\nneutron density, $\\rho_n=0.02$ fm$^{-3}$. It is seen that above  $\\rho_p=0.1$\nfm$^{-3}$ the proton chemical potential within TW is much softer and this\nseems to be the reason for the appearance of the third band in this model. \n\n\nFor each model TM1 and TW we  identified the crossing density,\nand corresponding pressure, of the EOS of $\\beta$-equilibrium stellar\nmatter and the corresponding spinodal for each value of the magnetic\nfield considered.  The EOS of state was obtained considering neutrons,\nprotons and electrons in $\\beta$-equilibrium. In order to illustrate\nwhat was done we represent in Fig.~\\ref{spztm1npd} the spinodal\nsections obtained within TM1 for $B^*=0, 10^5$ and $5\\times10^5$\nrespectively by full, dashed and dotted thick lines. We include in the\nsame figure, using thin lines with the same type of curve for each $B$ value, the corresponding EOS of  $\\beta$-equilibrium stellar matter. The crossing spinodal-EOS is identified by a big dot. Both the spinodal and the EOS are plotted in the $\\rho_p, \\rho_n$ plane. \n\nThe crossing density of the EOS with the thermodynamical spinodal gives a \nprevision of the transition density \\cite{bao, pasta1} to an homogeneous phase, \nand is always larger than the one obtained from the crossing of the EOS with the\ndynamical spinodal for $npe$ matter, which includes the Coulomb\ninteraction. In \\cite{link99} the authors have shown how the transition density\nand respective pressure were related to the fraction of the star's\nmoment of inertia contained in the solid crust, and obtained\n a relation between the radius and mass of compact stars. \n\nIn Tables~\\ref{table4} and \\ref{table5} the values of the crossing density and\nrespective pressure are given for stellar matter under different magnetic\nfields, respectively without and with the AMM.\nThe values of the crossing density for $B=0$, 0.069 fm$^{-3}$ for TM1 and  0.085 fm$^{-3}$  for TW,  can be compared with the\ncorresponding ones obtained from the crossing of the dynamical spinodal with\nthe EOS Ref. \\cite{camille08,pasta1}, respectively 0.06 fm$^{-3}$ for TM1  and 0.075 fm$^{-3}$ for TW. As expected they are a\nbit larger, with TW model having a larger crossing density than the other.\nThe effect of the magnetic field is to increase the values of the crossing\ndensity: at $B^*=10^5$ both models have similar transition densities of the\norder of $\\sim 0.1$ fm$^{-3}$ corresponding to a much larger pressure for \nTM1 than TW. For  $B^*=3\\times 10^5$ the transition densities increase to\n$\\rho\\sim 0.14-0.15$ fm$^{-3}$.\n\n\\begin{table*}[t]\n\\caption{\nPredicted density, proton fraction and pressure at the inner edge of the\ncrust of a compact star at zero temperature, as defined by the crossing\nbetween the thermodynamical instability region of $np$ matter and the\n$\\beta$-equilibrium EOS for homogeneous, neutrino-free stellar\nmatter in the $\\rho_p,\\rho_n$ plane. The AMM is not included.} \n\\label{table4}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\n$B^{*}$  & Models & $\\rho^{\\hbox{cross}}_b (\\hbox{ fm}^{-3}) $ & $Y_p$ & $P_{m}(\\hbox{ MeV}\\hbox{ fm}^{-3}) $ \\\\\n\\hline\n  $0$            & TM1 & 0.069509 & 0.024713 & 0.50288 \\\\\n                                & TW  & 0.084955 & 0.036690  & 0.52246 \\\\\n$10^{5}$         & TM1  & 0.097030 & 0.14645 & 0.95944  \\\\\n                                       &  TW  & 0.10099   & 0.14641   & 0.67321  \\\\\n$2\\times10^{5}$    & TM1 & 0.12266 & 0.24283 & 1.4008  \\\\\n                                             & TW  & 0.12786  & 0.23599 & 1.0156  \\\\\n$3\\times10^{5}$    & TM1 & 0.14085 & 0.31304 & 1.5944 \\\\\n                                             & TW  & 0.15159 & 0.30219 & 1.3795 \\\\\n$5\\times10^{5}$    & TM1 & 0.16783 & 0.40921  & 1.6324 \\\\\n                                             & TW &0.19784 & 0.40194 & 2.3310 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\nIn Table~\\ref{table5} we show the same data given in Table~\\ref{table4} but\nincluding the AMM in the calculation. The conclusions are similar: the\ntransition density increases with the increase of the magnitude of the magnetic field but not\nso fast. However, the corresponding pressures are larger than before. We conclude that the existence of a strong magnetic field at the crust gives rise to a larger crust.\n\n\\begin{table*}[t]\n\\caption{\nPredicted density, proton fraction and pressure at the inner edge of the crust of a compact star at zero temperature, as defined by the crossing between the thermodynamical instability region of $np$ matter and the $\\beta$-equilibrium condition for homogeneous, neutrino-free stellar matter. The case where the AMM is  included.}  \n\\begin{ruledtabular}\n\\label{table5}\n\\begin{tabular}{c c c c c}\n$B^{*}$ & Models & $\\rho^{\\hbox{cross}}_b (\\hbox{ fm}^{-3}) $ & $Y_p$ & $P_{m}(\\hbox{ MeV}\\hbox{ fm}^{-3}) $ \\\\\n\\hline\n           $10^{5}$             & TM1  & 0.086942 & 0.17670  & 1.3801  \\\\\n                                           &TW  & 0.091391 & 0.17829 & 1.2809 \\\\\n $2\\times10^{5}$ &TM1 & 0.096337 & 0.29468 & 1.5373 \\\\\n                                           &TW & 0.092438 & 0.30512 & 1.3549  \\\\\n$3\\times10^{5}$ &TM1 & 0.11046 & 0.37135 & 1.7091 \\\\\n                                           &TW  & 0.11251 & 0.38035 & 1.8047 \\\\\n $5\\times10^{5}$ & TM1  & 0.12822 & 0.47093 & 1.6616 \\\\\n                                            & TW    & 0.14880 & 0.48803 & 2.7717 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\n\\subsection{Spinodal section $npe$ matter}\n\nWe have studied the effect of the magnetic field on the instability region of\n$np$ matter in the previous section.\nFor $npe$ matter without magnetic field, NLWM models still present a small\nthermodynamical instability region but for DDRHM models there is no instability region \\cite{abmp06}. The\nincompressibility of the free electron gas is so high that the spinodal\ndisappears or almost disappears.     \n\n In Fig. \\ref{spin_npe_nlw} the spinodals for $npe$ matter are shown\n for TM1 and different magnetic fields. \nIn fact,  although including electrons, the instability region can become almost as\nlarge as the $B=0$ $np$-spinodal.\nThis is due both to the Landau quantization of the orbital motion of protons\nand electrons: the incompressibility of the electron gas is smaller than the\none of a magnetic free electron gas.\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure7.eps}\n\\caption{(Color online) Spinodal section in terms of $\\rho_p$ versus $\\rho_n$ \nfor TM1 for $npe$ neutral matter at $T = 0 \\hbox{ MeV}$ and for severals values of magnetic \nfields (a) without  and (b) with  AMM.}\n\\label{spin_npe_nlw}\n\\end{figure}\n\nFor TW, contrary to the $B=0$ case,  the inclusion of the magnetic field gives\nrise to a spinodal region as seen in  Fig. \\ref{spin_npe_tw}.\nThe behavior of this model with the magnetic field  is similar to\nTM1. We also point out that the\n inclusion of the AMM has a strong effect on the spinodal part corresponding to the first LL: it is drastically reduced or even disappears.\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure8.eps}\n\\caption{(Color online) Spinodal section in terms of $\\rho_p$ versus\n  $\\rho_n$ for TW for $npe$ neutral matter at $T = 0  \\hbox{ MeV}$ and\n  for severals values of magnetic fields (a) without  and  (b) with  AMM.}\n\\label{spin_npe_tw}\n\\end{figure}\n\n\n\\subsection{Direction of instability}\n\nThe eigenvector associated with the negative eigenvalue of the free energy curvature\nmatrix  defines the direction of the instability and tells us how does the system\nseparate into a dense liquid and a gas phase. It was shown in \\cite{abmp06,camille08} that\nin the absence of the magnetic field the direction of instability favors the\nreduction of the isospin asymmetry of the dense clusters of the system, and\nincreases the isospin asymmetry of the gas surrounding the clusters, the so\ncalled distillation effect. This effect is represented in Fig.~\\ref{spdtm2}\nwhere it is seen that for  the $B=0$ curve (thick full line)  the fraction $\\delta\n\\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$\nis larger than $\\rho_p\/\\rho_n$ below $y_p=0.5$ and the other way round above.\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure9.eps}\n\\caption{(Color online) $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ plotted as a\nfunction of the proton fraction with $T = 0  \\hbox{ MeV}$ and $\\rho= 0.06\n\\hbox{ fm}^{-3}$  for the TM1 model and several values of the magnetic\nfields  without  AMM. The fraction $\\rho_p\/\\rho_n$ is given by the thin dotted line.}\n\\label{spdtm2}\n\\end{figure}\n\nIn Fig. \\ref{spdtm2} we plot, for TM1,  the ratio  $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$  for $\\rho=0.06$ fm$^{-3}$ as a function of the proton fraction. Several results for different values of the magnetic field are shown by the thick lines. The thin lines represent the ratio $\\rho_p\/\\rho_n$ for reference and $y_p=0.5$ points corresponding to symmetric matter, as well as   $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}=1$, which is the ratio of density fluctuations for symmetric matter with no field. For the largest field considered the spinodal region contains a single\nLandau level and the curve varies smoothly starting at  $\\delta\n\\rho^{-}_{p}\/\\delta \\rho^{-}_{n}\\sim1.5$. We\npoint out the very large value of this fraction, always above 1, for\n$y_p<0.5$.  The magnetic field favors an increase of the proton fraction\nquite above the symmetric matter value. For $B^*=10^5$ the spinodal has two bands, see Fig.~\\ref{spztm1npn} and \\ref{spztwnp}\n, corresponding to the occupation of the\nfirst two Landau levels. The transition from one to the other is clearly seen\nwith  a large discontinuity of $\\delta\n\\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ at $y_p\\sim 0.7$. Above this $y_p$ value the\ncurve behaves like the previous ones. However for $y_p<0.7$ the behavior is\nquite different: the curve decreases from the value at $y_p$=0, which is\nindependent of the magnitude of the  magnetic field, to a value much smaller\nthan the corresponding value of the  fraction $\\rho_p\/\\rho_n$. The fluctuations will not drive the system\nout of the first Landau level and therefore the larger the proton fraction, the\ncloser the system comes to the top of the band and the smaller are the allowed\nproton fluctuations. For $y_p>0.7$ or for the larger magnetic fields the\nLandau levels are only partially filled and the fluctuations will never drive\nthe system out of the corresponding Landau level. \n\n\\begin{figure}[htb]\n\\begin{tabular}{c}\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure10.eps}\\vspace{1.5cm}\\\\\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure11.eps}\n\\end{tabular}\n\\caption{(Color online) $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ plotted as a\nfunction of the proton fraction with $T = 0  \\hbox{ MeV}$ and $\\rho= 0.06\n\\hbox{ fm}^{-3}$  for the TM1 (top) and TW (bottom) and for severals values of the magnetic\nfields  without (a) and (c) and with (b) and (d) AMM. The fraction $\\rho_p\/\\rho_n$ is given by the thin dotted line.}\n\\label{spdtm1}\n\\end{figure}\n\nSimilar features are obtained for TW and\/or including the AMM. In Fig. \\ref{spdtm1}\n  we show, respectively for TM1 (top)  and TW (bottom), the fraction $\\delta\n\\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ as a function of $y_p$ for a fixed baryonic\ndensity, $\\rho= 0.06 \\hbox{ fm}^{-3}$, chosen inside the instability region.\nFor $y_p>0.5$, TM1 and TW behave in a similar way, while below this value the main \ndifference is the smaller $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ for TW, corresponding to a smaller distillation effect. This behavior is also present for $B=0$ and it was shown that this was due to the presence of the rearrangement term. The inclusion of the AMM favors larger proton fractions because neutron polarization stiffens the EOS.\n\n\\begin{figure}[htb]\n\\vspace{1.5cm}\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure12.eps}\n\\caption{(Color online) $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$\n  plotted as a function of the density with $T = 0  \\hbox{ MeV}$ and\n  $y_p = 0.2$ with (thin lines) and without (thick lines) electrons\n  for the TM1 and TW models and for severals values of the magnetic\n  fields (a) and (c) without  and (b) and (d) with AMM.}\n\\label{drpdrnyp02}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.7\\linewidth,angle=0]{figure13.eps}\n\\caption{(Color online) $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$\n  plotted as a function of the density with $T = 0  \\hbox{ MeV}$ and\n  $y_p = 0.4$ with (thin lines) and without (thick lines) electrons\n  for the TM1 and TW models and for severals values of the magnetic\n  fields a)  without  and b)  with  AMM.}\n\\label{drpdrnyp04}\n\\end{figure}\n\nIn Figs.~\\ref{drpdrnyp02} and~\\ref{drpdrnyp04} we represent the fraction\n$\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ as a function of density\nrespectively for two\nvalues of $y_p$, 0.2 and 0.4, for $np$ matter (thick lines) and $npe$ matter\n(thin lines). We consider TM1 and TW.\n Both models have\na very similar behavior for finite values of $B$ although for $y_p=0.2$ and $B=0$ they differ: for\nTM1, $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$   increases with density while,\nfor TW, this fraction decreases for $\\rho>0.02$ fm$^{-3}$. This effect is not\nso strong for $y_p=0.4$ and for $npe$ matter the fraction is always quite small\ndue to  the presence of electrons which prevents large proton variations.\n\nFor $B=10^5$,  $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ decreases with\ndensity while for $B=5\\times 10^5$ the opposite occurs. In both cases only the first\nLandau level is occupied, however for the lower field the first Landau level is\nalmost full and the density\nfluctuations will occur in such a way that the system stays in the same Landau\nlevel: the larger the total density the smaller the fluctuations. For the\nlarger field the first Landau level is only partially filled, far the top of the band. For the same nuclear density, the larger the proton fraction\nthe lower the system energy and therefore the fraction $\\delta\n\\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ increases with density.\n\n\nIn Fig.~\\ref{drpdrnyp04} we give the same information with $y_p=0.4$. While for\n$B^* =5\\times 10^5$ for the range of densities considered, matter occupies only\none Landau level, for $B^* =10^5$ we may have two (TM1) or three (TW) Landau\nlevels, see Figs.~\\ref{spztm1npn} and~\\ref{spztwnp}.  This explains the\ndiscontinuities occurring for $\\sim 0.1$ fm$^{-3}$. For the highest magnetic\nfield only one Landau level partially filled comes into play and\ntherefore the\nfraction $\\delta \\rho^{-}_{p}\/\\delta \\rho^{-}_{n}$ \nincreases with density because that is favored energetically. For $B^* =10^5$\nthe presence of almost filled Landau levels prevents the existence of large\nproton density fluctuations. \n\n\n\\section{Conclusions and outlook}\n\nIn the present work we have studied the instabilities of $np$ matter and $npe$ neutral matter under\nvery strong magnetic fields. The fields considered are much stronger than the\nstrongest field measured until now at the surface of a magnetar which is $B^*\\sim 10^2$\nfor SGR 1806-20 \\cite{sgr}. However, it is expected that  fields in the interior of neutrons\nstars will be much larger. The present work shows how fields of the order\nof $B=5\\times 10^{18}$ G could affect the inner crust of a compact star.\n\nWe have considered  two relativistic nuclear models: one NLW model (TM1)  and one\nDDRH model (TW). For both models, we have determined the spinodal surface, from the curvature\nmatrix of the free energy, for different magnitudes of the magnetic field. It\nwas shown that the instability region could be divided into several bands\naccording to the magnitude of the magnetic field and the number of  the Landau\nlevels occupied.  The presence of the magnetic field will generally  increase the\ninstability region. Making a crude estimation of the transition density at the\ninner crust of a compact star under a strong magnetic field from the crossing\nof the EOS of $\\beta$-equilibrium stellar matter with the thermodynamical spinodal, it was shown that the transition\ndensity and associated pressure increases with the magnetic field. This will\naffect the structure of the star increasing the fraction of mass and of the\nstar' s moment of inertia  concentrated at the crust. These effects will be\nnoticeable if, for densities of the order of 0.1 fm$^{-3}$, the magnetic field\nis  of the order of $B^*=10^4$ or larger.\n\nThe TW model has larger instability regions than  the TM1 model for the larger\nproton densities. A smoother increase of the proton chemical potential for the\nfirst model justifies this behavior. This behavior of the symmetry energy may  even give rise to a larger\nnumber of bands in the spinodal of TW than the spinodal of TM1 for the same magnetic field.\n\nWe have also investigated the direction of instability. It was  shown\nthat if the Landau level is only partially occupied the density fluctuations\nare such that the system evolves for a state with dense clusters very proton\nrich immersed in a proton poor gas. A larger proton fraction is favored\nenergetically due to the degeneracy of the Landau levels. If on the other hand,\nwe study the fluctuation of particles occupying an almost complete Landau level, proton fluctuations\ncannot be so large and it may even occur an anti-distillation effect with a\ndecrease of the proton fraction of the dense clusters. This is due to the fact\nthat these fluctuations will keep the system in the same Landau level. \n   \n\n\\begin{acknowledgments}\n\nThis work was partially supported by FEDER and FCT (Portugal)\nunder the grant SFRH\/BPD\/14831\/2003, and projects  PTDC\/FP\/64707\/2006 and  POCI\/FP\/81923\/2007. \n\n\\end{acknowledgments}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\nA highlight in the theory of Drinfeld-Jimbo quantum groups is the construction of canonical bases by Lusztig and Kashiwara; cf.  \\cite{Lu93}. Let $(\\mbf U, {\\mbf U}^\\imath)$ be a quantum symmetric pair of finite type \\cite{Le99} (also cf. \\cite{BK15}).  A general theory of ($\\imath$-)canonical bases arising from quantum symmetric pairs of finite type was developed recently by Bao and the first author in \\cite{BW16}, where $\\imath$-canonical bases on based $\\mbf U$-modules and on the modified form of ${\\mbf U}^\\imath$ were constructed.\n\nOne notable feature of the definition of quantum symmetric pairs is its dependence on parameters; see \\cite{Le99, BK15, BW16} where various conditions on parameters are imposed at different levels of generalities for various constructions. In \\cite{BWW18} it is shown that the unequal parameters of type B Hecke algebras correspond under the $\\imath$-Schur duality to certain specializations of parameters in the type AIII\/AIV quantum symmetric pairs. \n\n\n\\vspace{3mm}\n\nFor the remainder of the paper we let $\\mbf U$ be the quantum group of $\\mathfrak{sl}_2$ over $\\mathbb Q(q)$ with generators $E, F, K^{\\pm 1}$. Let ${\\mbf U}^\\imath$ be the $\\mathbb Q(q)$-subalgebra of $\\mbf U$ (with a parameter $\\kappa$), which is a polynomial algebra in  $t$, where \n\\[\nt = F+ q^{-1}EK^{-1}  +\\kappa K^{-1}. \n\\]\nThen ${\\mbf U}^\\imath$ is a coideal subalgebra, and $(\\mbf U, {\\mbf U}^\\imath)$ is an example of quantum symmetric pairs; cf. \\cite{Ko93}. For the consideration of $\\imath$-canonical bases (which will be referred to as $\\imath$-divided powers from now on) the parameter $\\kappa$ is taken to be a bar invariant element in $\\mathcal A =\\mathbb Z[q,q^{-1}]$; cf. \\cite{BW16}. \n\nIn contrast to the usual quantum $\\mathfrak{sl}_2$ case where the divided powers are simple to define, finding explicit formulae for the $\\imath$-divided powers is a highly nontrivial problem. Closed formulae for the $\\imath$-divided powers in ${\\mbf U}^\\imath$ were known earlier only in two distinguished cases when $\\kappa=0$ or $1$ (cf. \\cite{BeW18}); this verified a conjecture in \\cite{BW13} when $\\kappa=1$.\n\n\nThe goal of this paper is to establish closed formulae  for the $\\imath$-divided powers in ${\\mbf U}^\\imath$  as polynomials in $t$ and also viewed as elements in $\\mbf U$ (via the embedding of ${\\mbf U}^\\imath$ to $\\mbf U$), when $\\kappa$ is an arbitrary $q$-integer which is clearly bar invariant. For arbitrary $\\overline{\\kappa} =\\kappa \\in \\mathcal A$, we are able to present a closed formula only for the {\\em second} $\\imath$-divided power (note the first $\\imath$-divided power is simply $t$ itself). \n\n\n\n\\vspace{3mm}\n\nIn contrast to the quantum group setting, there are {\\em two} $\\mathcal A$-forms, ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ and ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$, for ${\\mbf U}^\\imath$ corresponding to the parities $\\{\\rm ev, \\rm odd\\}$ of highest weights of finite-dimensional simple $\\mbf U$-modules \\cite{BW13, BW16}. As a very special case of a main theorem in \\cite{BW16}, ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ (and respectively, ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$) admits  $\\imath$-canonical bases ($= \\imath$-divided powers) for an arbitrary parameter $\\overline{\\kappa} =\\kappa \\in \\mathcal A$, which are invariant with respect to a bar map (which fixes $t$ and hence is not a restriction of Lusztig's bar map on $\\mbf U$ to ${\\mbf U}^\\imath$) and satisfy an asympototic compatibility with the $\\imath$-canonical bases on finite-dimensional simple $\\mbf U$-modules. \n\nComputations by hand and by Mathematica have\nled us to make an ansatz for the formulae for the $\\imath$-divided powers as polynomials in $t$ when $\\kappa$ is an arbitrary $q$-integer. In further discussions we need to separate the cases when $\\kappa$ is an even or odd $q$-integer, and let us restrict ourselves to the case for the $q$-integer $\\kappa$ being even in the remainder of the Introduction. Our ansatz is that the $\\imath$-divided powers $\\dvev{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ and $\\dv{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$ are given as follows: for $a\\in \\mathbb N$, \n\\begin{align}\n\\label{def:idp:evevKa}\n\\dvev{n} = \n\\begin{cases}\n\\frac{t}{[2a]!}  (t  - [-2a+2])(t -[-2a+4]) \\cdots (t -[2a-4]) (t  - [2a-2]), & \\text{if } n=2a, \\\\\n\\\\\n\\frac{1}{[2a+1]!} (t  - [-2a])(t -[-2a+2]) \\cdots (t -[2a-2]) (t  - [2a]), &\\text{if } n=2a+1.\n\\end{cases}\n\\end{align}\n\\begin{align}\n\\label{def:dv:evoddK}\n{\\small \\dv{n} = \n\\begin{cases}\n\\frac{1}{[2a]!}   (t  - [-2a+1] ) (t  - [-2a+3]) \\cdots (t  - [2a-3]) (t -[2a-1]), & \\text{if } n=2a, \\\\\n\\\\\n\\frac{t}{[2a+1]!}     (t  - [-2a+1] ) (t  - [-2a+3]) \\cdots (t  - [2a-3]) (t -[2a-1]), &\\text{if } n=2a+1.\n\\end{cases}\n}\n\\end{align}\nThat is, the $\\imath$-divided power formulas as polynomials in $t$ for $\\kappa$ being even $q$-integers are the same as for $\\kappa=0$  \\cite{BW13, BeW18}. \n\nTo verify the above formulae \\eqref{def:idp:evevKa}-\\eqref{def:dv:evoddK} are indeed the $\\imath$-canonical bases as defined and established in \\cite{BW16}, we need to verify 2 properties: (1) these polynomials in $t$ lie in the corresponding $\\mathcal A$-forms of ${\\mbf U}^\\imath$; (2) they satisfy the asymptotic compatibility with $\\imath$-canonical bases on finite-dimensional simple $\\mbf U$-modules. \n\nTo that end, we find the expansions for the polynomials in $t$ defined in \\eqref{def:idp:evevKa}-\\eqref{def:dv:evoddK}  in terms of $E, F, K^{\\pm 1}$ of $\\mbf U$ (via the embedding ${\\mbf U}^\\imath \\to \\mbf U$); they are given by explicit triple sum formulae. Once these explicit formulae are found, they are verified by lengthy inductions. In the formulation of the expansion formulae, a sequence of degree $n$ polynomials $p^{(n)} (x) =p_n(x)\/ [n]!$ which are defined recursively arise naturally; see \\S \\ref{subsec:pn}. The presence of $p^{(n)} (x)$ makes the formulation of the expansion formula and its proof much more difficult than the distinguished cases when $\\kappa=0$ or $1$ treated in \\cite{BeW18}. \n\nThe polynomials  $p^{(n)} (x)$ satisfy a crucial integrality property in the sense that $p^{(n)} (\\kappa)\\in \\mathcal A$ which leads to the integral property (1) above. To prove such an integrality we express these polynomials in terms of another sequence of polynomials (which are a reincarnation of the $\\imath$-divided powers viewed as polynomials) with some $q^2$-binomial coefficients. Note that $p_0(0)=1$ and $p_n(0)=0$ for $n\\ge 1$, our triple sum expansion formula reduces at $\\kappa=0$ to a double sum formula in \\cite{BeW18}. \n\nWhen applying the expansion formulas for the $\\imath$-divided powers to the highest weight vectors of the finite-dimensional simple $\\mbf U$-modules, we establish the asymptotic compatibility property (2) above explicitly. Note this form of compatibility cannot be made as strong as the compatibility between $\\imath$-divided powers on ${\\mbf U}^\\imath$ and simple $\\mbf U$-modules when $\\kappa=0$ or $1$ (as expected in \\cite{BW16}).\n\n\nWith Huanchen Bao's help, we compute the closed formulae for the second $\\imath$-divided power for an arbitrary parameter $\\overline{\\kappa} =\\kappa \\in \\mathcal A$; see Appendix~\\ref{sec:genK}. It is an interesting open problem to find closed formulae for higher $\\imath$-divided power with such an arbitrary parameter $\\kappa$. \n \n %\n\\vspace{3mm}\nThe paper is organized as follows. There are 4 sections depending on the parities of the weights and of the parameter $\\kappa$. \n\nIn Section~\\ref{sec:evevK}, we recall the basics of the quantum symmetric pair $(\\mbf U, {\\mbf U}^\\imath)$. Throughout the section we take $\\kappa$ to be an arbitrary even $q$-integer. We establish a key integral property of the polynomials  $p^{(n)} (x)$, and use it to formulate and establish the expansion formula for the $\\imath$-divided powers $\\dvev{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$. We prove that $\\{\\dvev{n} | n\\ge 0 \\}$ form an $\\imath$-canonical basis for ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ by showing $\\dvev{n} v^+_{2\\la} $ is an $\\imath$-canonical basis element on the finite-dimensional simple $\\mbf U$-modules $L(2\\lambda)$, for integers $\\lambda \\gg n$. \n\nIn Section~\\ref{sec:oddevK}, letting $\\kappa$ be an arbitrary even $q$-integer, we establish the expansion formula for the $\\imath$-divided powers $\\dv{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$. We prove that $\\{\\dv{n} | n \\ge 0\\}$ form an $\\imath$-canonical basis for ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$ by showing $\\dv{n}v^+_{2\\la+1} $ is an $\\imath$-canonical basis element on the finite-dimensional simple $\\mbf U$-modules $L(2\\lambda+1)$, for integers $\\lambda \\gg n$. \n\nIn Section~\\ref{sec:evoddK}, we take $\\kappa$ to be an arbitrary odd $q$-integer. We establish a key integral property of another sequence of polynomials  $\\mathfrak p^{(n)} (x)$, and use it to establish the expansion formula for the $\\imath$-divided powers $\\dvp{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$. We show that $\\{\\dvp{n} | n \\ge 0\\}$ form an $\\imath$-canonical basis for ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ and $\\dvp{n}v^+_{2\\la} $ is an $\\imath$-canonical basis element on the finite-dimensional simple $\\mbf U$-modules $L(2\\lambda)$, for integers $\\lambda \\gg n$.\n\nIn Section~\\ref{sec:oddoddK}, we take $\\kappa$ to be an arbitrary odd $q$-integer. We establish the expansion formula for the $\\imath$-divided powers $\\dvd{n}$ in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$. We show that $\\{\\dvd{n} | n \\ge 0\\}$ form an $\\imath$-canonical basis for ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$ and $\\dvd{n}v^+_{2\\la+1} $ is an $\\imath$-canonical basis element on the finite-dimensional simple $\\mbf U$-modules $L(2\\lambda+1)$, for integers $\\lambda \\gg n$.\n\n\nIn Appendix~\\ref{sec:genK}, for arbitrary $\\overline{\\kappa} =\\kappa \\in \\mathcal A$, we present closed formulae for the second $\\imath$-divided powers in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm ev}$ and ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm odd}$.\n\n\n\\vspace{.3cm}\n\n{\\bf Acknowledgement.} \nWW thanks Huanchen Bao for his insightful collaboration. The formula in the Appendix for the second $\\imath$-divided power with arbitrary parameter $\\kappa$ (which was obtained with help from Huanchen) was crucial to this project, and to a large extent this paper grows by exploring for what values for the parameter $\\kappa$ reasonable formulae for higher divided powers can be obtained. The research of WW  and the undergraduate research of CB are partially supported by a grant from National Science Foundation. Mathematica was used intensively in this work.\n\n\n\\section{The $\\imath$-divided powers $\\dvev{n}$ for even weights and even $\\kappa$}\n  \\label{sec:evevK}\n\n\\subsection{The quantum $\\mathfrak{sl}_2$}\n\nRecall  the quantum group $\\mbf U=\\mbf U_q(\\mathfrak{sl}_2)$ is the $\\mathbb Q(q)$-algebra generated by $F, E, K, K^{-1}$, subject to the relations:  \n\\[\nKK^{-1}=K^{-1}K=1,\n \\;\n EF -FE =\\frac{K-K^{-1}}{q-q^{-1}}, \n \\;\n K E =q^2 E K, \n \\;\n K F=q^{-2} FK.\n\\]\nThere is an anti-involution $\\varsigma$ of the $\\mathbb Q$-algebra $\\mbf U$:\n\\begin{align}\n \\label{eq:vs}\n\\varsigma: \\mbf U \\longrightarrow \\mbf U,\n\\qquad E\\mapsto E, \\quad F\\mapsto F, \\quad K\\mapsto K, \\quad q\\mapsto q^{-1}.\n\\end{align}\n\n Let $\\mathcal A =\\mathbb Z[q,q^{-1}]$ and $\\kappa \\in \\mathcal A$.  Set\n\\begin{equation}  \\label{eq:Y}\n\\check{E} :=q^{-1}EK^{-1}, \n\\qquad\nh :=\\frac{K^{-2}-1}{q^2-1}.\n\\end{equation} \n\n\n  Define, for $a\\in \\mathbb Z, n\\ge 0$, \n\\begin{equation}  \\label{kbinom}\n\\qbinom{h;a}{n} =\\prod_{i=1}^n \\frac{q^{4a+4i-4} K^{-2} -1}{q^{4i} -1},\n\\qquad \n[h;a]= \\qbinom{h;a}{1}.\n\\end{equation}\nThen we have, for $a\\in \\mathbb Z, n\\in \\mathbb N$, \n\\begin{align}\n  \\label{FEk}\n  \\begin{split}\nF \\check{E}  - & q^{-2} \\check{E}  F =h,\n\\\\\n\\qbinom{h;a}{n} F = F \\qbinom{h;a+1}{n}, &\n\\qquad \\qbinom{h;a}{n} \\check{E}  =\\check{E}  \\qbinom{h;a-1}{n}.\n\\end{split}\n\\end{align}\nIt follows by definition that\n\\begin{equation}  \\label{kbinom2}\n\\qbinom{h;a}{n} =\\qbinom{h;1+a}{n}\n -q^{4a}K^{-2} \\qbinom{h;1+a}{n-1}.\n\\end{equation}\n\nThe following formula holds for $n\\ge 0$:  \n\\begin{align}   \\label{FYn}\nF \\check{E}^{(n)} &=q^{-2n} \\check{E}^{(n)} F +  \\check{E}^{(n-1)} \\frac{q^{3-3n} K^{-2} - q^{1-n} }{q^2-1}.\n\\end{align} \n\n\n\n\nLet \n\\[\n\\check{E}^{(n)} =\\check{E}^n\/[n]!, \\quad\nF^{(n)} =F^n\/[n]!,\\quad \\text{ for } n\\ge 1.\n\\] \nThen $\\check{E}^{(n)}= q^{-n^2} E^{(n)} K^{-n}$. It is understood that $\\check{E}^{(n)}=0$ for $n<0$ and $\\check{E}^{(0)}=1$.\n\n\n\\subsection{The coideal subalgebra ${\\mbf U}^\\imath$}\n\nRecall $\\check{E}$ from \\eqref{eq:Y}. Let \n\\begin{align}\n  \\label{eq:t}\nt &=F + \\check{E} +\\kappa K^{-1}. \n\\end{align}\nDenote by ${\\mbf U}^\\imath$ the $\\mathbb Q(q)$-subalgebra of $\\mbf U$ generated by $t$. Then ${\\mbf U}^\\imath$ is a coideal subalgebra of $\\mbf U$ and $(\\mbf U, {\\mbf U}^\\imath)$ forms a quantum symmetric pair \\cite{Ko93, Le99}; also cf. \\cite{BW16}.\n\nDenote by $\\dot{\\mbf U}$ the modified quantum group of $\\mathfrak{sl}_2$ \\cite{Lu93}, which is the $\\mathbb Q(q)$-algebra generated by $E\\mathbf 1_\\lambda, F\\mathbf 1_\\lambda$ and the  idempotents $\\mathbf 1_\\lambda$, for $\\lambda\\in \\mathbb Z$. Let ${}_\\A{\\dot{\\mbf U}}$ (respectively, ${}_\\A{\\dot{\\mbf U}}_{\\rm{ev}}$, ${}_\\A{\\dot{\\mbf U}}_{\\text{odd}}$) be the $\\mathcal A$-subalgebra of $\\dot{\\mbf U}$  generated by $E^{(n)} \\mathbf 1_\\lambda, F^{(n)} \\mathbf 1_\\lambda, \\mathbf 1_\\lambda$, for all $n\\ge 0$ and for $\\lambda\\in \\mathbb Z$ (respectively, for $\\lambda$ even, for $\\lambda$ odd). Note ${}_\\A{\\dot{\\mbf U}} ={}_\\A{\\dot{\\mbf U}}_{\\rm{ev}} \\oplus {}_\\A{\\dot{\\mbf U}}_{\\text{odd}}$.\nThere is a natural left action of $\\mbf U$ (and hence ${\\mbf U}^\\imath$) on $\\dot{\\mbf U}$ such that $K \\mathbf 1_\\lambda =q^\\lambda \\mathbf 1_\\lambda$. \nFor $\\mu\\in \\mathbb N$, denote by $L(\\mu)$ the finite-dimensional simple $\\mbf U$-module of highest weight $\\mu$, and denote by $L(\\mu)_\\mathcal A$ its Lusztig $\\mathcal A$-form. \n\nFollowing \\cite{BW16}, we introduce the following $\\mathcal A$-subalgebras of ${\\mbf U}^\\imath$, for ${\\rm p}\\in \\{ {\\rm ev}, {\\rm odd} \\}$: \n\\begin{align*}\n{}_\\mathcal A {\\mbf U}^\\imath_{\\rm p}\n&=\\{x\\in {\\mbf U}^\\imath~|~x u \\in {}_\\A{\\dot{\\mbf U}}_{\\rm p}, \\forall u\\in {}_\\A{\\dot{\\mbf U}}_{\\rm p} \\}\n\\\\\n&=\\big \\{x\\in {\\mbf U}^\\imath~|~x v \\in L(\\mu)_\\mathcal A, \\forall v\\in L(\\mu)_\\mathcal A, \\forall {\\rm p}\\equiv\\mu\\pmod{2} \\big \\}. \n\\end{align*}\nBy \\cite{BW16}, ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm p}$ is a free $\\mathcal A$-submodule of ${\\mbf U}^\\imath$ such that ${\\mbf U}^\\imath =\\mathbb Q(q) \\otimes_\\mathcal A {}_\\mathcal A {\\mbf U}^\\imath_{\\rm p}$, for ${\\rm p}\\in \\{ {\\rm ev}, {\\rm odd} \\}$. \n\n\\subsection{Definition of $\\dvev{n}$ for even $\\kappa$}\n\nIn this section~\\ref{sec:evevK} we shall always take $\\kappa$ to be an even $q$-integer, i.e., \n\\begin{align}  \n   \\label{eq:evenK}\n\\kappa & =[2\\ell], \\quad  \\text{ for } \\ell  \\in \\mathbb Z.\n\\end{align}\nWe shall take the following as a definition of the $\\imath$-divided powers $\\dvev{n}$. \n\n\\begin{definition}\nSet $\\dvev{1} =t = F +\\check{E} +\\kappa K^{-1}$. The divided powers $\\dvev{n}$, for $n\\ge 1$, are defined by the recursive relations: \n\\begin{align}   \n  \\label{eq:tt}\n\\begin{split}\nt \\cdot \\dvev{2a-1} &=[2a] \\dvev{2a},\n\\\\\nt \\cdot \\dvev{2a} &=  [2a+1] \\dvev{2a+1} +   [2a] \\dvev{2a-1}, \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}   \nEquivalently, $\\dvev{n}$ is defined by the closed formula  \\eqref{def:idp:evevKa}. \n\\end{definition}\nThe bar involution $\\psi_\\imath$ on ${\\mbf U}^\\imath$, which fixed $t$ and sends $q\\mapsto q^{-1}$, clearly fixes the above $\\imath$-divided powers. \n\nThe following is a variant of \\cite[Lemma~2.2]{BeW18} (where $\\kappa=0$) with the same proof.\n\n\\begin{lem} \n  \\label{lem:anti}\nThe anti-involution $\\varsigma$ on $\\mbf U$\nsends $\nF\\mapsto F,  \\check{E} \\mapsto \\check{E},  K^{-1} \\mapsto K^{-1},  q \\mapsto q^{-1}$;  in addition, $\\varsigma$ sends \n\\[\nh\\mapsto -q^{2} h, \n\\quad\n\\dvev{n} \\mapsto \\dvev{n}, \n\\quad\n\\qbinom{h;a}{n}\\mapsto (-1)^n q^{2n(n+1)} \\qbinom{h;1-a-n}{n}, \\; \\forall a\\in\\mathbb Z, n\\in\\mathbb N.\n\\] \n\\end{lem}\n\n\\subsection{Polynomials $p_n(x)$ and $p^{(n)}(x)$}\n \\label{subsec:pn}\n \n\\begin{definition}\nFor $n\\in \\mathbb N$, the monic polynomial $p_n(x)$ in $x$ of degree $n$ is defined as\n\\begin{align}\n \\label{def:pn}\np_{n+1} =x p_n  + q^{1-2n} [n] [n-1] p_{n-1}, \\qquad  p_0 =1.\n\\end{align}\n\\end{definition}\nAlso set $p_n =0$ for all $n<0.$\nNote that $p_n$ is an odd polynomial for $n$ odd while it is an even polynomial for $n$ even. These polynomials $p_n$ will appear in the expansion formula for the $\\imath$-divided powers in $\\mbf U$. \n\n\\begin{example}\nHere are $p_n(x)$, for the first few $n\\ge 0$:\n\\begin{align*}\np_0& =1, \\qquad p_1 =x,\n\\qquad\np_2=x^2,\n\\qquad\np_3=x^3 +(q^{-4} +q^{-2})x,\n\\\\\np_4&=x^4 + (q^{-4} +q^{-2})(q^{-4} +q^{-2}+2)x^2,\n\\\\\np_5&=x^5 + (q^{-4} +q^{-2}) (q^{-8}+q^{-6}+3q^{-4} +2q^{-2}+3)x^3\n\\\\\n& \\qquad \\qquad + (q^{-4} +q^{-2})^2 (q^{-8}+q^{-6}+2q^{-4} +q^{-2}+1)x.\n\\end{align*}\n\\end{example}\n\nIntroduce the monic polynomial of degree $n$:\n\\begin{align*}\ng_n(x) =\n\\begin{cases}\n\\prod_{i=0}^{m-1} (x^2-[2i]^2), & \\text{ if } n=2m \\text{ is even};\n\\\\\nx \\prod_{i=1}^{m} (x^2-[2i]^2), & \\text{ if } n=2m+1 \\text{ is odd}.\n\\end{cases}\n\\end{align*}\nDefine \n\\begin{equation}  \\label{eq:divpg}\np^{(n)}(x) =p_n(x)\/[n]!, \\qquad\ng^{(n)}(x) =g_n(x)\/[n]!. \n\\end{equation} \nThe recursion for $p_n$ can be rewritten as\n\\begin{equation}   \\label{eq:f(n)}\n[n+1]p^{(n+1)} =x p^{(n)}  + q^{1-2n} [n-1] p^{(n-1)}.\n\\end{equation}\n\nOur next goal is to show that $p^{(n)}([2\\ell]) \\in \\mathcal A$ for all $\\ell \\in \\mathbb Z$; cf. Proposition~\\ref{prop:fg:ev}. This is carried out by relating $p^{(n)}$ to $g^{(n)}$ for various $n$, and showing that $g^{(n)}([2\\ell]) \\in \\mathcal A$ for all $\\ell \\in \\mathbb Z$.\n\n\n\\begin{lem} \\label{lem:ev:g-integral}\nFor $\\kappa =[2\\ell]$ with $\\ell \\in \\mathbb Z$ (see \\eqref{eq:evenK}), we have $g^{(n)}(\\kappa) \\in \\mathcal A$, for all $n\\ge 0$. \n\\end{lem}\n\n\\begin{proof} \nWe separate the cases for $n=2m+1$ and $n=2m$. \nNoting that\n\\begin{equation}  \\label{eq:A}\n\\kappa -[2i] =[2\\ell]-[2i] = [\\ell-i] (q^{\\ell+i} +q^{-\\ell-i}),\n\\end{equation} \nwe have\n \\begin{align*}\n g^{(2m+1)}(\\kappa) &= \\frac1{[2m+1]!} \\prod_{i=-m}^{m} (\\kappa-[2i])\n =\n \\qbinom{\\ell+m}{2m+1} \\prod_{i=-m}^{m}(q^{\\ell+i} +q^{-\\ell-i} ) \\in \\mathcal A.\n \\end{align*}\n\n\nSimilarly, we have\n \\begin{align*}\n g^{(2m)}(\\kappa) &= \\frac1{[2m]!} \\kappa \\prod_{i=1-m}^{m-1} (\\kappa-[2i])\n \\\\\n &= \\frac1{[2m]!} \\prod_{i=1-m}^{m} (\\kappa-[2i]) + \\frac1{[2m-1]!}  \\prod_{i=1-m}^{m-1} (\\kappa-[2i])\n \\\\\n &=\\qbinom{\\ell+m-1}{2m} \\prod_{i=1-m}^{m}(q^{\\ell+i} +q^{-\\ell-i} ) \n  + g^{(2m-1)}(\\kappa)  \\in \\mathcal A.\n  \\end{align*}\n  The lemma is proved. \n\\end{proof}\n\n\\begin{prop}\n \\label{prop:fg:ev}\n For $m \\in \\mathbb Z_{\\ge 1}$, we have\n\\begin{align*}\np^{(2m)} &= \\sum_{a=0}^{m-1} q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} g^{(2m-2a)},\n\\\\\np^{(2m-1)} &= \\sum_{a=0}^{m-1} q^{(3-2m)a} \\qbinom{m-1}{a}_{q^2} g^{(2m-2a-1)}.\n\\end{align*}\nIn particular, we have $p^{(n)} (\\kappa) \\in \\mathcal A$, for all $\\kappa =[2\\ell]$ with $\\ell \\in \\mathbb Z$ and all $n\\ge 0$. \n\\end{prop}\n\n\\begin{proof}  \n It follows from  these formulae for $p^{(n)}$ and Lemma~\\ref{lem:ev:g-integral} that $p^{(n)}([2\\ell]) \\in \\mathcal A$.\n\nIt remains to prove these formulae. Recall\n\\begin{align}  \\label{recursive:f}\n x \\cdot p^{(n)} = [n+1] p^{(n+1)}  -  q^{1-2n} [n-1] p^{(n-1)}, \\quad \\text{ for } n \\ge 1.\n\\end{align}\nAlso recall\n\\begin{align}  \\label{recursive:g}\n\\begin{split}\nx \\cdot g^{(2a-1)} &=[2a] g^{(2a)},\n\\\\\nx \\cdot g^{(2a)} &=  [2a+1] g^{(2a+1)} +   [2a] g^{(2a-1)}, \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}\n\nWe prove the formulae by induction on $m$, with the base cases for $p^{(1)}$ and $p^{(2)}$ being clear. We separate in two cases. \n\n(1). Let us prove the formula for $p^{(2m+1)}$, assuming the formulae for $p^{(2m-1)}$ and $p^{(2m)}$.\nBy \\eqref{recursive:f}--\\eqref{recursive:g} we have\n\\begin{align*}\n[2m+1]p^{(2m+1)} \n&= x p^{(2m)} +q^{1-4m} [2m-1] p^{(2m-1)}  \\\\\n&= \\sum_{a=0}^{m-1}q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} x \\cdot g^{(2m-2a)} \\\\\n&\\qquad\n  +\\sum_{a=0}^{m-1} q^{(3-2m)a+1-4m} [2m-1] \\qbinom{m-1}{a}_{q^2} g^{(2m-2a-1)}\n \\\\\n&= \\sum_{a=0}^{m-1}q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2}\n      \\left( [2m-2a+1]g^{(2m-2a+1)} +[2m-2a]g^{(2m-2a-1)}\n    \\right) \\\\\n&\\qquad\n  +\\sum_{a=0}^{m-1} q^{(3-2m)a+1-4m} [2m-1] \\qbinom{m-1}{a}_{q^2} g^{(2m-2a-1)}.\n\\end{align*}\nBy combining the like terms for $g^{(2m-2a-1)}$ above and using the identity\n\\[\nq^{(1-2m)a} [2m-2a] +q^{(3-2m)a+1-4m} [2m-1] =q^{(1-2m)(a+1)} [4m-2a-1], \n\\]\nwe obtain\n\\begin{align*}\n[2m+1]p^{(2m+1)} \n&= \\sum_{a=0}^{m-1}q^{(1-2m)a}  [2m-2a+1] \\qbinom{m-1}{a}_{q^2}  g^{(2m-2a+1)} \n     \\\\\n&\\qquad\n  +\\sum_{a=0}^{m-1} q^{(1-2m)(a+1)} [4m-2a-1] \\qbinom{m-1}{a}_{q^2} g^{(2m-2a-1)}\n  \\\\ %\n&= [2m+1]\\sum_{a=0}^{m}q^{(1-2m)a} \\qbinom{m}{a}_{q^2} g^{(2m-2a+1)}, \n\\end{align*}\nwhere the last equation results from shifting the second summation index from $a\\to a-1$ and using\nthe identity\n\\[\n[4m-2a+1] \\qbinom{m-1}{a}_{q^2} +q^{(1-2m)a} [4m-2a+1] \\qbinom{m-1}{a-1}_{q^2}\n=[2m+1] \\qbinom{m}{a}_{q^2}. \n\\]\n\n(2) We shall prove the formula for $p^{(2m+1)}$, assuming the formulae for $p^{(2m-1)}$ and $p^{(2m)}$.\nBy \\eqref{recursive:f}--\\eqref{recursive:g} we have\n\\begin{align*}\n&[2m+2]p^{(2m+2)} \n\\\\\n&= x p^{(2m+1)} +q^{-1-4m} [2m] p^{(2m)}  \\\\\n&= \\sum_{a=0}^{m}q^{(1-2m)a} \\qbinom{m}{a}_{q^2} x \\cdot g^{(2m-2a+1)}\n +\\sum_{a=0}^{m-1} q^{(1-2m)a-1-4m} [2m] \\qbinom{m-1}{a}_{q^2} g^{(2m-2a)}\n \\\\\n %\n&= \\sum_{a=0}^{m} q^{(1-2m)a} [2m-2a+2] \\qbinom{m}{a}_{q^2}  g^{(2m-2a+2)}\n +\\sum_{a=0}^{m-1} q^{(1-2m)a-1-4m} [2m] \\qbinom{m-1}{a}_{q^2} g^{(2m-2a)}\n \\\\\n &= [2m+2] \\sum_{a=0}^{m} q^{(-1-2m)a} \\qbinom{m}{a}_{q^2} g^{(2m-2a+2)},\n \\end{align*}\nwhere the last equation results from shifting the second summation index from $a\\to a-1$ and \n using the following identity\n \\[\n [2m-2a+2] \\qbinom{m}{a}_{q^2} \n+ q^{-2-2m} [2m] \\qbinom{m-1}{a-1}_{q^2}\n=  q^{-2a} [2m+2] \\qbinom{m}{a}_{q^2}. \n \\]\nThe proposition is proved.\n\\end{proof}\n\n\n\\begin{rem}\n\\label{rem:positive-p}\nNote that $p_n(x) \\in \\mathbb N [q^{-1}] [x]$ for all $n$. Therefore, for all non-negative even $q$-integer $\\kappa$, we have $p_n (\\kappa) \\in \\mathbb N [q,q^{-1}]$. \n\\end{rem}\n\n\\subsection{Formulae for $\\dvev{n}$ with even $\\kappa$}\n\nRecall $p^{(n)}(x)$ from \\eqref{def:pn}--\\eqref{eq:divpg}. \n\\begin{thm}   \n \\label{thm:dvev:evKappa}\nAssume $\\kappa$ is an even $q$-integer as in \\eqref{eq:evenK}. Then we have, for $m\\ge 1$,  \n\\begin{align}\n\\dvev{2m} &= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a)} \np^{(2m-b-2c)}(\\kappa) \n\\label{t2m:evev2} \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}, \n\\notag\n\\\\\n\\dvev{2m-1} &= \n\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a)}\n p^{(2m-b-2c-1)}(\\kappa) \n\\label{t2m-1:evev2}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c+1} F^{(b-a)}. \n\\notag\n\\end{align}\n\\end{thm}\nThe proof of Theorem~\\ref{thm:dvev:evKappa} will be given in \\S\\ref{subsec:proof:dvev:evKappa} below. \nLet us list some equivalent formulae here. \nBy applying the anti-involution $\\varsigma$ we convert the formulae in Theorem~\\ref{thm:dvev:evKappa} into the following.\n\n\\begin{thm}   \n \\label{thm:dvev:evKappa2}\nAssume $\\kappa$ is an even $q$-integer as in \\eqref{eq:evenK}. Then we have, for $m\\ge 1$,   \n\\begin{align}\n\\dvev{2m} &= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} (-1)^c q^{3c+b(2m-b-2c) +a(b-a)}  p^{(2m-b-2c)}(\\kappa) \n\\label{t2m:evevFhE} \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot  F^{(b-a)} K^{b-2m+2c} \\qbinom{h;m-c}{c}  \\check{E}^{(a)},\n\\notag\n\\\\ %\n\\dvev{2m-1} &= \n\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} (-1)^c q^{c +b(2m-b-2c-1) +a(b-a)}\n p^{(2m-b-2c-1)}(\\kappa) \n\\label{t2m-1:evevFhE}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot  F^{(b-a)} K^{b-2m+2c+1}  \\qbinom{h;m-c}{c}  \\check{E}^{(a)}.\n\\notag\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:anti}, the anti-involution $\\varsigma$ sends \n$\\qbinom{h;1-m}{c}\\mapsto (-1)^c q^{2c(c+1)} \\qbinom{h;m-c}{c}$, \n$q\\mapsto q^{-1}$, while fixing $\\kappa, K, \\check{E}^{(a)}, F^{(a)}$ and $\\dvev{n}$. The formulae \\eqref{t2m:evevFhE}--\\eqref{t2m-1:evevFhE} now follows by applying $\\varsigma$ to the formulae  \\eqref{t2m:evev2}--\\eqref{t2m-1:evev2} in Theorem~\\ref{thm:dvev:evKappa}. \n\\end{proof}\nFor $c\\in \\mathbb N, m\\in\\mathbb Z$, we define\n\\begin{align}   \\label{cbinom}\n  \\cbinom{m}{c} &= \\prod_{i=1}^c \\frac{q^{4(m+i-1)} -1}{q^{-4i}-1}  \\in \\mathcal A,\n  \\qquad \\cbinom{m}{0}=1.\n\\end{align}\n Note that $\\cbinom{m}{c}$ are $q^2$-binomial coefficients up to some $q$-powers.\nLet $\\lambda, m \\in \\mathbb Z$ with $m\\ge 1$.  We note that\n\\begin{align}\n  \\label{eq:qbinom-hc}\n\\qbinom{h;m-c}{c} \\mathbf 1_{2\\lambda} \n&= \\prod_{i=1}^c \\frac{q^{4(m-\\lambda-c+i-1)} -1}{q^{4i} -1}  \\mathbf 1_{2\\lambda}\n= (-1)^c q^{-2c(c+1)} \\cbinom{m-\\lambda-c}{c}  \\mathbf 1_{2\\lambda}.\n\\end{align}\n\nThe following corollary is immediate from \\eqref{eq:qbinom-hc}, Proposition~\\ref{prop:fg:ev}, and Theorem~\\ref{thm:dvev:evKappa}. \n\\begin{cor}  \nWe have $\\dvev{n} \\in {}_\\mathcal A {\\mbf U}^\\imath_{\\rm{ev}}$, for all $n$. \n\\end{cor}\n\n\\begin{rem}   \nNote $p_n(0)=0$ for $n>0$, and $p_0(0)=1$. \nThe formulae in Theorems~\\ref{thm:dvev:evKappa} and  \\ref{thm:dvev:evKappa2} in the special case for $\\kappa=0$ recover the formulae in \\cite[Theorem~2.5, Proposition~2.7]{BeW18}.\n\\end{rem}\n\n\n\n\nFor $n\\in \\mathbb N$, we denote\n\\begin{align}  \\label{eq:divB}\nb^{(n)}  = \\sum_{a=0}^n  q^{-a(n-a)}  \\check{E}^{(a)}F^{(n-a)}.\n\\end{align}\n\n\\begin{example}\nThe formulae for $\\dvev{n}$ in Theorem~\\ref{thm:dvev:evKappa}, for $1\\le n \\le 3$, read as follows:\n\\begin{align*}\n\\dvev{1} \n&= F +\\check{E} + \\kappa K^{-1},\n\\\\\n\\dvev{2} \n&= \\check{E}^{(2)} +q^{-1} \\check{E} F + F^{(2)}  + q [h;0] \n + \\kappa (q^{-1}K^{-1}F + q^{-1} \\check{E} K^{-1})  + \\kappa^2 \\frac{K^{-2}}{[2]},\n\\\\\n\\dvev{3}  &   = b^{(3)} + q^3[h;-1]F+ q^3 \\check{E} [h;-1]\n\\\\\n&\\qquad \n+ \\big(q^{-2} \\check{E}^{(2)}K^{-1} +q^{-3} \\check{E} K^{-1}F + q^{-2} K^{-1}F^{(2)} \n+ q^3 [h;-1] K^{-1} \\big) \\kappa\n\\\\\n&\\qquad + \\frac{q^{-2}}{[2]} (\\check{E} K^{-2} +K^{-2}F) \\kappa^2\n+\\frac{\\kappa^3 + (q^{-4} +q^{-2})\\kappa}{[3]!} K^{-3}. \n\\end{align*}\n\\end{example}\n\n\n\\subsection{The $\\imath$-canonical basis for $\\dot{\\mbf U}^\\imath_{\\rm{ev}}$ for even $\\kappa$}\n  \\label{sec:iCB:ev}\n  \nDenote by $L(\\mu)$ the \nsimple $\\mbf U$-module of highest weight $\\mu \\in \\mathbb N$, with highest weight vector $v^+_\\mu$.\nThen $L(\\mu)$ admits a canonical basis $\\{ F^{(a)} v^+_\\mu\\mid 0\\le a \\le \\mu\\}$. \nFollowing \\cite{BW13,BW16}, there exists a new bar involution $\\psi_\\imath$ on $L(\\mu)$, which, in our current rank one setting, can be defined simply by requiring $\\dvev{n} v^+_\\mu$ to be $\\psi_\\imath$-invariant for all $n$. \nAs a very special case of the general results in \\cite[Corollary~F]{BW16} (also cf. \\cite{BW13}), we know that\n the $\\imath$-canonical basis $\\{b^{\\mu}_{a} \\}_{0\\le a \\le \\mu}$ of $L(\\mu)$ exists and is characterized by the following 2 properties: \n\n(iCB1) $b^{\\mu}_{a}$ is $\\psi_\\imath$-invariant; \\qquad \n\n(iCB2) $b^{\\mu}_{a} \\in F^{(a)} v^+_\\mu + \\sum_{0\\le r< a} q^{-1} \\mathbb Z[q^{-1}] F^{(r)} v^+_\\mu$. \n\n \n \\begin{thm}  \\label{thm:iCB:ev}\n   $\\quad$\n\\begin{enumerate}\n\\item\nLet $n\\in \\mathbb N$. \nFor each integer $\\lambda \\gg n$, the element $\\dvev{n} v^+_{2\\la} $ is an $\\imath$-canonical basis element for $L(2\\lambda)$. \n \n \\item\nThe set $\\{\\dvev{n} \\mid n \\in \\mathbb N \\}$ forms the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ (and an $\\mathcal A$-basis in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm{ev}}$). \n\\end{enumerate}\n \\end{thm}\n \n \\begin{proof}\nLet $\\lambda, m \\in \\mathbb N$ with $m\\ge 1$.  \nWe compute by Theorem~\\ref{thm:dvev:evKappa2}(1) and \\eqref{eq:qbinom-hc} that \n\\begin{align} \n\\dvev{2m} v^+_{2\\la}  \n&= \\sum_{b=0}^{2m} \\sum_{c \\geq 0} (-1)^c q^{3c+b(2m-b-2c)}  p^{(2m-b-2c)}(\\kappa) \nF^{(b)} K^{b-2m+2c} \\qbinom{h;m-c}{c} v^+_{2\\la}  \n\\notag \\\\\n&= \\sum_{b=0}^{2m} \\sum_{c \\geq 0} q^{(b-2\\lambda)(2m-b-2c)-2c^2+c}  p^{(2m-b-2c)}(\\kappa) \n\\cbinom{m-\\lambda-c}{c}F^{(b)}   v^+_{2\\la}  \n  \\label{dvev:2m-mod}.\n\\end{align}\nSimilarly we compute by Theorem~\\ref{thm:dvev:evKappa2}(2) that \n\\begin{align} \n\\dvev{2m-1} v^+_{2\\la}  \n&= \n\\sum_{b=0}^{2m-1} \\sum_{c \\geq 0} (-1)^c q^{c +b(2m-b-2c-1)}\n p^{(2m-b-2c-1)}(\\kappa) \nF^{(b)} K^{b-2m+2c+1}  \\qbinom{h;m-c}{c}  v^+_{2\\la} \n\\notag\n\\\\\n&= \n\\sum_{b=0}^{2m-1} \\sum_{c \\geq 0}  q^{(b-2\\lambda)(2m-b-2c-1)-2c^2-c}\n p^{(2m-b-2c-1)}(\\kappa) \n\\cbinom{m-\\lambda-c}{c} F^{(b)}  v^+_{2\\la} . \n \\label{dvev:2m-1-mod}\n\\end{align}\n\nNote that\n\\begin{equation}\n  \\label{eq:q-1}\n\\cbinom{m-\\lambda-c}{c} \\in \\mathbb N[q^{-1}], \\qquad \\text{ for }\\lambda \\ge m, c\\ge 0.\n\\end{equation}  \n\nLet $n\\in \\mathbb N$. Recall $\\kappa =[2\\ell]$, for $\\ell \\in \\mathbb Z$.  \nOne computes that $\\deg_q p^{(n)} (\\kappa) =(2\\ell-1)n - {n \\choose 2}$ and hence $q^{(b-2\\lambda)(n-b-2c)-2c^2 \\pm c}\n p^{(n-b-2c)}(\\kappa) \\in q^{-1} \\mathbb N[q^{-1}]$ when $\\lambda \\gg n>b+2c$. It follows by \\eqref{dvev:2m-mod}, \\eqref{dvev:2m-1-mod} and \\eqref{eq:q-1} that \n \\begin{align}\n   \\label{eq:lattice}\n \\dvev{n} v^+_{2\\la}   & \\in F^{(n)}  v^+_{2\\la}  + \\sum_{b<n}q^{-1} \\mathbb Z[q^{-1}] F^{(b)} v^+_{2\\la} ,\n \\qquad \\text{ for } \\lambda \\gg n.\n \\end{align}\n Therefore, the first statement follows by the characterization properties (iCB1)--(iCB2) of the $\\imath$-canonical basis for $L(2\\lambda)$.\n \n The second statement follows now from the definition of  the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ using\nthe projective system $\\{ L(2\\lambda) \\}_{\\lambda\\in \\mathbb N}$ with ${\\mbf U}^\\imath$-homomorphisms $L(2\\lambda+2) \\rightarrow L(2\\lambda)$; cf. \\cite[\\S6]{BW16}; this set $\\{\\dvev{n} \\mid n \\in \\mathbb N \\}$ forms an $\\mathcal A$-basis for  ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm{ev}}$. \nThe theorem is proved. \n \\end{proof}\n \n \\begin{rem}\nIt follows by \\eqref{dvev:2m-mod} that \n\\[\n\\dvev{2m} v^+_{2\\la}    \\in F^{(2m)}  v^+_{2\\la}  +  q^{2m-2\\lambda-1} \\kappa F^{(2m-1)}  v^+_{2\\la}  +\\sum_{k\\ge 2} \\mathcal A \\cdot F^{(2m-k)}  v^+_{2\\la} .\n\\]\nRecall $\\kappa=[2\\ell]$. When $\\ell -1 \\ge \\lambda -m \\ge 0$, we have $q^{2m-2\\lambda-1} \\kappa \\not \\in q^{-1} \\mathbb Z[q^{-1}]$, and therefore, the nonzero element $\\dvev{2m} v^+_{2\\la} $ is not an $\\imath$-canonical basis element in $L(2\\lambda)$. So we cannot expect to have a stronger form of Theorem~\\ref{thm:iCB:ev} as \\cite[Theorem 2.10(1)]{BeW18} in  the case when $\\kappa=0$ (i.e., $\\ell=0$). \n \\end{rem}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:dvev:evKappa} }\n\\label{subsec:proof:dvev:evKappa}\nWe prove the formulae for $\\dvev{n}$ by induction on $n$, in two steps (1)--(2) below.  The base cases when $n=1,2$ are clear. \n\n(1) We shall prove the formula \\eqref{t2m:evev2} for $\\dvev{2m}$, assuming the formula \\eqref{t2m-1:evev2} for $\\dvev{2m-1}$.\n\nRecall $[2m] \\dvev{2m} = t \\cdot \\dvev{2m-1}$, and $t= F+ \\check{E}  +\\kappa K^{-1}$. \nLet us compute \n\\[\nI=\\check{E}  \\dvev{2m-1},\n\\qquad\nII=F \\dvev{2m-1},\n\\qquad\nIII=\\kappa K^{-1} \\dvev{2m-1}.\n\\]\nFirst we have\n\\begin{align*}\nI\n&=\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a)}\n    [a+1] p^{(2m-b-2c-1)}(\\kappa) \\\\\n  & \\qquad\\qquad \\cdot \\check{E}^{(a+1)}  \\qbinom{h;1-m}{c} K^{b-2m+2c+1} F^{(b-a)}\n\\\\\n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -(b-1)(2m-b-2c) -(a-1)(b-a)}\n    [a] p^{(2m-b-2c)}(\\kappa) \\\\\n  & \\qquad\\qquad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)},\n\\end{align*}\nwhere the last equation is obtained by shifting indices $a\\to a-1$, $b\\to b-1$ (and then adding some zero terms to make the bounds of the summations uniform throughout).\nUsing \\eqref{FYn} we  have\n\\begin{align*}\nII \n&=\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a)}\n   p^{(2m-b-2c-1)}(\\kappa) \\\\\n  & \\qquad \\cdot \n  \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n   \\qbinom{h;1-m}{c} K^{b-2m+2c+1} F^{(b-a)} =II^1 +II^2,\n\\end{align*}\nwhere $II^1$ and $II^2$ are the two natural summands associated to the plus sign.\nBy shifting index $b\\to b-1$ and then adding some zero terms, we further have\n\\begin{align*}\nII^1&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c - (b-1)(2m-b-2c) -a(b-a-1)-2a+2b+4c-4m}\n   [b-a] p^{(2m-b-2c)}(\\kappa) \\\\\n  & \\qquad \\qquad \\cdot  \\check{E}^{(a)}  \\qbinom{h;-m}{c} K^{b-2m+2c} F^{(b-a)}.\n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\nII^2\n &=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c-2}{2}+2c - (b+1)(2m-b-2c) - (a+1)(b-a)-2}\n  p^{(2m-b-2c)}(\\kappa) \\\\\n  & \\qquad \\qquad \\cdot  \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1} K^{b-2m+2c} F^{(b-a)}.\n\\end{align*}\nUsing \\eqref{recursive:f} we also compute\n\\begin{align*}\nIII\n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a) -2a}\n    \\kappa \\cdot p^{(2m-b-2c-1)}(\\kappa) \\\\\n  & \\qquad\\qquad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\\\\n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a) -2a}\n    [2m-b-2c] p^{(2m-b-2c)}(\\kappa)  \\\\\n  & \\qquad\\qquad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\n  \\\\\n & \\quad+\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a) -2a -4m}\n    [2m-b-2c] p^{(2m-b-2c)}(\\kappa)  \\\\\n  & \\qquad\\qquad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c-1} K^{b-2m+2c-2} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $I, II^1, II^2$ and $III$  gives us\n\\[\nt \\cdot \\dvev{2m-1} = \n \\sum_{0 \\le a \\le b \\le 2m}  \\sum_{c\\ge 0}  p^{(2m-b-2c)}(\\kappa)  \\check{E}^{(a)} H_{a,b,c}  K^{b-2m+2c} F^{(b-a)}, \n\\]\nwhere   \n\\begin{align*}\nH_{a,b,c} & :=\n q^{\\binom{2c}{2}+2c -(b-1)(2m-b-2c) -(a-1)(b-a)} [a]  \\qbinom{h;1-m}{c} \n  \\\\\n& + q^{\\binom{2c}{2}+2c - (b-1)(2m-b-2c) -a(b-a-1)-2a+2b+4c-4m} [b-a]  \\qbinom{h;-m}{c} \n\\\\\n& + q^{\\binom{2c}{2}-2c+1 - (b+1)(2m-b-2c) - (a+1)(b-a)}\n  \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1}    \n\\\\\n&+ q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a) -2a} [2m-b-2c] \\qbinom{h;1-m}{c}  \n  \\\\\n& - q^{\\binom{2c}{2}+2c -b(2m-b-2c-1) -a(b-a) -2a -4m} [2m-b-2c] \\qbinom{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nRecall $[2m] \\dvev{2m} = t \\cdot \\dvev{2m-1}$. To prove the formula \\eqref{t2m:evev2},  by the PBW basis theorem it suffices to prove the following identity, for all $a,b,c$:\n\\begin{align}  \\label{eq:H}\nH_{a,b,c} = q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a)} [2m]  \\qbinom{h;1-m}{c}.\n\\end{align}\nThanks to $q^{2m-a} [a] -[2m] =q^{-a} [a-2m]$, we can combine the RHS with the first summand of LHS \\eqref{eq:H}.\nHence, after canceling out the $q$-powers $q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a)-a}$ on both sides, we see that \\eqref{eq:H} is equivalent to the following identity, for all $a, b, c$:\n\\begin{equation}  \\label{ABCD}\nA+B+C+D_1 +D_2=0,\n\\end{equation}\nwhere   \n\\begin{align*}\nA &= [a-2m]  \\qbinom{h;1-m}{c},\n  \\\\\nB &=    q^{4c-2m+b} [b-a] \\qbinom{h;-m}{c},\n\\\\\nC   &=  q^{2a-2m+1} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1},\n\\\\\nD_1 &=  q^{2c+b-a} [2m-b-2c] \\qbinom{h; 1-m}{c},\n   \\\\\nD_2 &= -  q^{2c-4m+b-a} [2m-b-2c] \\qbinom{h; 1-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us prove the identity \\eqref{ABCD}. Using \\eqref{kbinom2}, we can write $B=B_1+B_2$, where \n\\begin{align*}\nB_1 &=  q^{4c-2m+b} [b-a]  \\qbinom{h; 1-m}{c},\n   \\quad\nB_2 = -  q^{4c-6m+b} [b-a] \\qbinom{h; 1-m}{c-1} K^{-2}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} = q^{4m-4c-3a}\\frac{( q^{4c-4m}K^{-2}-1)}{q^2-1}\n +  q^{2m-2c-2a-1} [2m-2c-a],\n \\] \nwe rewrite $C=C_1+C_2$, where\n\\begin{align*}\nC_1 &= q^{-2c+2m-a} [2c] \\qbinom{h; 1-m}{c},\n   \\quad\nC_2 =   q^{-2c} [2m-2c-a]  \\qbinom{h; 1-m}{c-1}.\n\\end{align*}\n\nA direct computation gives us\n\\begin{align*}\nA+B_1+C_1+D_1 \n&= (q-q^{-1})  [2c] [2m-2c-a]  \\qbinom{h; 1-m}{c},\n\\\\\nC_2 +(B_2+D_2)  \n&= C_2 -  q^{2c-4m} [2m-2c-a] \\qbinom{h; 1-m}{c-1} K^{-2}\n\\\\\n&= - (q-q^{-1}) [2c] [2m-2c-a] \\qbinom{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we have $A+B+C+D_1+D_2=0,$ whence \\eqref{ABCD}, completing Step~(1). \n\n\\vspace{3mm}\n\n(2) Assuming the formulae for $\\dvev{n}$ with $n\\le 2m$, we shall now prove the following formula for $\\dvev{2m+1}$ (obtained from \\eqref{t2m-1:evev2} with $m$ replaced by $m+1$):\n\\begin{align}\n\\dvev{2m+1} &= \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}+2c -b(2m-b-2c+1) -a(b-a)}\n p^{(2m-b-2c+1)}(\\kappa) \n\\label{t2m+1:evev2}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\notag\n\\end{align}\nThe proof is based on the recursion $t \\cdot \\dvev{2m} =[2m+1] \\dvev{2m+1} +[2m] \\dvev{2m-1}$. \nRecall $t= F +\\check{E} +\\kappa K^{-1}$ and $\\dvev{2m}$ from \\eqref{t2m:evev2}. We shall compute \n\\[\n\\texttt I=\\check{E}  \\dvev{2m},\n\\qquad\n\\texttt{II}=F \\dvev{2m},\n\\qquad\n\\texttt{III} =\\kappa K^{-1} \\dvev{2m}, \n\\]\nrespectively. First we have\n\\begin{align*}\n\\texttt{I}\n&= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a)} \n [a+1] p^{(2m-b-2c)}(\\kappa)  \n \\check{E}^{(a+1)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\\\\\n&= \n \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -(b-1)(2m-b-2c+1) -(a-1)(b-a)} \n[a] p^{(2m-b-2c+1)}(\\kappa) \\\\\n&\\qquad\\qquad\\qquad\\qquad \n\\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)},\n\\end{align*}\nwhere the last equation is obtained by shifting indices $a\\to a-1$, $b\\to b-1$. We also have\n\\begin{align*}\n\\texttt{II}\n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a)} \np^{(2m-b-2c)}(\\kappa) \n\\\\\n& \\qquad \\cdot \n  \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n   \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)} \n=\\texttt{II}_1 +\\texttt{II}_2,\n\\end{align*}\nwhere $\\texttt{II}_1$ and $\\texttt{II}_2$ are the two natural summands associated to the plus sign.\nBy shifting the index $b\\to b-1$, we have\n\\begin{align*}\n\\texttt{II}_1&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -(b-1)(2m-b-2c+1) -a(b-a-1)-2a +2(b+2c-2m-1)} \n[b-a] p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\qquad\\qquad \\qquad \\cdot \n  \\check{E}^{(a)}   \n      \\qbinom{h;-m}{c} K^{b-2m+2c-1} F^{(b-a)}.\n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\n\\texttt{II}_2\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c-2}{2} -(b+1)(2m-b-2c+1) -(a+1)(b-a)} \np^{(2m-b-2c+1)}(\\kappa) \n\\\\\n& \\qquad\\qquad \\qquad \\cdot \n  \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n   \\qbinom{h;1-m}{c-1} K^{b-2m+2c-1} F^{(b-a)}.\n\\end{align*}\nUsing \\eqref{recursive:f} we also compute\n\\begin{align*}\n\\texttt{III} \n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a) -2a} \n[2m-b-2c+1]  \n\\\\\n&\\qquad\\qquad\\qquad\\qquad \\cdot p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\\n& \\quad + \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a) -2a-4m} \n[2m-b-2c+1]  \n\\\\\n&\\qquad\\qquad\\qquad\\qquad \\cdot p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\qbinom{h;1-m}{c-1} K^{b-2m+2c-3} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $\\texttt{I}, \\texttt{II}_1, \\texttt{II}_2$, and $\\texttt{III}$, we obtain \n\\[\nt \\cdot \\dvev{2m} = \n \\sum_{0 \\le a \\le b \\le 2m+1}  \\sum_{c\\ge 0} p^{(2m-b-2c+1)}(\\kappa)  \\check{E}^{(a)} L_{a,b,c} K^{b-2m+2c-1} F^{(b-a)},\n\\]\nwhere  \n\\begin{align*}\nL_{a,b,c} := &\nq^{\\binom{2c}{2} -(b-1)(2m-b-2c+1) -(a-1)(b-a)} [a] \\qbinom{h;1-m}{c}  \n\\\\\n&+ q^{\\binom{2c}{2} -(b-1)(2m-b-2c+1) -a(b-a-1)-2a +2(b+2c-2m-1)} [b-a]  \\qbinom{h;-m}{c}  \n\\\\\n&+ q^{\\binom{2c}{2}-4c+3 -(b+1)(2m-b-2c+1) -(a+1)(b-a)} \n \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}   \\qbinom{h;1-m}{c-1}   \\\\\n&+ q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a) -2a} [2m-b-2c+1]  \\qbinom{h;1-m}{c}  \n\\\\\n&+ q^{\\binom{2c}{2} -b(2m-b-2c) -a(b-a) -2a-4m} [2m-b-2c+1]  \\qbinom{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nOn the other hand, using \\eqref{t2m-1:evev2} (with an index shift $c\\to c-1$) and \\eqref{t2m+1:evev2} we write \n\\begin{align*}\n[2m+1] & \\dvev{2m+1} +[2m] \\dvev{2m-1} \n\\\\\n&=\\sum_{0 \\le a \\le b \\le 2m+1} \\sum_{c \\geq 0} p^{(2m-b-2c+1)}(\\kappa) \n\\check{E}^{(a)} R_{a,b,c} K^{b-2m+2c-1} F^{(b-a)}, \n\\end{align*}\nwhere  \n\\begin{align*}\nR_{a,b,c} :=&\nq^{\\binom{2c}{2}+2c -b(2m-b-2c+1) -a(b-a)}   [2m+1] \\qbinom{h;-m}{c} \n\\\\\n&+ q^{\\binom{2c}{2}-2c+1 -b(2m-b-2c+1) -a(b-a)} [2m]  \n \\qbinom{h;1-m}{c-1}.  \n\\end{align*}\n\nTo prove the formula \\eqref{t2m+1:evev2} for $\\dvev{2m+1}$, it suffices to show that, for all $a,b,c$,\n\\begin{equation}\n  \\label{L=R}\nL_{a,b,c}=R_{a,b,c}.\n\\end{equation}\nCanceling the $q$-powers $q^{\\binom{2c}{2}+2bc-b(2m-b+1)-a(b-a)}$ on both sides, we see that the identity \\eqref{L=R} is equivalent to the following identity, for all $a,b,c$:\n\\begin{align}\n \\label{l=r}\n \\begin{split}\nq^{2m-2c-a+1} [a] \\qbinom{h;1-m}{c}  \n&+ q^{2c-2m+b-a-1} [b-a] \\qbinom{h;-m}{c}  \n\\\\\n&+ q^{-2m-2c+a+2} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}   \\qbinom{h;1-m}{c-1}    \\\\\n&+ q^{b -2a} [2m-b-2c+1]  \\qbinom{h;1-m}{c}  \n\\\\\n&+ q^{b -2a -4m} [2m-b-2c+1] \\qbinom{h;1-m}{c-1} K^{-2}\n\\\\\n=& q^{2c} [2m+1] \\qbinom{h;-m}{c}  \n + q^{-2c+1} [2m] \\qbinom{h;1-m}{c-1}.  \n   \\end{split}\n\\end{align}\n\nBy combining the second summand of LHS with the first summand of RHS as well as combining the third summand of LHS with the second summand of RHS, the identity \\eqref{l=r} is reduced to the following equivalent identity, for all $a, b, c$:\n\\begin{equation}  \\label{WXYZ}\nW+X+Y+Z_1+Z_2=0,\n\\end{equation}\nwhere  \n\\begin{align*}\nW&=  [a]  \\qbinom{h;1-m}{c},\n\\\\\nX&= q^{4c-2m+b-1} [-2m+b-a-1] \\qbinom{h;-m}{c},\n\\\\\nY&= \\frac{q^{-4m-a+1}K^{-2}-q^{a+1}}{q^2-1}   \\qbinom{h;1-m}{c-1},  \\\\\nZ_1&=  q^{2c-2m+b -a-1} [2m-b-2c+1] \\qbinom{h;1-m}{c},\n  \\\\\nZ_2&=  q^{2c-6m+b-a-1} [2m-b-2c+1] \\qbinom{h;1-m}{c-1} K^{-2}.  \n\\end{align*}\n\nLet us prove the identity \\eqref{WXYZ}. Using \\eqref{kbinom2}, we can write $X=X_1+X_2$, where \n\\begin{align*}\nX_1 &=q^{4c-2m+b-1} [-2m+b-a-1] \\qbinom{h; 1-m}{c},\n   \\\\\nX_2 &= - q^{4c-6m+b-1} [-2m+b-a-1] \\qbinom{h; 1-m}{c-1} K^{-2}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{-4m-a+1}K^{-2}-q^{a+1}}{q^2-1}\n= q^{-4c-a+1}\\frac{( q^{4c-4m}K^{-2}-1)}{q^2-1}\n +  q^{-2c} [-2c-a],\n \\] \nwe rewrite $Y=Y_1+Y_2$, where\n\\begin{align*}\nY_1 &=q^{-2c-a} [2c] \\qbinom{h; 1-m}{c},\n   \\qquad\nY_2 = q^{-2c} [-2c-a] \\qbinom{h; 1-m}{c-1}.\n\\end{align*}\nA direct computation shows that\n\\begin{align*}\nW+X_1+Y_1+Z_1 &=\n- (q-q^{-1}) [2c+a] [2c]  \\qbinom{h; 1-m}{c},\n\\\\\n(X_2+Z_2) +Y_2 \n&= q^{2c-4m} [2c+a] \\qbinom{h; 1-m}{c-1} K^{-2} +Y_2\n\\\\\n&= (q-q^{-1}) [2c+a] [2c] \\qbinom{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we obtain $W+X+Y+Z_1+Z_2=0$, whence \\eqref{WXYZ}, completing Step ~(2).\n\nThe proof of Theorem~\\ref{thm:dvev:evKappa} is completed. \\qed\n\n\n\n  \n\\section{The $\\imath$-divided powers $\\dv{n}$ for odd weights and even $\\kappa$}\n  \\label{sec:oddevK}\n\nIn this section we shall always take $\\kappa$ to be an even $q$-integer, i.e., \n\\[\n\\kappa =[2\\ell], \\qquad  \\text{ for } \\ell  \\in \\mathbb Z.\n\\]  \n\n\\subsection{Definition of $\\dv{n}$ for even $\\kappa$}\n\n\nWe introduce some notations. Set, for $n\\ge 1, a\\in \\mathbb Z$, \n\\begin{equation}   \\label{brace}\n\\LR{h;a}{0}=1, \n\\qquad\n\\LR{h;a}{n}= \\prod_{i=1}^n \\frac{q^{4a+4i-4} K^{-2}-q^2}{q^{4i}-1}, \n\\qquad \\llbracket h;a\\rrbracket = \\LR{h;a}{1}.\n\\end{equation}\nIt follows from \\eqref{brace} that, for $n\\ge 0$ and $a\\in \\mathbb Z$, \n\\begin{align}\n \\label{eq:commLR}\n \\begin{split}\n\\LR{h;a}{n} F &= F  \\LR{h;a+1}{n},\n\\qquad\n\\LR{h;a}{n} \\check{E} =\\check{E} \\LR{h;a-1}{n},\n\\\\\n\\LR{h;a}{n} &=\\LR{h;1+a}{n} -q^{4a}K^{-2} \\LR{h;1+a}{n-1}.  \n  \\end{split}\n\\end{align}\n\nWe shall take the following as a definition of the $\\imath$-divided powers $\\dv{n}$. \n\n\\begin{definition}\nSet $\\dv{1} =t= F +\\check{E} +\\kappa K^{-1}$. The divided powers $\\dv{n}$, for $n\\ge 1$, are defined by the  recursive relations: \n\\begin{align}   \n  \\label{eq:tt:oddevKa}\n\\begin{split}\nt \\cdot \\dv{2a-1} &=[2a] \\dv{2a}+   [2a-1] \\dv{2a-2},\n\\\\\nt \\cdot \\dv{2a} &=  [2a+1] \\dv{2a+1} , \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}   \nEquivalently, $\\dv{n}$ is defined by the closed formula \\eqref{def:dv:evoddK}.\n\\end{definition}\n\nThe following lemma is a variant of \\cite[Lemma~3.3]{BeW18} (where $\\kappa=0$) with the same proof.\n\n\\begin{lem}   \n  \\label{lem:anti2}\nThe anti-involution $\\varsigma$ on $\\mbf U$ fixes $F,  \\check{E},  K,  \\dvd{n}$, respectively. Moreover, $\\varsigma$ sends \n$q \\mapsto q^{-1}$,  and \n\\[\n\\LR{h;a}{n}\\mapsto (-1)^n q^{2n(n-1)}  \\LR{h;2-a-n}{n}, \\quad \\forall a \\in \\mathbb Z,\\; n\\in \\mathbb N. \n\\]\n\\end{lem}\n\n\\subsection{Formulae of $\\dv{n}$ with even $\\kappa$}\n\nRecall the polynomials $p^{(n)}$, for $n\\ge 0$, from \\S\\ref{subsec:pn}.\n\n\\begin{thm}  \n  \\label{thm:dv:evenKappa}\nLet $\\kappa$ be an even $q$-integer. Then we have, for $m\\ge 0$,  \n\\begin{align}\n\\dv{2m} &= \n\\sum_{b=0}^{2m}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)} \np^{(2m-b-2c)}(\\kappa) \n\\label{t2m:oddev} \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}, \n\\notag\n\\\\%\n\\dv{2m+1} &= \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)}\n p^{(2m-b-2c+1)}(\\kappa) \n\\label{t2m+1:oddev}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\notag\n\\end{align}\n\\end{thm}\n\n\nThe proof of Theorem~\\ref{thm:dv:evenKappa} will be given in \\S\\ref{subsec:proof:dv:evKappa} below. \nBy applying the anti-involution $\\varsigma$ we convert the formulae in Theorem~\\ref{thm:dv:evenKappa} into the following.\n\n\\begin{thm}  \n  \\label{thm:dv:evenKappa2}\nLet $\\kappa$ be an even $q$-integer. Then we have, for $m\\ge 0$,  \n\\begin{align}\n\\dv{2m} &= \n\\sum_{b=0}^{2m}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} (-1)^c q^{-c +b(2m-b-2c) +a(b-a)} \np^{(2m-b-2c)}(\\kappa) \n\\label{t2m:oddevFhE}  \\\\\n&\\qquad \\qquad \\qquad\\qquad \\cdot F^{(b-a)} K^{b-2m+2c} \\LR{h;1+m-c}{c}  \\check{E}^{(a)}\n\\notag\n\\\\\n\\dv{2m+1} &=  \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} (-1)^c q^{c +b(2m-b-2c+1) +a(b-a)}\n p^{(2m-b-2c+1)}(\\kappa) \n\\label{t2m+1:oddevFhE}  \\\\\n&\\qquad\\qquad\\qquad\\qquad \\cdot F^{(b-a)} K^{b-2m+2c-1} \\LR{h;1+m-c}{c}  \\check{E}^{(a)}.\n\\notag\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nRecall from Lemma~\\ref{lem:anti2} that the anti-involution $\\varsigma$ on $\\mbf U$   \nfixes $F,  \\check{E},  K,  \\dvd{n}, \\kappa$ while sending\n$q \\mapsto q^{-1}$, \n$\\LR{h;1-m}{c}\\mapsto (-1)^c q^{2c(c-1)}  \\LR{h;1+m-c}{c}$.\nThe formulae  \\eqref{t2m:oddevFhE}--\\eqref{t2m+1:oddevFhE} now follows from \\eqref{t2m:oddev}--\\eqref{t2m+1:oddev}. \n\n\n\n\n\\end{proof}\n\n  Let $\\lambda, m \\in \\mathbb Z$ and $c\\in \\mathbb N$.  Recall $\\cbinom{m}{c}$ from \\eqref{cbinom}.\n We note that\n\\begin{align}\n  \\label{eq:LRhc}\n\\LR{h; 1+m-c}{c} \\mathbf 1_{2\\lambda+1}  \n&= \\prod_{i=1}^c \\frac{q^{4m-4c+4i} K^{-2} -q^2}{q^{4i} -1} \\mathbf 1_{2\\lambda+1}   \n= (-1)^c q^{-2c^2} \\cbinom{m-\\lambda-c}{c} \\mathbf 1_{2\\lambda+1}.\n\\end{align}\n\nThe following corollary is immediate from \\eqref{eq:LRhc}, Proposition~\\ref{prop:fg:ev}, and Theorem~\\ref{thm:dv:evenKappa}. \n\\begin{cor}  \nWe have $\\dv{n} \\in {}_\\mathcal A {\\mbf U}^\\imath_{\\rm{odd}}$, for all $n$. \n\\end{cor}\n\n\n\n\\begin{example}\nThe formulae of $\\dv{n}$, for $1\\le n\\le 3$, in Theorem~\\ref{thm:dv:evenKappa} reads as follows. \n\\begin{align*}\n\\dv{1} \n&= F +\\check{E} + \\kappa K^{-1},\n\\\\%\n\\dv{2} \n&=  b^{(2)}  + q\\llbracket h;0 \\rrbracket + \\kappa (q^{-1}K^{-1}F + q^{-1} \\check{E} K^{-1})  +  \\frac{\\kappa^{2}}{[2]} K^{-2},  \n \\\\%\n \\dv{3}  &   = b^{(3)} + q^{-1} \\llbracket h;0 \\rrbracket F + q^{-1} \\check{E} \\llbracket h;0 \\rrbracket\n\\\\\n & \\qquad +(q^{-2} \\check{E}^{(2)} K^{-1} + q^{-3} \\check{E} K^{-1} F + q^{-2} K^{-1} F^{(2)} + q^{-1} \\llbracket h;0 \\rrbracket K^{-1})\\kappa\n\\\\\n & \\qquad + \\frac{q^{-2}}{[2]}(\\check{E} K^{-2} + K^{-2} F)\\kappa^2\n + \\frac{K^{-3}}{[3]!}\\big( \\kappa^3 + (q^{-4} + q^{-2})\\kappa \\big).\n\\end{align*}\n\\end{example}\n\n\\begin{rem}\nThe formula for $\\dv{2m}$ is formally obtained from $\\dvev{2m}$ in Theorem~\\ref{thm:dvev:evKappa},\nwith $\\qbinom{h;a}{c}$ replaced by $\\LR{h;a}{c}$.\nThe formula for $\\dv{2m-1}$ is formally obtained from $\\dvev{2m-1}$ in Theorem~\\ref{thm:dvev:evKappa},\nwith $\\qbinom{h;a}{c}$ replaced by $q^{-4c}\\LR{h;a+1}{c}$.\n\\end{rem}\n\n\\begin{rem}   \nNote $p_n(0)=0$ for $n>0$, and $p_0(0)=1$. \nThe formulae in Theorems~\\ref{thm:dv:evenKappa} and  \\ref{thm:dv:evenKappa2} in the special case for $\\kappa=0$ recover the formulae in \\cite[Theorem~3.1, Proposition~3.4]{BeW18}.\n\\end{rem}\n\n\n\n\n\n\n\\subsection{The $\\imath$-canonical basis for $\\dot{\\mbf U}_{\\text{odd}}$ for even $\\kappa$}\n  \\label{sec:iCB:odd}\n  \nRecall from \\S\\ref{sec:iCB:ev} the $\\imath$-canonical basis on simple $\\mbf U$-modules $L(\\mu)$, for $\\mu \\in \\mathbb N$. \n  \n\\begin{thm}  \\label{thm:iCB:odd}\n    $\\quad$\n\\begin{enumerate}\n\\item\nLet $n\\in \\mathbb N$. \nFor each integer $\\lambda \\gg n$, the element $\\dv{n} v^+_{2\\la+1} $ is an $\\imath$-canonical basis element for $L(2\\lambda+1)$. \n \n \\item\nThe set $\\{\\dv{n} \\mid n \\in \\mathbb N \\}$ forms the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ (and an $\\mathcal A$-basis in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm{odd}}$). \n\\end{enumerate}\n\\end{thm}\n \n \\begin{proof}\n  Recall $\\cbinom{m}{c}$ from \\eqref{cbinom} and $\\LR{h; a}{c}$ from \\eqref{brace}.  Let $\\lambda, m \\in \\mathbb N$.  \nIt follows by a direct computation using Theorem~\\ref{thm:dv:evenKappa2} and \\eqref{eq:LRhc} that \n\\begin{align}\n\\dv{2m} v^+_{2\\la+1}  &= \n\\sum_{b=0}^{2m} \\sum_{c \\geq 0} (-1)^c q^{-c +b(2m-b-2c)} \np^{(2m-b-2c)}(\\kappa) \n\\notag\n \\\\\n&\\qquad \\qquad \\qquad\\qquad \\cdot F^{(b)} K^{b-2m+2c} \\LR{h;1+m-c}{c}  v^+_{2\\la+1} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m} \\sum_{c \\geq 0} q^{-2c^2-c +(b-2\\lambda-1)(2m-b-2c)}  p^{(2m-b-2c)}(\\kappa) \\cbinom{m-\\lambda-c}{c}  F^{(b)}  v^+_{2\\la+1} .\n\\label{t2m:oddevF}  \n\\end{align}\nSimilarly using Theorem~\\ref{thm:dv:evenKappa2} we have \n\\begin{align}\n\\dv{2m+1} v^+_{2\\la+1}  &=  \n\\sum_{b=0}^{2m+1} \\sum_{c \\geq 0} (-1)^c q^{c +b(2m-b-2c+1)}\n p^{(2m-b-2c+1)}(\\kappa) \n\\notag  \\\\\n&\\qquad\\qquad\\qquad\\qquad \\cdot F^{(b)} K^{b-2m+2c-1} \\LR{h;1+m-c}{c} v^+_{2\\la+1} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m+1} \\sum_{c \\geq 0} q^{-2c^2+c +(b-2\\lambda-1)(2m-b-2c+1)} p^{(2m-b-2c+1)}(\\kappa) \\cbinom{m-\\lambda-c}{c} F^{(b)} v^+_{2\\la+1} . \n\\label{t2m+1:oddevF} \n\\end{align}\n\nBy a similar argument for \\eqref{eq:lattice}, using \\eqref{t2m:oddevF}--\\eqref{t2m+1:oddevF}  we obtain\n \\begin{align*}\n \\dv{n} v^+_{2\\la+1}   & \\in F^{(n)}  v^+_{2\\la+1}  + \\sum_{b<n}q^{-1} \\mathbb Z[q^{-1}] F^{(b)} v^+_{2\\la+1} ,\n \\qquad \\text{ for } \\lambda \\gg n.\n \\end{align*}\n \nThe second statement follows now from the definition of the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ using\nthe projective system $\\{ L(2\\lambda+1) \\}_{\\lambda\\ge 0}$; cf. \\cite[\\S6]{BW16}. \n \\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:dv:evenKappa} }\n   \\label{subsec:proof:dv:evKappa}\n\n \nWe prove the formulae for $\\dv{n}$ by induction on $n$, in two steps (1)--(2) below. The base cases when $n=1,2$ are clear. \n\n(1) We shall prove the formula \\eqref{t2m+1:oddev} for $\\dv{2m+1}$, assuming the formula \\eqref{t2m:oddev} for $\\dv{2m}$.\n\nRecall $[2m+1] \\dvev{2m+1} = t \\cdot \\dv{2m}$, and $t= F +\\check{E} +\\kappa K^{-1}$. \nLet us compute \n\\[\nI=\\check{E}  \\dv{2m},\n\\qquad\nII=F \\dv{2m},\n\\qquad\nIII=\\kappa K^{-1} \\dv{2m}.\n\\]\n\nFirst we have\n\\begin{align*}\nI\n&= \\sum_{b=0}^{2m}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)} [a+1] p^{(2m-b-2c)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\LR{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\\\%\n&= \\sum_{b=0}^{2m+1}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - (b-1)(2m-b-2c+1) -(a-1)(b-a)} [a] p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)},\n\\end{align*}\nwhere the last equation is obtained by shifting indices $a\\to a-1$, $b\\to b-1$ (and then adding some zero terms to make the bounds of the summations uniform throughout).\nUsing \\eqref{FYn} we  have\n\\begin{align*}\nII \n&=   \\sum_{b=0}^{2m}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)}  p^{(2m-b-2c)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n    \\LR{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\end{align*}  \nwhere $II^1$ and $II^2$ are the two natural summands associated to the plus sign.\nBy shifting index $b\\to b-1$, we further have\n\\begin{align*}\nII^1\n&=\\sum_{b=0}^{2m+1}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - (b+1)(2m-b-2c+1) -a(b-a+1)} [b-a] p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n \\check{E}^{(a)} \n       \\LR{h;-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\nII^2 \n&=\\sum_{b=0}^{2m+1}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2} - (b+1)(2m-b-2c+1) -(a+1)(b-a)}  p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n    \\LR{h;1-m}{c-1} K^{b-2m+2c-1} F^{(b-a)}. \n\\end{align*}  \nUsing \\eqref{recursive:f}\nwe also compute\n\\begin{align*}\nIII\n&= \\sum_{b=0}^{2m}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)-2a} \n\\kappa \\cdot p^{(2m-b-2c)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n \\\\%%\n&= \\sum_{b=0}^{2m+1}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)-2a} \n[2m-b-2c+1]  \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\%\n&\\quad \n- \\sum_{b=0}^{2m+1}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2} - (b+2)(2m-b-2c+2) -a(b-a)-2a+1} \n[2m-b-2c+1] \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\LR{h;1-m}{c-1} K^{b-2m+2c-3} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $I, II^1, II^2$ and $III$  gives us\n\\[\nt \\cdot \\dv{2m} = \n \\sum_{0 \\le a \\le b \\le 2m+1}  \\sum_{c\\ge 0}  p^{(2m-b-2c+1)}(\\kappa)  \\check{E}^{(a)} \n \\mathcal H_{a,b,c}  K^{b-2m+2c-1} F^{(b-a)}, \n\\]\nwhere   \n\\begin{align*}\n\\mathcal H_{a,b,c}  :=\n& \\; q^{\\binom{2c}{2} - (b-1)(2m-b-2c+1) -(a-1)(b-a)} [a] \\LR{h;1-m}{c} \n\\\\\n&+ q^{\\binom{2c}{2} - (b+1)(2m-b-2c+1) -a(b-a+1)} [b-a] \\LR{h;-m}{c} \n\\\\\n&+ q^{\\binom{2c-2}{2} - (b+1)(2m-b-2c+1) -(a+1)(b-a)}\n \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;1-m}{c-1} \n\\\\\n&+ q^{\\binom{2c}{2} - b(2m-b-2c) -a(b-a)-2a} \n[2m-b-2c+1]  \\LR{h;1-m}{c}\n\\\\\n&- q^{\\binom{2c-2}{2} - (b+2)(2m-b-2c+2) -a(b-a)-2a+1} \n[2m-b-2c+1] \\LR{h;1-m}{c-1} K^{-2}.\n\\end{align*}\nRecall $[2m+1] \\dv{2m+1} = t \\cdot \\dv{2m}$. To prove the formula \\eqref{t2m+1:oddev} for $\\dv{2m+1}$,  by the PBW basis theorem and the inductive assumption it suffices to prove the following identity, for all $a,b,c$:\n\\begin{align}  \\label{eq:H3}\n\\mathcal H_{a,b,c} = q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)} [2m+1] \\LR{h;1-m}{c}.\n\\end{align}\nThanks to $q^{2m-a+1} [a] -[2m+1] = q^{-a} [a-2m-1]$, we can combine the RHS with the first summand of LHS \\eqref{eq:H3}.\nHence, after canceling out the $q$-powers $q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a) -a}$\non both sides, we see that \\eqref{eq:H3} is equivalent to the following identity, for all $a, b, c$:\n\\begin{equation}  \\label{ABCD3}\n\\mathcal A+\\mathcal B+\\mathcal C+\\mathcal D_1 +\\mathcal D_2=0,\n\\end{equation}\nwhere \n\\begin{align*}\n\\mathcal A &= [a-2m-1] \\LR{h;1-m}{c},\n\\\\\n\\mathcal B &= q^{4c-2m+b-1} [b-a]  \\LR{h;-m}{c},   \n\\\\\n\\mathcal C &= q^{2a-2m+2} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;1-m}{c-1},   \n\\\\\n\\mathcal D_1 &= q^{2c+b-a} [2m-b-2c+1]  \\LR{h;1-m}{c},\n\\\\\n\\mathcal D_2 &= - q^{2c-4m+b-a} [2m-b-2c+1] \\LR{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us prove the identity \\eqref{ABCD3}. Using \\eqref{eq:commLR}, we can write $\\mathcal B=\\mathcal B_1+\\mathcal B_2$, where \n\\begin{align*}\n\\mathcal B_1 &=  q^{4c-2m+b-1} [b-a]  \\LR{h;-m}{c},\n   \\quad\n\\mathcal B_2 = -  q^{4c-6m+b-1} [b-a]  \\LR{h;-m}{c}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} = q^{4m-4c-3a}\\frac{( q^{4c-4m}K^{-2}-q^2)}{q^2-1}\n +  q^{2m-2c-2a} [2m-2c-a+1],\n \\] \nwe rewrite $\\mathcal C=\\mathcal C_1+\\mathcal C_2$, where\n\\begin{align*}\n\\mathcal C_1 &= q^{2m-2c-a+1} [2c] \\LR{h; 1-m}{c},\n   \\quad\n\\mathcal C_2 =   q^{2-2c} [2m-2c-a+1]  \\LR{h; 1-m}{c-1}.\n\\end{align*}\n\nA direct computation gives us\n\\begin{align*}\n\\mathcal A+\\mathcal B_1+\\mathcal C_1+\\mathcal D_1 \n&= (q-q^{-1}) [2c] [2m-2c-a+1]  \\LR{h; 1-m}{c},\n\\\\\n\\mathcal C_2 +(\\mathcal B_2+\\mathcal D_2)  \n&= \\mathcal C_2 -  q^{2c-4m} [2m-2c-a+1] \\LR{h; 1-m}{c-1} K^{-2}\n\\\\\n&= - (q-q^{-1}) [2c] [2m-2c-a+1]  \\LR{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we have $\\mathcal A+\\mathcal B+\\mathcal C+\\mathcal D_1+\\mathcal D_2=0,$ whence \\eqref{ABCD3}, completing Step~(1). \n\n\\vspace{3mm}\n(2) Assuming the formulae for $\\dv{n}$ with $n\\le 2m+1$, we shall now prove the following formula \\eqref{t2m+2:oddev} for $\\dv{2m+2}$ (obtained with $m$ replaced by $m+1$ in \\eqref{t2m:oddev}):\n\\begin{align}\n\\dv{2m+2} &= \n\\sum_{b=0}^{2m+2}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c+2) -a(b-a)} \np^{(2m-b-2c+2)}(\\kappa) \n  \\label{t2m+2:oddev}\n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;-m}{c} K^{b-2m+2c-2} F^{(b-a)}. \n\\notag\n\\end{align}\nThe proof is based on the recursion $t \\cdot \\dv{2m+1} =[2m+2] \\dv{2m+2} +[2m+1] \\dv{2m}$. \nRecall $t= F +\\check{E} +\\kappa K^{-1}$ and $\\dv{2m+1}$ from \\eqref{t2m+1:oddev}. We shall compute \n\\[\n\\texttt{I}=\\check{E}  \\dv{2m+1},\n\\qquad\n\\texttt{II}=F \\dv{2m+1},\n\\qquad\n\\texttt{III} =\\kappa K^{-1} \\dv{2m+1}, \n\\]\nrespectively. First by by shifting indices $a\\to a-1$ and $b\\to b-1$, we have\n\\begin{align*}\n\\texttt{I}\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)}\n[a+1] p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\\n&=\\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c - (b-1)(2m-b-2c+2) -(a-1)(b-a)}\n[a] p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}.\n\\end{align*}\nUsing \\eqref{FYn} we   also have\n\\begin{align*}\n\\texttt{II}\n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)}\n p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\quad \\cdot \n \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n=\\texttt{II}_1 +\\texttt{II}_2,\n\\end{align*}\nwhere $\\texttt{II}_1$ and $\\texttt{II}_2$ are the two natural summands associated to the plus sign.\nBy shifting the index $b\\to b-1$, we have\n\\begin{align*}\n\\texttt{II}_1 \n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c - (b+1)(2m-b-2c+2) -a(b-a-1)-2a}\n[b-a] p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n    \\check{E}^{(a)} \n    \\LR{h;-m}{c} K^{b-2m+2c-2} F^{(b-a)}. \n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\n\\texttt{II}_2 \n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2}-2c - (b+1)(2m-b-2c+2) -(a+1)(b-a)+2}\n p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n \\LR{h;1-m}{c-1} K^{b-2m+2c-2} F^{(b-a)}.\n \\end{align*}\n\nUsing \\eqref{eq:pn2div} we also compute\n\\begin{align*}\n\\texttt{III} \n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)-2a}\n\\kappa \\cdot p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}\n\\\\\n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)-2a}\n[2m-b-2c+2]  \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \np^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}\n\\\\%\n& - \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2}-2c - (b+2)(2m-b-2c+3) -a(b-a)-2a+3}\n[2m-b-2c+2] \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \np^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)}  \\LR{h;1-m}{c-1} K^{b-2m+2c-4} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $\\texttt{I}, \\texttt{II}_1, \\texttt{II}_2$, and $\\texttt{III}$, we obtain \n\\[\nt \\cdot \\dv{2m+1} = \n \\sum_{0 \\le a \\le b \\le 2m+2}  \\sum_{c\\ge 0} p^{(2m-b-2c+2)}(\\kappa)  \n  \\check{E}^{(a)} \\mathcal L_{a,b,c} K^{b-2m+2c-2} F^{(b-a)},\n\\]\nwhere  \n\\begin{align*}\n\\mathcal L_{a,b,c} := \n& \\; q^{\\binom{2c}{2}-2c - (b-1)(2m-b-2c+2) -(a-1)(b-a)} \n[a] \\LR{h;1-m}{c}\n\\\\\n&+q^{\\binom{2c}{2}-2c - (b+1)(2m-b-2c+2) -a(b-a-1)-2a} \n[b-a]  \\LR{h;-m}{c} \n\\\\\n&+q^{\\binom{2c-2}{2}-2c - (b+1)(2m-b-2c+2) -(a+1)(b-a)+2}\n\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;1-m}{c-1} \n\\\\\n&+q^{\\binom{2c}{2}-2c - b(2m-b-2c+1) -a(b-a)-2a}\n[2m-b-2c+2]  \\LR{h;1-m}{c}\n\\\\\n&- q^{\\binom{2c-2}{2}-2c - (b+2)(2m-b-2c+3) -a(b-a)-2a+3}\n[2m-b-2c+2]  \\LR{h;1-m}{c-1} K^{-2}. \n\\end{align*}\n\nOn the other hand, using \\eqref{t2m:oddev} (with an index shift $c\\to c-1$) and \\eqref{t2m+2:oddev} we write \n\\begin{align*}\n [2m+2] & \\dv{2m+2} +[2m+1] \\dv{2m} \n\\\\\n&=\\sum_{0 \\le a \\le b \\le 2m+2} \\sum_{c \\geq 0} \np^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)} \\mathcal R_{a,b,c} K^{b-2m+2c-2} F^{(b-a)}, \n\\end{align*}\nwhere  \n\\begin{align*}\n\\mathcal R_{a,b,c} \n&:= q^{\\binom{2c}{2} - b(2m-b-2c+2) -a(b-a)} \n[2m+2]  \\LR{h;-m}{c}  \n\\\\\n&\\qquad + q^{\\binom{2c-2}{2} - b(2m-b-2c+2) -a(b-a)} \n[2m+1]  \\LR{h;1-m}{c-1}.    \n\\end{align*}\n\nTo prove the formula \\eqref{t2m+2:oddev} for $\\dv{2m+2}$, it suffices to show that, for all $a,b,c$,\n\\begin{equation}\n  \\label{L=R3}\n\\mathcal L_{a,b,c}= \\mathcal R_{a,b,c}.\n\\end{equation}\nCanceling the $q$-powers $q^{\\binom{2c}{2}-2c - (b-1)(2m-b-2c+2) -(a-1)(b-a)}$ on both sides, we see that the identity \\eqref{L=R3} is equivalent to the following identity, for all $a,b,c$:\n\\begin{align}\n \\label{l=r3}\n \\begin{split}\n[a] \\LR{h;1-m}{c}  \n&+q^{4c-4m+b-4}  [b-a]  \\LR{h;-m}{c}   \n\\\\\n&+q^{2a-4m+1} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;1-m}{c-1}   \n\\\\\n&+q^{2c-2m+b-a-2} [2m-b-2c+2]  \\LR{h;1-m}{c}\n\\\\\n&- q^{2c-6m+b-a-2} [2m-b-2c+2]  \\LR{h;1-m}{c-1} K^{-2}\n\\\\\n&= q^{4c-2m+a-2}  [2m+2]  \\LR{h;-m}{c}  \n+ q^{a-2m+1}  [2m+1]  \\LR{h;1-m}{c-1}.  \n   \\end{split}\n\\end{align}\n\nBy combining the second summand of LHS with the first summand of RHS as well as combining the third summand of LHS with the second summand of RHS, the identity \\eqref{l=r3} is reduced to the following equivalent identity, for all $a, b, c$:\n\\begin{equation}  \\label{WXYZ3}\n\\mathcal W+\\mathcal X+\\mathcal Y+\\mathcal Z_1+\\mathcal Z_2=0,\n\\end{equation}\nwhere  \n\\begin{align*}\n\\mathcal W & =[a] \\LR{h;1-m}{c},\n\\\\\n\\mathcal X & =q^{4c-2m+b-2}  [b-a-2m-2]  \\LR{h;-m}{c},   \n\\\\\n\\mathcal Y & =  \\frac{q^{-a-4m+1}K^{-2}-q^{a+3}}{q^2-1} \\LR{h;1-m}{c-1},    \n\\\\\n\\mathcal Z_1 & =q^{2c-2m+b-a-2} [2m-b-2c+2]  \\LR{h;1-m}{c},\n\\\\\n\\mathcal Z_2 & = - q^{2c-6m+b-a-2} [2m-b-2c+2]  \\LR{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us finally prove the identity \\eqref{WXYZ3}. Using \\eqref{eq:commLR}, we can write $\\mathcal X=\\mathcal X_1+\\mathcal X_2$, where \n\\begin{align*}\n\\mathcal X_1 &=q^{4c-2m+b-2}  [b-a-2m-2]  \\LR{h; 1-m}{c},\n   \\\\\n\\mathcal X_2 &= - q^{4c-6m+b-2}  [b-a-2m-2]  \\LR{h; 1-m}{c-1} K^{-2}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{-a-4m+1}K^{-2}-q^{a+3}}{q^2-1}  \n= q^{-4c-a+1}\\frac{( q^{4c-4m}K^{-2}-q^2)}{q^2-1}\n +  q^{2-2c} [-2c-a],\n \\] \nwe rewrite $\\mathcal Y=\\mathcal Y_1+\\mathcal Y_2$, where\n\\begin{align*}\n\\mathcal Y_1 &=q^{-2c-a} [2c] \\LR{h; 1-m}{c},\n   \\qquad\n\\mathcal Y_2 = q^{2-2c} [-2c-a] \\LR{h; 1-m}{c-1}.\n\\end{align*}\nA direct computation shows that\n\\begin{align*}\n\\mathcal W+\\mathcal X_1+\\mathcal Y_1+\\mathcal Z_1 &=\n- (q-q^{-1}) [2c+a] [2c]  \\LR{h; 1-m}{c},\n\\\\\n(\\mathcal X_2+\\mathcal Z_2) +\\mathcal Y_2 \n&= q^{2c-4m} [2c+a] \\LR{h; 1-m}{c-1} K^{-2} +\\mathcal Y_2\n\\\\\n&= (q-q^{-1}) [2c+a] [2c] \\LR{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we obtain $\\mathcal W+\\mathcal X+\\mathcal Y+\\mathcal Z_1+\\mathcal Z_2=0$, whence \\eqref{WXYZ3}, completing Step ~(2).\n\nThe proof of Theorem~\\ref{thm:dv:evenKappa} is completed. \\qed\n\n\n\\section{The $\\imath$-divided powers  $\\dvp{n}$ for even weights and odd $\\kappa$}\n  \\label{sec:evoddK}\n\nIn this section~\\ref{sec:evoddK} we shall always take $\\kappa$ to be an odd $q$-integer, i.e., \n\\begin{align}  \n   \\label{eq:oddK}\n\\kappa & =[2\\ell-1], \\quad  \\text{ for } \\ell  \\in \\mathbb Z.\n\\end{align}\n\\subsection{Definition of $\\dvp{n}$ for odd $\\kappa$}\n\nWe shall take the following as a definition of the $\\imath$-divided powers  $\\dvp{n}$ with odd $\\kappa$, which is formally identical to the formulae for $\\dv{n}$ with even $\\kappa$. \n\n\\begin{definition}\nSet $\\dvp{1} = t =  F +\\check{E} +\\kappa K^{-1}$. The divided powers $\\dvp{n}$, for $n\\ge 1$, are defined by the recursive relations: \n\\begin{align}   \n  \\label{eq:tt:oddevKa}\n\\begin{split}\nt \\cdot \\dvp{2a-1} &=[2a] \\dvp{2a}+   [2a-1] \\dvp{2a-2},\n\\\\\nt \\cdot \\dvp{2a} &=  [2a+1] \\dvp{2a+1} , \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}   \n\\end{definition}\nEquivalently, we have the following closed formula for $\\dvp{n}$, for $a\\in\\mathbb N$:\n\\begin{align*}\n\\dvp{n} = \n\\begin{cases}\n\\frac{1}{[2a]!}  (t  - [-2a+1] ) (t  - [-2a+3]) \\cdots (t  - [2a-3]) (t -[2a-1]), & \\text{if } n=2a, \\\\\n\\\\\n\\frac{t}{[2a+1]!}  (t  - [-2a+1] ) (t  - [-2a+3]) \\cdots (t  - [2a-3]) (t -[2a-1]), &\\text{if } n=2a+1.\n\\end{cases}\n\\end{align*}\nNote the above formulae are formally the same as \\eqref{def:dv:evoddK} for $\\dv{n} \\in {}_\\mathcal A{\\mbf U}^\\imath_{\\rm odd}$ for  $\\kappa$ an even $q$-integer.\n\n\n\\subsection{Polynomials $\\mathfrak p_n(x)$  and $\\mathfrak p^{(n)}(x)$}\n  \\label{subsec:pn2}\n\nWe define a sequence of polynomials $\\mathfrak p_n(x)$ in a variable $x$, for $n\\in \\mathbb N$,  by letting\n\\begin{align}\n  \\label{eq:pn2}\n\\mathfrak p_{n+1} =x \\mathfrak p_n  + q^{2-2n} [n] [n-2] \\mathfrak p_{n-1}, \\qquad \\mathfrak p_0=1.\n\\end{align}\nNote $\\mathfrak p_n$ is a monic polynomial in $x$ of degree $n$. Also set $\\mathfrak p_n=0$ for $n<0$. These polynomials $\\mathfrak p_n$ will appear in the expansion formula for the $\\imath$-divided powers in $\\mbf U$. \n\n\n\\begin{example}\nHere are the polynomials $\\mathfrak p_n(x)$, for $1\\le n \\le 5$:\n\\begin{align*}\n& \\mathfrak p_1(x)  =x,\n\\qquad\n\\mathfrak p_2(x) =x^2-1,\n\\qquad\n\\mathfrak p_3(x) =x^3-x,\n\\\\\n\\mathfrak p_4(x) &=x^4 +(q^{-4}[3]-1)x^2 -q^{-4}[3],  \n\\quad\n\\mathfrak p_5(x) = x (x^2 - 1) (x^2+q^{-5}[3]! + q^{-6}[5]).\n\\end{align*}\n\\end{example}\nA simple induction using the recursive formula \\eqref{eq:pn2} shows that $\\mathfrak p_n\/(x-1) \\in \\mathbb N[q^{-1}][x]$, for $n\\ge 2$. In particular,  for $\\kappa$ any positive odd $q$-integer, we always have $\\mathfrak p_n (\\kappa) \\in \\mathbb N[q,q^{-1}]$. \n\nIntroduce the monic polynomials $\\mathfrak g_n(x)$ of degree $n$:\n\\begin{align}   \\label{eq:gn-ood}\n\\mathfrak g_n(x) =\n\\begin{cases}\n\\prod_{i=1}^{m} (x^2-[2i-1]^2), & \\text{ if } n=2m \\text{ is even};\n\\\\\nx \\prod_{i=1}^{m} (x^2-[2i-1]^2), & \\text{ if } n=2m+1 \\text{ is odd}.\n\\end{cases}\n\\end{align}\n\nDefine \n\\[\n\\mathfrak p^{(n)}(x) =\\mathfrak p_n(x)\/[n]!, \n\\qquad\n\\mathfrak g^{(n)}(x) =\\mathfrak g_n(x)\/[n]!. \n\\]\nThen Equation \\eqref{eq:pn2} implies that\n\\begin{align}\n  \\label{eq:pn2div}\nx \\cdot \\mathfrak p^{(n)}  =[n+1] \\mathfrak p^{(n+1)}-q^{2-2n} [n-2] \\mathfrak p^{(n-1)} \\quad (n\\ge 1).\n\\end{align}\n\n\nOur next goal is to prove that $\\mathfrak p^{(n)}(\\kappa)$ are integral, i.e., $\\mathfrak p^{(n)}([2\\ell-1]) \\in \\mathcal A$, for all $n, \\ell \\in \\mathbb Z$; see Proposition~\\ref{prop:fg:odd}. This is achieved by relating $\\mathfrak p^{(n)}$ to $\\mathfrak g^{(n)}$ for varied $n$. \n\n\\begin{lem} \\label{lem:odd:g-integral}\nFor $\\kappa =[2\\ell-1]$ as in \\eqref{eq:oddK}, we have $\\mathfrak g^{(n)} (\\kappa) \\in \\mathcal A$, for all $n\\in \\mathbb N$. \n\\end{lem}\n\n\\begin{proof}\nWe separate the cases for $n=2m+1$ and $n= 2m$. \nNoting that\n\\begin{equation}  \\label{eq:A2}\n\\kappa -[2i-1] =[2\\ell-1]-[2i-1] = [\\ell-i] (q^{\\ell+i-2} +q^{-\\ell-i+2}),\n\\end{equation} \nwe have\n \\begin{align*}\n \\mathfrak g^{(2m)}(\\kappa) &= \\frac1{[2m]!} \\prod_{i=1-m}^{m} (\\kappa-[2i-1])\n =  \n \\qbinom{\\ell+m-1}{2m} \\prod_{i=1-m}^{m}(q^{\\ell+i-2} +q^{-\\ell-i+2} ) \\in \\mathcal A.\n \\end{align*}\n \nSimilarly, using \\eqref{eq:A2} we have\n \\begin{align*}\n \\mathfrak g^{(2m+1)}(\\kappa) &= \\frac1{[2m+1]!} \\kappa \\prod_{i=1-m}^{m} (\\kappa-[2i-1])\n \\\\\n &= \\frac1{[2m+1]!} \\prod_{i=-m}^{m} (\\kappa-[2i-1]) - \\frac1{[2m]!}  \\prod_{i=1-m}^{m} (\\kappa-[2i-1])\n \\\\\n &=\\qbinom{\\ell+m}{2m+1} \\prod_{i=-m}^{m}(q^{\\ell+i-2} +q^{-\\ell-i+2} ) \n  + \\qbinom{\\ell+m-1}{2m} \\prod_{i=1-m}^{m}(q^{\\ell+i-2} +q^{-\\ell-i+2} ) \\in \\mathcal A.\n  \\end{align*}\n  The lemma is proved. \n\\end{proof}\n\n\\begin{prop}\n \\label{prop:fg:odd}\nFor $m \\in \\mathbb N$, we have\n\\begin{align*}\n\\mathfrak p^{(2m)} &= \\sum_{a=0}^{m-1} q^{(3-2m)a} \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a)},\n\\\\\n\\mathfrak p^{(2m+1)} &= \\sum_{a=0}^{m-1} q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a+1)}.\n\\end{align*}\nIn particular, we have $\\mathfrak p^{(n)} (\\kappa) \\in \\mathcal A$, for all $n\\in \\mathbb N$ and all $\\kappa =[2\\ell-1]$ with $\\ell \\in \\mathbb Z$. \n\\end{prop}\n\n\\begin{proof}\n It follows from  these formulae for $\\mathfrak p^{(n)}$ and Lemma~\\ref{lem:odd:g-integral} that $\\mathfrak p^{(n)} ([2\\ell-1]) \\in \\mathcal A$.\n\nLet us prove these formulae. Note that \n\\begin{align}  \\label{recursive:g2}\n\\begin{split}\nx \\cdot \\mathfrak g^{(2a-1)} &=[2a] \\mathfrak g^{(2a)} +   [2a-1] \\mathfrak g^{(2a-2)},\n\\\\\nx \\cdot \\mathfrak g^{(2a)} &=  [2a+1] \\mathfrak g^{(2a+1)}, \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}\n\nWe prove by induction on $m$, with the base cases for $\\mathfrak p^{(1)}$ and $\\mathfrak p^{(2)}$ being clear. We separate in two cases. \n\n(1) Let us prove the formula for $\\mathfrak p^{(2m+1)}$, assuming the formulae for $\\mathfrak p^{(2m-1)}$ and $\\mathfrak p^{(2m)}$.\nBy \\eqref{eq:pn2div} and \\eqref{recursive:g2} we have \n\\begin{align*} \n[2m+1] \\mathfrak p^{(2m+1)} \n&= x \\mathfrak p^{(2m)} +q^{2-4m} [2m-2] \\mathfrak p^{(2m-1)}  \\\\\n&= \\sum_{a=0}^{m-1} q^{(3-2m)a} \\qbinom{m-1}{a}_{q^2} x \\cdot \\mathfrak g^{(2m-2a)}\n\\\\&\\qquad \n  +\\sum_{a=0}^{m-2} q^{(3-2m)a+2-4m}  [2m-2] \\qbinom{m-2}{a}_{q^2} \\mathfrak g^{(2m-2a-1)}\n  \\\\\n&= \\sum_{a=0}^{m-1} q^{(3-2m)a} [2m-2a+1] \\qbinom{m-1}{a}_{q^2}   \\mathfrak g^{(2m-2a+1)}\n\\\\&\\qquad \n  +\\sum_{a=0}^{m-2} q^{(3-2m)a+2-4m}  [2m-2] \\qbinom{m-2}{a}_{q^2} \\mathfrak g^{(2m-2a-1)}\n  \\\\\n&= \\sum_{a=0}^{m-1} q^{(3-2m)a} [2m-2a+1] \\qbinom{m-1}{a}_{q^2}   \\mathfrak g^{(2m-2a+1)}\n\\\\&\\qquad \n  +\\sum_{a=1}^{m-1} q^{(3-2m)(a-1)+2-4m}  [2m-2] \\qbinom{m-2}{a-1}_{q^2} \\mathfrak g^{(2m-2a+1)}\n  \\\\\n  &=[2m+1] \\sum_{a=0}^{m-1} q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a+1)},\n\\end{align*}\nwhere the last equation results from shifting the second summation by $a\\to a-1$ and using\nthe identity \n\\[\nq^{2a} [2m-2a+1] \\qbinom{m-1}{a}_{q^2} +q^{2a-2m-1}  [2m-2] \\qbinom{m-2}{a-1}_{q^2}\n=[2m+1] \\qbinom{m-1}{a}_{q^2}. \n\\]\n\\vspace{3mm} \n\n(2) We shall prove the formula for $\\mathfrak p^{(2m+2)}$, assuming the formulae for $\\mathfrak p^{(2m+1)}$ and $\\mathfrak p^{(2m)}$.\nBy \\eqref{eq:pn2div} and \\eqref{recursive:g2} we have  \n\\begin{align*} \n&[2m+2] \\mathfrak p^{(2m+2)} \n\\\\\n&= x \\mathfrak p^{(2m+1)} +q^{-4m} [2m-1] \\mathfrak p^{(2m)} \n \\\\\n&= \\sum_{a=0}^{m-1} q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} x\\cdot \\mathfrak g^{(2m-2a+1)}\n+ \\sum_{a=0}^{m-1} q^{(3-2m)a-4m} [2m-1] \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a)}\n\\\\%\n&= \\sum_{a=0}^{m-1} q^{(1-2m)a} \\qbinom{m-1}{a}_{q^2} \n\\left( [2m-2a+2] \\mathfrak g^{(2m-2a+2)} +[2m-2a+1]  \\mathfrak g^{(2m-2a)} \\right)\n\\\\& \\qquad \n+ \\sum_{a=0}^{m-1} q^{(3-2m)a-4m} [2m-1] \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a)}. \n\\end{align*}\nBy combining the like terms for $\\mathfrak g^{(2m-2a)}$ and using the identity\n\\[\nq^{(1-2m)a} [2m-2a+1] + q^{(3-2m)a-4m} [2m-1]\n=q^{(1-2m)(a+1)} [4m-2a],\n\\]\nwe have\n\\begin{align*} \n[2m+2] \\mathfrak p^{(2m+2)} \n&= \\sum_{a=0}^{m-1} q^{(1-2m)a}  [2m-2a+2] \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a+2)}  \n\\\\& \\qquad \n+ \\sum_{a=0}^{m-1} q^{(1-2m)(a+1)} [4m-2a] \\qbinom{m-1}{a}_{q^2} \\mathfrak g^{(2m-2a)}\n\\\\%\n&= [2m+2]  \\sum_{a=0}^{m} q^{(1-2m)a} \\qbinom{m}{a}_{q^2} \\mathfrak g^{(2m-2a+2)},\n\\end{align*}\nwhere the last equation results from shifting the second summation index from $a\\to a-1$ and using the identity\n\\[\n[2m-2a+2] \\qbinom{m-1}{a}_{q^2} + [4m-2a+2] \\qbinom{m-1}{a-1}_{q^2} \n= [2m+2] \\qbinom{m}{a}_{q^2}.\n\\]\nThe proposition is proved.\n\\end{proof}\n\n\n\\subsection{Formulae for $\\dvp{n}$ with odd $\\kappa$}\n\n\\begin{thm}   \n  \\label{thm:dvp:oddKappa}\nLet $\\kappa$ be an odd $q$-integer. Then, for $m\\ge 0$, we have\n\\begin{align}\n\\dvp{2m} &= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c\\geq 0} q^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)} \n\\mathfrak p^{(2m-b-2c)}(\\kappa) \n\\label{t2m:evodd} \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}, \n\\notag\n\\\\\n\\dvp{2m+1} &= \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} q^{\\binom{2c}{2} -b(2m-b-2c+1) -a(b-a)}\n \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n\\label{t2m+1:evodd}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\notag\n\\end{align}\n\\end{thm}\n\nThe proof of Theorem~\\ref{thm:dvp:oddKappa} will be given in \\S\\ref{subsec:proof:dvp:oddKappa} below. \nBy applying the anti-involution $\\varsigma$ we convert the formulae in Theorem~\\ref{thm:dvp:oddKappa} as follows.\n\n\\begin{thm}  \n  \\label{thm:dvp:oddKappa2}\nLet $\\kappa$ be an odd $q$-integer. Then we have, for $m\\ge 1$,  \n\\begin{align}\n\\dvp{2m} &= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \n(-1)^c q^{c +b(2m-b-2c) +a(b-a)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) \n \\label{t2m:oddevKa}  \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot  F^{(b-a)} K^{b-2m+2c} \\qbinom{h;m-c}{c}  \\check{E}^{(a)}\n\\notag\n\\\\\n\\dvp{2m+1}   &= \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \n(-1)^c q^{3c +b(2m-b-2c+1) +a(b-a)} \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n \\label{t2m+1:oddevKa} \\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot F^{(b-a)} K^{b-2m+2c-1} \\qbinom{h;m-c}{c}  \\check{E}^{(a)}.\n\\notag\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:anti}, the anti-involution $\\varsigma$ sends \n$\\qbinom{h;1-m}{c}\\mapsto (-1)^c q^{2c(c+1)} \\qbinom{h;m-c}{c}$, \n$q\\mapsto q^{-1}$, while fixing $\\kappa, K, \\check{E}^{(a)}, F^{(a)}$ and $\\dvev{n}$. The formulae \\eqref{t2m:oddevKa}--\\eqref{t2m+1:oddevKa}  now follows from \\eqref{t2m:evodd}--\\eqref{t2m+1:evodd} in Theorem~\\ref{thm:dvp:oddKappa}. \n\\end{proof}\n\nThe following corollary is immediate from \\eqref{eq:qbinom-hc}, Proposition~\\ref{prop:fg:odd}, and Theorem~\\ref{thm:dvp:oddKappa}. \n\\begin{cor}  \nWe have $\\dvp{n} \\in {}_\\mathcal A {\\mbf U}^\\imath_{\\rm{ev}}$, for all $n$. \n\\end{cor}\n\n\\begin{rem}   \nNote $\\mathfrak p_n(1)=0$ for $n\\ge 2$, and $\\mathfrak p_0(1)=\\mathfrak p_1(1)=1$. \nThe formulae in Theorems~\\ref{thm:dvp:oddKappa} and  \\ref{thm:dvp:oddKappa2} in the special case for $\\kappa=1$ recover the formulae in \\cite[Theorem~4.1, Proposition~4.3]{BeW18}.\n\\end{rem}\n\n\n\\begin{example}\nThe formulae of $\\dvp{n}$, for $1\\le n\\le 3$, in Theorem~\\ref{thm:dvp:oddKappa} reads as follows. \n\\begin{align*}\n\\dvp{1} \n&= F+ \\check{E} + K^{-1},\n\\\\\n\\dvp{2} \n&= b^{(2)}  + q^3 [h;0]+\\kappa( q^{-1}K^{-1}F +q^{-1}\\check{E} K^{-1} ) +  \\frac{\\kappa^{2}-1}{[2]} K^{-2},\n\\\\\n\\dvp{3} \n&=  b^{(3)} + q \\check{E} [h;0] + q [h;0] F \n+ \\kappa( q^{-2} \\check{E}^{(2)} K^{-1} + q^{-3} \\check{E} K^{-1} F + q^{-2} K^{-1} F^{(2)} + q [h;0] K^{-1})\n\\\\\n & \\qquad + q^{-2}\\frac{\\kappa^{2}-1}{[2]}(\\check{E} K^{-2} + K^{-2} F)  \n + \\frac{\\kappa^3 -\\kappa}{[3]!} K^{-3}.\n\\end{align*}\n\\end{example}\n \n\\subsection{The $\\imath$-canonical basis for $\\dot{\\mbf U}_{\\rm{ev}}$ with odd $\\kappa$}\n  \\label{sec:iCB:evwt-oddK}\n  \n\nRecall from \\S\\ref{sec:iCB:ev} the $\\imath$-canonical basis on simple $\\mbf U$-modules $L(\\mu)$, for $\\mu \\in \\mathbb N$. \n  \n\\begin{thm}  \\label{thm:iCB:evwt-oddK}\n    $\\quad$\n\\begin{enumerate}\n\\item\nLet $n\\in \\mathbb N$. \nFor each integer $\\lambda \\gg n$, the element $\\dvp{n} v^+_{2\\la} $ is an $\\imath$-canonical basis element for $L(2\\lambda)$. \n \n \\item\nThe set $\\{\\dvp{n} \\mid n \\in \\mathbb N \\}$ forms the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ (and an $\\mathcal A$-basis in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm{ev}}$). \n\\end{enumerate}\n\\end{thm}\n \n \\begin{proof}\nLet $\\lambda, m \\in \\mathbb N$. Recall $\\cbinom{m}{c}$ from \\eqref{cbinom} and $\\qbinom{h; a}{c}$ from \\eqref{kbinom}. It follows by a direct computation using Theorem~\\ref{thm:dvp:oddKappa2} and \\eqref{eq:qbinom-hc} that \n\\begin{align}\n\\dvp{2m} v^+_{2\\la}  &= \n\\sum_{b=0}^{2m} \\sum_{c\\geq 0} \n(-1)^c q^{c +b(2m-b-2c)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) F^{(b)} K^{b-2m+2c} \\qbinom{h;m-c}{c}  v^+_{2\\la} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m} \\sum_{c \\geq 0} q^{-2c^2-c +(b-2\\lambda)(2m-b-2c)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) \\cbinom{m-\\lambda-c}{c}  F^{(b)}  v^+_{2\\la} .\n\\label{t2m:evwt-oF}  \n\\end{align}\nSimilarly using Theorem~\\ref{thm:dvp:oddKappa2} we have \n\\begin{align}\n\\dvp{2m+1} v^+_{2\\la}  &=  \n\\sum_{b=0}^{2m+1} \\sum_{c\\geq 0} \n(-1)^c q^{3c +b(2m-b-2c+1)} \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n F^{(b)} K^{b-2m+2c-1} \\qbinom{h;m-c}{c} v^+_{2\\la} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m+1} \\sum_{c \\geq 0} q^{-2c^2+c +(b-2\\lambda)(2m-b-2c+1)} \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\cbinom{m-\\lambda-c}{c} F^{(b)} v^+_{2\\la} . \n\\label{t2m+1:evwt-oF} \n\\end{align}\n\nBy a similar argument for \\eqref{eq:lattice}, using \\eqref{t2m:evwt-oF}--\\eqref{t2m+1:evwt-oF}   we obtain\n \\begin{align*}\n \\dvp{n} v^+_{2\\la}   & \\in F^{(n)}  v^+_{2\\la}  + \\sum_{b<n}q^{-1} \\mathbb Z[q^{-1}] F^{(b)} v^+_{2\\la} ,\n \\qquad \\text{ for } \\lambda \\gg n.\n \\end{align*}\n \nThe second statement follows now from the definition of the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ using\nthe projective system $\\{ L(2\\lambda) \\}_{\\lambda\\ge 0}$; cf. \\cite[\\S6]{BW16}. \n \\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:dvp:oddKappa} }\n  \\label{subsec:proof:dvp:oddKappa}\nWe prove the formulae for $\\dvp{n}$ by induction on $n$, in two steps (1)--(2) below. The base cases when $n=1,2$ are clear. \n\n\n(1) We shall prove the formula \\eqref{t2m+1:evodd} for $\\dvp{2m+1}$, assuming the formula \\eqref{t2m:evodd} for $\\dvp{2m}$.\n\nRecall $[2m+1] \\dvp{2m+1} = t \\cdot \\dvp{2m}$, and $t= F + \\check{E}  +\\kappa K^{-1}$. \nLet us compute \n\\[\nI=\\check{E}  \\dvp{2m},\n\\qquad\nII=F \\dvp{2m},\n\\qquad\nIII=\\kappa K^{-1} \\dvp{2m}.\n\\]\n\nFirst we have\n\\begin{align*}\nI\n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)} \n[a+1] \\mathfrak p^{(2m-b-2c)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}, \n\\\\%\n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c}{2}+2c - (b-1)(2m-b-2c+1) -(a-1)(b-a)} \n[a] \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)},\n\\end{align*}\nwhere the last equation is obtained by shifting indices $a\\to a-1$, $b\\to b-1$.\nUsing \\eqref{FYn} we  have\n\\begin{align*}\nII \n&=  \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c\\geq 0} q^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)} \n\\mathfrak p^{(2m-b-2c)}(\\kappa) \n \\\\\n& \\qquad\\quad \\cdot \n \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n\\qbinom{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}\n=II^1 +II^2, \n\\\\\n\\end{align*}  \nwhere $II^1$ and $II^2$ are the two natural summands associated to the plus sign.\nBy shifting index $b\\to b-1$, we further have\n\\begin{align*}\nII^1\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c}{2}+2c -(b+1)(2m-b-2c+1) -a(b-a+1)} \n[b-a] \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n   \\check{E}^{(a)}  \\qbinom{h;-m}{c} K^{b-2m+2c-1} F^{(b-a)}.\n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\nII^2 \n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c-2}{2}+2c - (b+1)(2m-b-2c+1) -(a+1)(b-a)-2} \n\\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n \\qbinom{h;1-m}{c-1} K^{b-2m+2c-1} F^{(b-a)}.\n\\end{align*}  \nUsing \\eqref{eq:pn2div} we also compute\n\\begin{align*}\nIII\n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)-2a} \n\\kappa \\cdot \\mathfrak p^{(2m-b-2c)}(\\kappa) \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\%%\n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)-2a}  [2m-b-2c+1]  \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \n\\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\%\n&\\quad - \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c\\geq 0} \nq^{\\binom{2c-2}{2}+2c - (b+2) (2m-b-2c+2) -a(b-a)-2a}  [2m-b-2c]  \n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot\n\\mathfrak p^{(2m-b-2c+1)}(\\kappa)  \\check{E}^{(a)}  \\qbinom{h;1-m}{c-1} K^{b-2m+2c-3} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $I, II^1, II^2$ and $III$  gives us\n\\[\nt \\cdot \\dvp{2m} = \n \\sum_{0 \\le a \\le b \\le 2m+1}  \\sum_{c\\ge 0}  \\mathfrak p^{(2m-b-2c+1)}(\\kappa)  \\check{E}^{(a)} \n \\mathscr H_{a,b,c}  K^{b-2m+2c-1} F^{(b-a)}, \n\\]\nwhere   \n\\begin{align*}\n\\mathscr H_{a,b,c} := \n& \\; q^{\\binom{2c}{2}+2c - (b-1)(2m-b-2c+1) -(a-1)(b-a)} \n[a]  \\qbinom{h;1-m}{c} \n\\\\\n&+q^{\\binom{2c}{2}+2c -(b+1)(2m-b-2c+1) -a(b-a+1)} \n[b-a] \\qbinom{h;-m}{c}\n\\\\\n&+q^{\\binom{2c-2}{2}+2c - (b+1)(2m-b-2c+1) -(a+1)(b-a)-2} \n \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1} \n\\\\\n&+q^{\\binom{2c}{2}+2c - b(2m-b-2c) -a(b-a)-2a}  \n[2m-b-2c+1]    \\qbinom{h;1-m}{c}\n\\\\\n&+q^{\\binom{2c-2}{2}+2c - (b+2) (2m-b-2c+2) -a(b-a)-2a}  \n[2m-b-2c]   \\qbinom{h;1-m}{c-1} K^{-2}.\n\\end{align*}\nRecall $[2m+1] \\dv{2m+1} = t \\cdot \\dvp{2m}$. To prove the formula \\eqref{t2m+1:evodd} for $\\dv{2m+1}$,  by the PBW basis theorem and the inductive assumption it suffices to prove the following identity, for all $a,b,c$:\n\\begin{align}  \\label{eq:H4}\n\\mathscr H_{a,b,c} = q^{\\binom{2c}{2} -b(2m-b-2c+1) -a(b-a)}\n [2m+1] \\qbinom{h;1-m}{c}.\n\\end{align}\nThanks to $q^{2m-a+1} [a] -[2m+1] = q^{-a} [a-2m-1]$, we can combine the RHS\\eqref{eq:H4} with the first summand of LHS\\eqref{eq:H4}.\nHence, after canceling out the $q$-powers $q^{\\binom{2c}{2}- b(2m-b-2c+1) -a(b-a) -a}$\non both sides, we see that \\eqref{eq:H4} is equivalent to the following identity, for all $a, b, c$:\n\\begin{equation}  \\label{ABCD4}\n\\mathscr A+\\mathscr B+\\mathscr C+\\mathscr D_1 +\\mathscr D_2=0,\n\\end{equation}\nwhere \n\\begin{align*}\n\\mathscr A &= [a-2m-1] \\qbinom{h;1-m}{c},\n\\\\\n\\mathscr B &= q^{4c-2m+b-1} [b-a]  \\qbinom{h;-m}{c},   \n\\\\\n\\mathscr C &= q^{2a-2m} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1},   \n\\\\\n\\mathscr D_1 &= q^{2c+b-a} [2m-b-2c+1]  \\qbinom{h;1-m}{c},\n\\\\\n\\mathscr D_2 &= - q^{2c-4m+b-a-1} [2m-b-2c] \\qbinom{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us prove the identity \\eqref{ABCD4}. Using \\eqref{kbinom2}, we can write $\\mathscr B=\\mathscr B_1+\\mathscr B_2$, where \n\\begin{align*}\n\\mathscr B_1 &=  q^{4c-2m+b-1} [b-a]  \\qbinom{h;-m}{c},\n   \\quad\n\\mathscr B_2 = -  q^{4c-6m+b-1} [b-a]  \\qbinom{h;-m}{c}.\n\\end{align*}\nNoting that  \n\\[\nq^{2a-2m}\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} = q^{2m-4c-a}\\frac{( q^{4c-4m}K^{-2}-1)}{q^2-1}\n +  q^{-2c-1} [2m-2c-a],\n \\] \nwe rewrite $\\mathscr C=\\mathscr C_1+\\mathscr C_2$, where\n\\begin{align*}\n\\mathscr C_1 &= q^{2m-2c-a-1} [2c] \\qbinom{h; 1-m}{c},\n   \\quad\n\\mathscr C_2 =   q^{-2c-1} [2m-2c-a]  \\qbinom{h; 1-m}{c-1}.\n\\end{align*}\n\nA direct computation gives us\n\\begin{align*}\n\\mathscr A+\\mathscr B_1+\\mathscr C_1+\\mathscr D_1 \n&=(1-q^{-2}) [2c] [2m-2c-a]  \\qbinom{h; 1-m}{c},\n\\\\\n\\mathscr C_2 +(\\mathscr B_2+\\mathscr D_2)  \n&= \\mathscr C_2 -  q^{2c-4m-1} [2m-2c-a] \\qbinom{h; 1-m}{c-1} K^{-2}\n\\\\\n&= - (1-q^{-2}) [2c] [2m-2c-a]  \\qbinom{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we have $\\mathscr A+\\mathscr B+\\mathscr C+\\mathscr D_1+\\mathscr D_2=0,$ whence \\eqref{ABCD4}, completing Step~(1). \n\n\\vspace{3mm}\n\n(2) Assuming the formulae for $\\dvp{n}$ with $n\\le 2m+1$, we shall now prove the following formula \\eqref{t2m+2:evodd} for $\\dvp{2m+2}$ (obtained with $m$ replaced by $m+1$ in \\eqref{t2m:evodd}):\n\\begin{align}\n\\dvp{2m+2} &= \n\\sum_{b=0}^{2m+2}  \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} +2c- b(2m-b-2c+2) -a(b-a)} \n\\mathfrak p^{(2m-b-2c+2)}(\\kappa) \n  \\label{t2m+2:evodd}\n \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;-m}{c} K^{b-2m+2c-2} F^{(b-a)}. \n\\notag\n\\end{align}\nThe proof is based on the recursion $t \\cdot \\dvp{2m+1} =[2m+2] \\dvp{2m+2} +[2m+1] \\dvp{2m}$. \nRecall $t= F +\\check{E} +\\kappa K^{-1}$ and $\\dvp{2m+1}$ from \\eqref{t2m+1:evodd}. We shall compute \n\\[\n\\texttt{I}=\\check{E}  \\dvp{2m+1},\n\\qquad\n\\texttt{II}=F \\dvp{2m+1},\n\\qquad\n\\texttt{III} =\\kappa K^{-1} \\dvp{2m+1}, \n\\]\nrespectively. First by by shifting indices $a\\to a-1$ and $b\\to b-1$, we have\n\\begin{align*}\n\\texttt{I}\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c+1) -a(b-a)}\n[a+1] \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\\n&=\\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - (b-1)(2m-b-2c+2) -(a-1)(b-a)}\n[a] \\mathfrak p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}.\n\\end{align*}\nUsing \\eqref{FYn} we   also have\n\\begin{align*}\n\\texttt{II}\n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c+1) -a(b-a)}\n \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\quad \\cdot \n \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n \\qbinom{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n=\\texttt{II}_1 +\\texttt{II}_2,\n\\end{align*}\nwhere $\\texttt{II}_1$ and $\\texttt{II}_2$ are the two natural summands associated to the plus sign.\nBy shifting the index $b\\to b-1$, we further have\n\\begin{align*}\n\\texttt{II}_1 \n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - (b+1)(2m-b-2c+2) -a(b-a-1)-2a}\n[b-a] \\mathfrak p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n    \\check{E}^{(a)} \n    \\qbinom{h;-m}{c} K^{b-2m+2c-2} F^{(b-a)}. \n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\n\\texttt{II}_2 \n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2} - (b+1)(2m-b-2c+2) -(a+1)(b-a)}\n \\mathfrak p^{(2m-b-2c+2)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n \\qbinom{h;1-m}{c-1} K^{b-2m+2c-2} F^{(b-a)}.\n \\end{align*}\n\nUsing \\eqref{eq:pn2div} we also compute\n\\begin{align*}\n\\texttt{III} \n&= \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c+1) -a(b-a)-2a}\n\\kappa \\cdot \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}\n\\\\\n&= \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c+1) -a(b-a)-2a}\n[2m-b-2c+2]  \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n\\mathfrak p^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)}  \\qbinom{h;1-m}{c} K^{b-2m+2c-2} F^{(b-a)}\n\\\\%\n& - \\sum_{b=0}^{2m+2} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2} - (b+2)(2m-b-2c+3) -a(b-a)-2a+2}\n[2m-b-2c+1] \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n\\mathfrak p^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)}  \\qbinom{h;1-m}{c-1} K^{b-2m+2c-4} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $\\texttt{I}, \\texttt{II}_1, \\texttt{II}_2$, and $\\texttt{III}$, we obtain \n\\[\nt \\cdot \\dvp{2m+1} = \n \\sum_{0 \\le a \\le b \\le 2m+2}  \\sum_{c\\ge 0} \\mathfrak p^{(2m-b-2c+2)}(\\kappa)  \n  \\check{E}^{(a)} \\mathscr L_{a,b,c} K^{b-2m+2c-2} F^{(b-a)},\n\\]\nwhere  \n\\begin{align*}\n\\mathscr L_{a,b,c} := \n& \\; q^{\\binom{2c}{2} - (b-1)(2m-b-2c+2) -(a-1)(b-a)} \n[a] \\qbinom{h;1-m}{c}\n\\\\\n&+q^{\\binom{2c}{2} - (b+1)(2m-b-2c+2) -a(b-a-1)-2a} \n[b-a]  \\qbinom{h;-m}{c} \n\\\\\n&+q^{\\binom{2c-2}{2} - (b+1)(2m-b-2c+2) -(a+1)(b-a)}\n\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1} \n\\\\\n&+q^{\\binom{2c}{2} - b(2m-b-2c+1) -a(b-a)-2a}\n[2m-b-2c+2]  \\qbinom{h;1-m}{c}\n\\\\\n&- q^{\\binom{2c-2}{2} - (b+2)(2m-b-2c+3) -a(b-a)-2a+2}\n[2m-b-2c+1]  \\qbinom{h;1-m}{c-1} K^{-2}. \n\\end{align*}\n\nOn the other hand, using \\eqref{t2m:evodd} (with an index shift $c\\to c-1$) and \\eqref{t2m+2:evodd} we write \n\\begin{align*}\n [2m+2] & \\dvp{2m+2} +[2m+1] \\dvp{2m} \n\\\\\n&=\\sum_{0 \\le a \\le b \\le 2m+2} \\sum_{c \\geq 0} \n\\mathfrak p^{(2m-b-2c+2)}(\\kappa) \\check{E}^{(a)} \\mathscr R_{a,b,c} K^{b-2m+2c-2} F^{(b-a)}, \n\\end{align*}\nwhere  \n\\begin{align*}\n\\mathscr R_{a,b,c} \n&:= q^{\\binom{2c}{2} +2c- b(2m-b-2c+2) -a(b-a)} \n[2m+2]  \\qbinom{h;-m}{c}  \n\\\\\n&\\qquad + q^{\\binom{2c-2}{2} +2c- b(2m-b-2c+2) -a(b-a)-2} \n[2m+1]  \\qbinom{h;1-m}{c-1}.    \n\\end{align*}\n\nTo prove the formula \\eqref{t2m+2:oddev} for $\\dvp{2m+2}$, it suffices to show that, for all $a,b,c$,\n\\begin{equation}\n  \\label{L=R4}\n\\mathscr L_{a,b,c}= \\mathscr R_{a,b,c}.\n\\end{equation}\nCanceling the $q$-powers $q^{\\binom{2c}{2} - (b-1)(2m-b-2c+2) -(a-1)(b-a)}$ on both sides, we see that the identity \\eqref{L=R4} is equivalent to the following identity, for all $a,b,c$:\n\\begin{align}\n \\label{l=r4}\n \\begin{split}\n[a] \\qbinom{h;1-m}{c}  \n&+q^{4c-4m+b-4}  [b-a]  \\qbinom{h;-m}{c}   \n\\\\\n&+q^{2a-4m-1} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\qbinom{h;1-m}{c-1}   \n\\\\\n&+q^{2c-2m+b-a-2} [2m-b-2c+2]  \\qbinom{h;1-m}{c}\n\\\\\n&- q^{2c-6m+b-a-3} [2m-b-2c+1]  \\qbinom{h;1-m}{c-1} K^{-2}\n\\\\\n&= q^{4c-2m+a-2}  [2m+2]  \\qbinom{h;-m}{c}  \n+ q^{a-2m-1}  [2m+1]  \\qbinom{h;1-m}{c-1}.  \n   \\end{split}\n\\end{align}\n\nBy combining the second summand of LHS with the first summand of RHS as well as combining the third summand of LHS with the second summand of RHS, the identity \\eqref{l=r4} is reduced to the following equivalent identity, for all $a, b, c$:\n\\begin{equation}  \\label{WXYZ4}\n\\mathscr W+\\mathscr X+\\mathscr Y+\\mathscr Z_1+\\mathscr Z_2=0,\n\\end{equation}\nwhere  \n\\begin{align*}\n\\mathscr W & =[a] \\qbinom{h;1-m}{c},\n\\\\\n\\mathscr X & =q^{4c-2m+b-2}  [b-a-2m-2]  \\qbinom{h;-m}{c},   \n\\\\\n\\mathscr Y & =  \\frac{q^{-a-4m-1}K^{-2}-q^{a+1}}{q^2-1} \\qbinom{h;1-m}{c-1},    \n\\\\\n\\mathscr Z_1 & =q^{2c-2m+b-a-2} [2m-b-2c+2]  \\qbinom{h;1-m}{c},\n\\\\\n\\mathscr Z_2 & = - q^{2c-6m+b-a-3} [2m-b-2c+1]  \\qbinom{h;1-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us finally prove the identity \\eqref{WXYZ4}. Using \\eqref{kbinom2}, we can write $\\mathscr X=\\mathscr X_1+\\mathscr X_2$, where \n\\begin{align*}\n\\mathscr X_1 &=q^{4c-2m+b-2}  [b-a-2m-2]  \\qbinom{h; 1-m}{c},\n   \\\\\n\\mathscr X_2 &= - q^{4c-6m+b-2}  [b-a-2m-2]  \\qbinom{h; 1-m}{c-1} K^{-2}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{-a-4m-1}K^{-2}-q^{a+1}}{q^2-1}  \n= q^{-4c-a-1}\\frac{( q^{4c-4m}K^{-2}-1)}{q^2-1}\n +  q^{-2c-1} [-2c-a-1],\n \\] \nwe rewrite $\\mathscr Y=\\mathscr Y_1+\\mathscr Y_2$, where\n\\begin{align*}\n\\mathscr Y_1 &=q^{-2c-a-2} [2c] \\qbinom{h; 1-m}{c},\n   \\qquad\n\\mathscr Y_2 = -q^{-2c-1} [2c+a+1] \\qbinom{h; 1-m}{c-1}.\n\\end{align*}\nA direct computation shows that\n\\begin{align*}\n\\mathscr W+\\mathscr X_1+\\mathscr Y_1+\\mathscr Z_1 &=\n- (1-q^{-2}) [2c+a+1] [2c] \\qbinom{h; 1-m}{c},\n\\\\\n(\\mathscr X_2+\\mathscr Z_2) +\\mathscr Y_2  \n&= q^{2c-4m-1} [2c+a+1] \\qbinom{h; 1-m}{c-1} K^{-2} +\\mathscr Y_2\n\\\\\n&= (1-q^{-2}) [2c+a+1] [2c] \\qbinom{h; 1-m}{c}.\n\\end{align*}\nSumming up these two equations, we obtain $\\mathscr W+\\mathscr X+\\mathscr Y+\\mathscr Z_1+\\mathscr Z_2=0$, whence \\eqref{WXYZ4}, completing Step ~(2).\n\nThe proof of Theorem~\\ref{thm:dvp:oddKappa} is completed. \\qed\n\n\n\n\n\n\n\\section{The $\\imath$-divided powers $\\dvd{n}$ for odd weights and odd $\\kappa$}\n  \\label{sec:oddoddK}\n  \nIn this section~\\ref{sec:oddoddK} we always take $\\kappa$ to be an odd $q$-integer, i.e., \n\\begin{equation*}\n\\kappa =[2\\ell-1], \\qquad  \\text{ for } \\ell  \\in \\mathbb Z.\n\\end{equation*} \n\n\\subsection{Definition of $\\dvd{n}$ for odd $\\kappa$}\n\n\n\\begin{definition}\nSet $\\dvd{1} =t= F +\\check{E} +\\kappa K^{-1}$. \nThe divided powers $\\dvd{n}$, for $n\\ge 1$, are defined by the recursive relations: \n\\begin{align}   \n  \\label{eq:tt:evoddKa}\n\\begin{split}\nt \\cdot \\dvd{2a-1} &=[2a] \\dvd{2a},\n\\\\\nt \\cdot \\dvd{2a} &=  [2a+1] \\dvd{2a+1} +   [2a] \\dvd{2a-1}, \\quad \\text{ for } a\\ge 1.\n\\end{split}\n\\end{align}   \n\\end{definition}\nEquivalently, we have the following closed formula for $\\dvd{n}$:\n\\begin{align}\n\\label{def:dvd:evoddKa}\n\\dvd{n} = \n\\begin{cases}\n\\frac{t}{[2a]!} (t  - [-2a+2])(t -[-2a+4]) \\cdots (t -[2a-4]) (t  - [2a-2]), & \\text{if } n=2a, \\\\\n\\\\\n\\frac{1}{[2a+1]!} (t  - [-2a])(t -[-2a+2]) \\cdots (t -[2a-2]) (t  - [2a]), &\\text{if } n=2a+1.\n\\end{cases}\n\\end{align}\nNote the formulae for $\\dvd{n}$ with odd $\\kappa$ is formally identical to the formulae for $\\dvev{n}$ with even $\\kappa$. \n\n\\subsection{Formulae for $\\dvd{n}$ with odd $\\kappa$}\n\nRecall the polynomials $\\mathfrak p^{(n)}$, for $n\\ge 0$, from \\S\\ref{subsec:pn2}.\n\n\\begin{thm}  \n   \\label{thm:dvd:oddKappa}\nAssume $\\kappa$ is an odd $q$-integer. Then we have, for $m\\ge 1$,   \n\\begin{align}\n\\dvd{2m} &= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)} \n\\mathfrak p^{(2m-b-2c)}(\\kappa)\n\\label{t2moddodd} \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}, \n\\notag\n\\\\\n\\dvd{2m-1} &= \n\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)}\n \\mathfrak p^{(2m-b-2c-1)}(\\kappa)\n\\label{t2m-1oddodd}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c+1} F^{(b-a)}. \n\\notag\n\\end{align}\n\\end{thm}\n\nThe proof of Theorem~\\ref{thm:dvd:oddKappa} will be given in \\S\\ref{subsec:proof:dvd:oddKappa} below. \nBy applying the anti-involution $\\varsigma$ we convert the formulae in Theorem~\\ref{thm:dvd:oddKappa} as follows.\n\n\\begin{thm}   \n \\label{thm:dvd:oddKappa2}\nAssume $\\kappa$ is an odd $q$-integer. Then we have, for $m\\ge 1$,   \n\\begin{align}\n\\dvd{2m} \n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \n(-1)^c  q^{c +b(2m-b-2c) +a(b-a)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) \n\\label{t2moddodd2} \\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot F^{(b-a)} K^{b-2m+2c} \\LR{h;m-c}{c}  \\check{E}^{(a)}, \n\\notag\n\\\\%\n\\dvd{2m-1} &= \n\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \n(-1)^c q^{-c+ b(2m-b-2c-1) +a(b-a)}\n \\mathfrak p^{(2m-b-2c-1)}(\\kappa)\n\\label{t2m-1oddodd2} \\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot F^{(b-a)} K^{b-2m+2c+1}   \\LR{h;m-c}{c}  \\check{E}^{(a)}.\n\\notag\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nRecall from Lemma~\\ref{lem:anti2} that the anti-involution $\\varsigma$ on $\\mbf U$   \nfixes $F,  \\check{E},  K,  \\dvd{n}, \\kappa$ while sending\n$q \\mapsto q^{-1}$, \n$\\LR{h;2-m}{c}\\mapsto (-1)^c q^{2c(c-1)}  \\LR{h;m-c}{c}$.\nThe formulae \\eqref{t2moddodd2}--\\eqref{t2m-1oddodd2} now follow by applying $\\varsigma$ to the formulae \\eqref{t2moddodd}--\\eqref{t2m-1oddodd} in Theorem~\\ref{thm:dvd:oddKappa}. \n\\end{proof}\n\nThe following corollary is immediate from \\eqref{eq:LRhc}, Proposition~\\ref{prop:fg:odd}, and Theorem~\\ref{thm:dvd:oddKappa}. \n\n\\begin{cor}  \nWe have $\\dvd{n} \\in {}_\\mathcal A {\\mbf U}^\\imath_{\\rm{odd}}$, for all $n$. \n\\end{cor}\n\n\n\\begin{rem}   \nNote $\\mathfrak p_n(1)=0$ for $n\\ge 2$, and $\\mathfrak p_0(1)=\\mathfrak p_1(1)=1$. \nThe formulae in Theorems~\\ref{thm:dvd:oddKappa} and  \\ref{thm:dvd:oddKappa2} in the special case for $\\kappa=1$ recover the formulae in \\cite[Theorem~5.1, Proposition~5.3]{BeW18}.\n\\end{rem}\n\n\n\\begin{example}\nThe formulae of $\\dvd{n}$, for $1\\le n\\le 3$, in Theorem~\\ref{thm:dvd:oddKappa} read as follows. \n\\begin{align*}\n\\dvd{1} \n&= F +\\check{E} + \\kappa K^{-1},\n\\\\\n\\dvd{2} \n&= \\check{E}^{(2)} +q^{-1} \\check{E} F + F^{(2)}  + q^{-1} [[h;1]] \n + \\kappa (q^{-1}K^{-1}F + q^{-1} \\check{E} K^{-1})  +  \\frac{\\kappa^2-1}{[2]} K^{-2},\n\\\\\n\\dvd{3}  &   = b^{(3)} + q[[h;0]]F+ q \\check{E} [[h;0]]\n\\\\\n&\\qquad \n+ \\big(q^{-2} \\check{E}^{(2)}K^{-1} +q^{-3} \\check{E} K^{-1}F + q^{-2} K^{-1}F^{(2)} \n+ q [[h;0]] K^{-1} \\big) \\kappa\n\\\\\n&\\qquad + \\frac{(\\kappa^2-1)}{[2]} (q^{-2}\\check{E} K^{-2} +q^{-2}K^{-2}F) \n+\\frac{\\kappa^3 -\\kappa}{[3]!} K^{-3}.\n\\end{align*}\n\\end{example}\n\n\n\\subsection{The $\\imath$-canonical basis for $\\dot{\\mbf U}_{\\text{odd}}$ with odd $\\kappa$}\n  \\label{sec:iCB:evwt-oddK}\n  \n\nRecall from \\S\\ref{sec:iCB:ev} the $\\imath$-canonical basis on simple $\\mbf U$-modules $L(\\mu)$, for $\\mu \\in \\mathbb N$. \n  \n\\begin{thm}  \\label{thm:iCB:evwt-oddK}\n    $\\quad$\n\\begin{enumerate}\n\\item\nLet $n\\in \\mathbb N$. \nFor each integer $\\lambda \\gg n$, the element $\\dvd{n} v^+_{2\\la+1} $ is an $\\imath$-canonical basis element for $L(2\\lambda+1)$. \n \n \\item\nThe set $\\{\\dvd{n} \\mid n \\in \\mathbb N \\}$ forms the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ (and an $\\mathcal A$-basis in ${}_\\mathcal A {\\mbf U}^\\imath_{\\rm{odd}}$). \n\\end{enumerate}\n\\end{thm}\n \n \\begin{proof}\n Let $\\lambda, m \\in \\mathbb N$. Recall $\\cbinom{m}{c}$ from \\eqref{cbinom}. It follows by a direct computation using Theorem~\\ref{thm:dvd:oddKappa2} and \\eqref{eq:LRhc} that \n\\begin{align}\n\\dvd{2m} v^+_{2\\la+1}  &= \n\\sum_{b=0}^{2m} \\sum_{c \\geq 0} (-1)^c  q^{c +b(2m-b-2c)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) \nF^{(b)} K^{b-2m+2c} \\LR{h;m-c}{c}  v^+_{2\\la+1} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m} \\sum_{c \\geq 0} q^{-2c^2+c +(b-2\\lambda)(2m-b-2c)}  \\mathfrak p^{(2m-b-2c)}(\\kappa) \\cbinom{m-\\lambda-c}{c}  F^{(b)}  v^+_{2\\la+1} .\n\\label{t2m:oddoF}  \n\\end{align}\nSimilarly using Theorem~\\ref{thm:dvd:oddKappa2} we have \n\\begin{align}\n\\dvd{2m-1} v^+_{2\\la+1}  &=  \n\\sum_{b=0}^{2m-1} \\sum_{c \\geq 0} \n(-1)^c q^{-c+ b(2m-b-2c-1)} \\mathfrak p^{(2m-b-2c-1)}(\\kappa) F^{(b)} K^{b-2m+2c+1}   \\LR{h;m-c}{c} v^+_{2\\la+1} \n\\notag\n\\\\\n&=  \\sum_{b=0}^{2m-1} \\sum_{c \\geq 0} q^{-2c^2-c +(b-2\\lambda)(2m-b-2c+1)} \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\cbinom{m-\\lambda-c}{c} F^{(b)} v^+_{2\\la+1} . \n\\label{t2m-1:oddoF} \n\\end{align}\n\nBy a similar argument for \\eqref{eq:lattice}, using \\eqref{t2m:oddoF}--\\eqref{t2m-1:oddoF}  we obtain\n \\begin{align*}\n \\dvd{n} v^+_{2\\la+1}   & \\in F^{(n)}  v^+_{2\\la+1}  + \\sum_{b<n}q^{-1} \\mathbb Z[q^{-1}] F^{(b)} v^+_{2\\la+1} ,\n \\qquad \\text{ for } \\lambda \\gg n.\n \\end{align*}\n \nThe second statement follows now from the definition of the $\\imath$-canonical basis for ${\\mbf U}^\\imath$ using\nthe projective system $\\{ L(2\\lambda+1) \\}_{\\lambda\\ge 0}$; cf. \\cite[\\S6]{BW16}. \n \\end{proof}\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:dvd:oddKappa} }\n  \\label{subsec:proof:dvd:oddKappa}\n  \nWe prove the formulae for $\\dvd{n}$ by induction on $n$, in two separate cases (1)--(2) below. The base cases when $n=1,2$ are clear. \n\n\n(1) We shall prove the formula \\eqref{t2moddodd} for $\\dvd{2m}$, assuming the formula \\eqref{t2m-1oddodd} for $\\dvd{2m-1}$.\n\nRecall $[2m] \\dvd{2m} = t \\cdot \\dvd{2m-1}$, and $t= F +\\check{E} +\\kappa K^{-1}$. \nLet us compute \n\\[\nI=\\check{E} \\dvd{2m-1},\n\\qquad\nII=F \\dvd{2m-1},\n\\qquad\nIII=\\kappa K^{-1} \\dvd{2m-1}.\n\\]\n\nWe compute\n\\begin{align*}\nI\n&= \n\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)} [a+1] \\mathfrak p^{(2m-b-2c-1)}(\\kappa)\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\LR{h;2-m}{c} K^{b-2m+2c+1} F^{(b-a)}\n\\\\\n &= \n\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} - (b-1)(2m-b-2c) -(a-1)(b-a)} [a] \\mathfrak p^{(2m-b-2c)}(\\kappa)\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}. \n\\end{align*}\nwhere the last equation is obtained by shifting indices $a\\to a-1$, $b\\to b-1$.\nUsing \\eqref{FYn} we  have\n\\begin{align*}\nII \n&=\\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)}\\mathfrak p^{(2m-b-2c-1)}(\\kappa) \\\\\n  & \\qquad \\cdot \n  \\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n    \\LR{h;2-m}{c} K^{b-2m+2c+1} F^{(b-a)}\n   =II^1 +II^2,\n\\end{align*}\nwhere $II^1$ and $II^2$ are the two natural summands associated to the plus sign.\nBy shifting the index $b\\to b-1$ and then adding some zero terms, we obtain\n\\begin{align*}\nII^1\n\n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - (b+1)(2m-b-2c) -a(b-a+1)} [b-a] \\mathfrak p^{(2m-b-2c)}(\\kappa)  \\\\\n&\\qquad\\qquad\\qquad \\cdot \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c} F^{(b-a)}.\n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and then adding some zero terms, we also obtain\n\\begin{align*}\nII^2 \n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c-2}{2} - (b+1)(2m-b-2c) -(a+1)(b-a)}\\mathfrak p^{(2m-b-2c)}(\\kappa) \\\\\n  & \\qquad\\qquad\\qquad \\cdot \n \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n    \\LR{h;2-m}{c-1} K^{b-2m+2c} F^{(b-a)}.\n\\end{align*}\nBy the identity \\eqref{eq:pn2div} we have\n\\begin{align*}\nIII\n&= \\sum_{b=0}^{2m-1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)-2a}\n \\kappa \\cdot \\mathfrak p^{(2m-b-2c-1)}(\\kappa)\n\\\\ \n&\\qquad\\qquad\\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\\\%%\n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)-2a}\n[2m-b-2c]  \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \\mathfrak p^{(2m-b-2c)} \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\\\\n&\\quad - \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \n q^{\\binom{2c-2}{2} -(b+2)(2m-b-2c+1) -a(b-a)-2a +2}\n[2m-b-2c-1]  \n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot \n\\mathfrak p^{(2m-b-2c)} \\check{E}^{(a)}  \\LR{h;2-m}{c-1} K^{b-2m+2c-2} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $I, II^1, II^2, III$ gives us\n\\[\nt \\cdot \\dvd{2m-1} = \\sum_{0 \\le a \\le b \\le 2m}  \\sum_{c\\ge 0} \n\\mathfrak p^{(2m-b-2c)}(\\kappa) \\check{E}^{(a)} \\mathfrak H_{a,b,c}  K^{b-2m+2c} F^{(b-a)}, \n\\]\nwhere \n\\begin{align*}\n\\mathfrak H_{a,b,c} := \n& \\; q^{\\binom{2c}{2} - (b-1)(2m-b-2c) -(a-1)(b-a)} [a]  \\LR{h;2-m}{c} \n\\\\\n&+ q^{\\binom{2c}{2} - (b+1)(2m-b-2c) -a(b-a+1)} [b-a] \\LR{h;1-m}{c} \n\\\\\n&+ q^{\\binom{2c-2}{2} - (b+1)(2m-b-2c) -(a+1)(b-a)} %\n   \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;2-m}{c-1}  \n\\\\%\n&+ q^{\\binom{2c}{2} - b(2m-b-2c-1) -a(b-a)-2a}[2m-b-2c] \\LR{h;2-m}{c} \n\\\\\n&- q^{\\binom{2c-2}{2} -(b+2)(2m-b-2c+1) -a(b-a)-2a +2}\n[2m-b-2c-1] \\LR{h;2-m}{c-1} K^{-2}, \n\\end{align*}\n\nRecall $[2m] \\dvd{2m} = t \\cdot \\dvd{2m-1}$. To prove the formula \\eqref{t2moddodd} for $\\dvd{2m}$,  by the PBW basis theorem and the inductive assumption it suffices to prove the following identity, for all $a,b,c$:\n\\begin{align}  \\label{eq:Hoo}\n\\mathfrak H_{a,b,c} = q^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)} [2m] \\LR{h;2-m}{c}.\n\\end{align}  \nThanks to $q^{2m-a} [a] -[2m] =q^{-a} [a-2m]$, we can combine the RHS\\eqref{eq:Hoo} with the first summand of the LHS\\eqref{eq:Hoo}.\nHence, after canceling out the $q$-powers $q^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)-a}$ on both sides, we see that \\eqref{eq:Hoo} is equivalent to the following identity, for all $a, b, c$:\n\\begin{equation}  \\label{ABCD2}\n\\mathfrak A+\\mathfrak B+\\mathfrak C+\\mathfrak D_1 +\\mathfrak D_2=0,\n\\end{equation}\nwhere  \n\\begin{align*}  \n\\mathfrak A &=  [a-2m] \\LR{h;2-m}{c},\n  \\\\\n\\mathfrak B &=  q^{b+4c-2m} [b-a] \\LR{h;1-m}{c},\n\\\\\n\\mathfrak C  &=  q^{2a-2m+3} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;2-m}{c-1},\n\\\\\n\\mathfrak D_1 &=  q^{2c+b-a} [2m-b-2c] \\LR{h;2-m}{c},\n   \\\\\n\\mathfrak D_2 &= - q^{2c-4m+b-a+3} [2m-b-2c-1] \\LR{h;2-m}{c-1} K^{-2}.\n\\end{align*}\n\nLet us prove the identity \\eqref{ABCD2}. Using \\eqref{eq:commLR}, \nwe can write $\\mathfrak B=\\mathfrak B_1+\\mathfrak B_2$, where \n\\begin{align*}\n\\mathfrak B_1 &=  q^{b+4c-2m} [b-a] \\LR{h;2-m}{c},\n   \\quad\n\\mathfrak B_2 = - q^{b+4c-6m+4} [b-a] \\LR{h;2-m}{c-1} K^{-2}.\n\\end{align*}\n Noting that  \n\\[\n\\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} = q^{4m-4c-3a-4}\\frac{( q^{4c-4m+4}K^{-2}-q^2)}{q^2-1}\n +  q^{2m-2c-2a-2} [2m-2c-a-1],\n \\] \nwe rewrite $\\mathfrak C=\\mathfrak C_1+\\mathfrak C_2$, where \n\\begin{align*}\n\\mathfrak C_1 &= q^{2m-a-2c-2} [2c] \\LR{h;2-m}{c},\n   \\qquad\n\\mathfrak C_2 =   q^{1-2c} [2m-2c-a-1] \\LR{h;2-m}{c-1}.\n\\end{align*}\n\nA direct computation gives us  \n\\begin{align*}\n\\mathfrak A+\\mathfrak B_1+\\mathfrak C_1+\\mathfrak D_1 \n&=  (1-q^{-2}) [2c] [2m-2c-a-1] p^{(2m-b-2c)}(\\kappa)   \\LR{h;2-m}{c},\n\\\\\n\\mathfrak C_2 +(\\mathfrak B_2+\\mathfrak D_2)  \n&= \\mathfrak C_2 -  q^{2c-4m+3} [2m-2c-a-1]  p^{(2m-b-2c)}(\\kappa)   \\LR{h;2-m}{c-1} K^{-2}\n\\\\\n&=  - (1-q^{-2}) [2c] [2m-2c-a-1] p^{(2m-b-2c)}(\\kappa)   \\LR{h;2-m}{c}.\n\\end{align*}\nSumming up these two equations, we have $\\mathfrak A+\\mathfrak B+\\mathfrak C+\\mathfrak D_1+\\mathfrak D_2=0,$ whence \\eqref{ABCD2}, completing Step~(1). \n\n\\vspace{3mm}\n\n\n\n(2) Assuming the formulae for $\\dvd{n}$ with $n\\le 2m$, we shall now prove the following formula for $\\dvd{2m+1}$ (obtained from \\eqref{t2m-1oddodd} with $m$ replaced by $m+1$):\n\\begin{align}\n\\dvd{2m+1} &= \n\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2} -b(2m-b-2c+1) -a(b-a)} p^{(2m-b-2c+1)}(\\kappa) \n\\label{t2m+1oddodd}\n\\\\\n&\\qquad\\qquad\\qquad\\quad \\cdot  \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\notag\n\\end{align}\nWe first need to compute $t \\cdot \\dvd{2m}$, where we recall $t= F +\\check{E}  +\\kappa K^{-1}$ and $\\dvd{2m}$ from \\eqref{t2moddodd}. We shall compute \n\\[\n\\texttt I=\\check{E}  \\dvd{2m},\n\\qquad\n\\texttt{II}=F \\dvd{2m},\n\\qquad\n\\texttt{III} =\\kappa K^{-1} \\dvd{2m}, \n\\]\nrespectively. First we have\n\\begin{align*}\n\\texttt{I}\n&=\\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)} \n[a+1] \\mathfrak p^{(2m-b-2c)}(\\kappa)\n  \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a+1)}  \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}\n\\\\ %\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c -(b-1)(2m-b-2c+1)-(a-1)(b-a)}  [a] \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n  \\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c-1} F^{(b-a)},\n\\end{align*}\nwhere the last equation is obtained by shifting the indices $a\\to a-1$, $b\\to b-1$. We also have\n\\begin{align*}\n\\texttt{II} \n&= \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)} \\mathfrak p^{(2m-b-2c)}(\\kappa)\n  \\\\\n&\\qquad    \\cdot \n\\left( \n   q^{-2a} \\check{E}^{(a)} F + \\check{E}^{(a-1)} \\frac{q^{3-3a}K^{-2}-q^{1-a}}{q^2-1}\n  \\right )\n \\LR{h;2-m}{c} K^{b-2m+2c} F^{(b-a)}\n=\\texttt{II}_1 +\\texttt{II}_2,\n\\end{align*}\nwhere $\\texttt{II}_1$ and $\\texttt{II}_2$ are the two natural summands associated to the plus sign.\nBy shifting the index $b\\to b-1$, we further have\n\\begin{align*}\n\\texttt{II}_1\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c -(b+1)(2m-b-2c+1) -a(b-a+1)} [b-a] \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n\\\\\n&\\qquad \\qquad \\qquad\\quad \\cdot\n \\check{E}^{(a)}  \\LR{h;1-m}{c} K^{b-2m+2c-1} F^{(b-a)}. \n\\end{align*}\nBy shifting the indices $a\\to a+1$, $b\\to b+1$ and $c\\to c-1$, we further have\n\\begin{align*}\n\\texttt{II}_2\n&=\\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c-2}{2}-2c -(b+1)(2m-b-2c+1)-(a+1)(b-a)+2} \\mathfrak p^{(2m-b-2c+1)}(\\kappa)\n  \\\\\n&\\qquad    \\cdot  \\check{E}^{(a)} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1}\n \\LR{h;2-m}{c-1} K^{b-2m+2c-1} F^{(b-a)}.\n\\end{align*}\nUsing the identity \\eqref{eq:pn2div} we also compute\n\\begin{align*}\n \\texttt{III} =\n\n& \\sum_{b=0}^{2m} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)-2a} \\kappa \\cdot \\mathfrak p^{(2m-b-2c)}(\\kappa)\n  \\\\\n& \\qquad\\quad \\cdot \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n  \\\\%\n= & \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} \nq^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)-2a} [2m-b-2c+1]  \n  \\\\\n& \\qquad\\quad \\cdot \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\LR{h;2-m}{c} K^{b-2m+2c-1} F^{(b-a)}\n\\\\\n& -  \\sum_{b=0}^{2m+1} \\sum_{a=0}^{b} \\sum_{c \\geq 0} q^{\\binom{2c-2}{2}-2c -(b+2)(2m-b-2c+2)-a(b-a)-2a+4} \n [2m-b-2c]  \n  \\\\\n& \\qquad\\quad \\cdot \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)}  \\LR{h;2-m}{c-1} K^{b-2m+2c-3} F^{(b-a)}.\n\\end{align*}\n(Note that we have shifted the index $c\\to c-1$ in the last summand above.)\n\nCollecting the formulae for $\\texttt{I}, \\texttt{II}_1, \\texttt{II}_2, \\texttt{III}$, we obtain an expression of the form\n\\[\nt \\cdot \\dvd{2m} = \n \\sum_{0 \\le a \\le b \\le 2m+1}  \\sum_{c\\ge 0}  \n \\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)} \\mathfrak L_{a,b,c} K^{b-2m+2c-1} F^{(b-a)},\n\\]\nwhere\n\\begin{align*}\n\\mathfrak L_{a,b,c} :=& \\;\nq^{\\binom{2c}{2}-2c -(b-1)(2m-b-2c+1)-(a-1)(b-a)}  [a]  \\LR{h;2-m}{c}\n\\\\\n&+q^{\\binom{2c}{2}-2c -(b+1)(2m-b-2c+1) -a(b-a+1)} [b-a] \\LR{h;1-m}{c}\n\\\\\n&+q^{\\binom{2c-2}{2}-2c -(b+1)(2m-b-2c+1)-(a+1)(b-a)+2}  \n \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;2-m}{c-1}\n\\\\\n&+q^{\\binom{2c}{2}-2c -b(2m-b-2c)-a(b-a)-2a} [2m-b-2c+1] \\LR{h;2-m}{c}\n\\\\\n&-q^{\\binom{2c-2}{2}-2c -(b+2)(2m-b-2c+2)-a(b-a)-2a+4} \n [2m-b-2c] \\LR{h;2-m}{c-1} K^{-2}.\n\\end{align*}\nOn the other hand, using \\eqref{t2m-1oddodd} (with an index shift $c\\to c-1$) and \\eqref{t2m+1oddodd} we write \n\\begin{align*}\n[2m+1] & \\dvd{2m+1} +[2m] \\dvd{2m-1} \n\\\\\n& =\\sum_{0 \\le a \\le b \\le 2m+1} \\sum_{c \\geq 0} \n\\mathfrak p^{(2m-b-2c+1)}(\\kappa) \\check{E}^{(a)} \\mathfrak R_{a,b,c} K^{b-2m+2c-1} F^{(b-a)}, \n\\end{align*}\nwhere  \n\\begin{align*}\n\\mathfrak R_{a,b,c} :=&\nq^{\\binom{2c}{2} -b(2m-b-2c+1) -a(b-a)} [2m+1]  \\LR{h;1-m}{c}  \n\\\\\n&\\quad + q^{\\binom{2c-2}{2} - b(2m-b-2c+1) -a(b-a)} [2m]  \\LR{h;2-m}{c-1}.\n\\end{align*}\n\n\nTo prove the formula \\eqref{t2m+1oddodd} for $\\dvd{2m+1}$, it suffices to show that, for all $a,b,c$, \n\\begin{equation}\n  \\label{L=R2}\n \\mathfrak L_{a,b,c}=\\mathfrak R_{a,b,c}.\n \\end{equation}\nCanceling the $q$-powers $q^{\\binom{2c}{2}-2c -(b-1)(2m-b-2c+1)-(a-1)(b-a)}$  on both sides, we see that the identity \\eqref{L=R2} is equivalent to the following identity, for all $a,b,c$:\n\\begin{align}\n  \\label{l=r2}\n \\begin{split}\n[a]  \\LR{h;2-m}{c} \n&+q^{4c-4m+b-2} [b-a] \\LR{h;1-m}{c}\n\\\\\n&+q^{2a-4m+3} \\frac{q^{-3a}K^{-2}-q^{-a}}{q^2-1} \\LR{h;2-m}{c-1}\n\\\\\n&+q^{2c-2m+b-a-1} [2m-b-2c+1] \\LR{h;2-m}{c}\n\\\\\n&-q^{2c-6m+b-a+2} [2m-b-2c] \\LR{h;2-m}{c-1} K^{-2}\n \\\\\n =& \\; q^{4c-2m+a-1} [2m+1]  \\LR{h;1-m}{c}  \n+ q^{a-2m+2} [2m]   \\LR{h;2-m}{c-1}.\n \\end{split}\n\\end{align}\n\nBy combining the second summand of the LHS\\eqref{l=r2} with the first summand of the RHS\\eqref{l=r2} as well as combining the third summand of the LHS with the second summand of the RHS, the identity \\eqref{l=r2} is reduced to the following equivalent identity, for all $a, b, c$:\n\\begin{equation}  \\label{WXYZ2}\n\\mathfrak W+ \\mathfrak X+\\mathfrak Y+\\mathfrak Z_1+\\mathfrak Z_2=0,\n\\end{equation}\nwhere \n\\begin{align*}\n\\mathfrak W&= [a]  \\LR{h;2-m}{c},\n\\\\\n\\mathfrak X&= q^{4c-2m+b-1} [b-a-2m-1] \\LR{h;1-m}{c},\n\\\\\n\\mathfrak Y&=  \\frac{q^{3-a-4m}K^{-2}-q^{3+a}}{q^2-1} \\LR{h;2-m}{c-1},  \n\\\\\n\\mathfrak Z_1&= q^{2c-2m+b-a-1} [2m-b-2c+1] \\LR{h;2-m}{c},\n\\\\\n\\mathfrak Z_2&= -q^{2c-6m+b-a+2} [2m-b-2c] \\LR{h;2-m}{c-1} K^{-2}.\n\\end{align*}\nLet us finally prove the identity \\eqref{WXYZ2}. Using \\eqref{eq:commLR}, we can write $\\mathfrak X=\\mathfrak X_1+\\mathfrak X_2$, where \n\\begin{align*}\n\\mathfrak X_1 &=q^{4c-2m+b-1} [b-a-2m-1] \\LR{h; 2-m}{c},\n   \\\\\n\\mathfrak X_2 &= - q^{4c-6m+b+3} [b-a-2m-1] \\LR{h; 2-m}{c-1} K^{-2}.\n\\end{align*}\nNoting that  \n\\[\n\\frac{q^{3-a-4m}K^{-2}-q^{3+a}}{q^2-1}\n= q^{-4c-a-1}\\frac{( q^{4c-4m+4}K^{-2}-q^2)}{q^2-1}\n +  q^{1-2c} [-2c-a-1],\n \\] \nwe rewrite $\\mathfrak Y=\\mathfrak Y_1+\\mathfrak Y_2$, where\n\\begin{align*}\n\\mathfrak Y_1 &=q^{-2c-a-2} [2c] \\LR{h; 2-m}{c},\n   \\qquad\n\\mathfrak Y_2 = -q^{1-2c} [2c+a+1] \\LR{h; 2-m}{c-1}.\n\\end{align*}\nA direct computation shows that\n\\begin{align*}\n\\mathfrak W+\\mathfrak X_1+\\mathfrak Y_1+\\mathfrak Z_1 &=\n-(1-q^{-2}) [2c+a+1] [2c]  \\LR{h; 2-m}{c},\n\\\\\n(\\mathfrak X_2+\\mathfrak Z_2) +\\mathfrak Y_2 \n&= q^{-2m-a+1} [2c+a] \\LR{h; 2-m}{c-1} K^{-2} +\\mathfrak Y_2\n\\\\\n&= (1-q^{-2}) [2c+a+1] [2c]  \\LR{h; 2-m}{c}.\n\\end{align*}\nSumming up these two equations, we obtain $\\mathfrak W+\\mathfrak X+\\mathfrak Y+\\mathfrak Z_1+\\mathfrak Z_2=0$, whence \\eqref{WXYZ2}, completing Step ~(2).\n\nThe proof of Theorem~\\ref{thm:dvd:oddKappa} is completed. \\qed\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe upcoming measurements at LHC are renewing considerable interest \nregarding predictions for total \ncross-sections. The model \\cite{corsetti,ggps}\n we shall describe in the following, \nattempts to \nlink the rate with which total cross-sections rise, to   \nthe infrared behaviour of the strong coupling constant \n$\\alpha_s$  and to QCD hard parton-parton scattering, using \n known phenomenological entities such as the available QCD parton density \nfunctions (PDFs). \n\n\\section{The model}\nThe energy behaviour of the total cross-section exhibits the following \nproperties \\cite{PDG}\n\\begin{itemize}\n\\item an initial decrease\n\\item a sharp change in curvature occurring somewhere between  \n20 and 50 GeV in the c.m. of the scattering hadrons\n\\item a smooth rise which asymptotically follows a   \n$\\ln s$ or $\\ln^2 {s}$ type \nincrease in consonance with the Froissart bound \\cite{froissart,blockfroissart}\n\\end{itemize}\nThe model we use is based on\n\\begin{enumerate}\n\\item hard component of scattering responsible for the rise of the total \ncross-section \\cite{allhalzen,minijets}\n\\item soft gluon emission from scattering particles which softens \nthe rise \\cite{corsetti}\n\\item eikonal transformation which implies \nmultiple scattering and requires \nimpact parameter distributions inside scattering particles and basic \nscattering cross-sections \\cite{durand}\n\\end{enumerate}\n\nAccording to our model, soft gluon emission is responsible for  the initial \ndecrease in $p p $, as well as for the transformation of the sharp rise due\nto the increase in gluon-gluon interactions\ninto \na smooth behavior.\nThus soft gluon emission plays a crucial role, with care taken to \nextend resummation to the zero energy modes, in complete analogy for what \nis  required by the Bloch-Nordsieck \ntheorem for QED\\cite{BN}. \nThe model can then be referred to as the BN model, for reasons which will \nalso be clearer in the following.\n\n\\subsection{Details of the BN model} \nWe use the following eikonal expression for the total inelastic \ncross-section:\n\\begin{equation}\n\\label{siginel}\n\\sigma_{inel}=\\int d^2 {\\vec b} [1-e^{-n(b,s)}]\n\\end{equation}\nwhere $n(b,s)$ corresponds to the average number of inelastic \ncollisions at  any given value of the impact parameter b. Neglecting the\nreal part of the eikonal, we then calculate the total cross-section as\n\\begin{equation}\n\\label{sigtot}\n\\sigma_{total}=2 \\int d^2 {\\vec b} [1-e^{-n(b,s)\/2}]\n\\end{equation}\nIn our model \n$n(b,s)$\nis split as\n\\begin{equation}\nn(b,s)=n_{soft}(b,s)+n_{hard}(b,s)\n\\end{equation}\nwhere we postulate the following factorization \n\\begin{equation}\nn_{soft\/hard}(b,s)=A_{BN}^{soft\/hard}(b,s) \\sigma_{soft\/hard}(s)\n\\end{equation}\nwith \n\\begin{equation}\nA_{BN}(b,s)=N \\int d^2 K_{\\perp}\\  e^{-iK_\\perp\\cdot b}\n {{d^2P(K_\\perp)}\\over{d^2 K_\\perp}} \n\\end{equation}\nwhere N is a normalization factor such that $\\int d^2 {\\vec b} A(b) =1$  and\n\\begin{equation}\n\\label{dpdk}\n {{d^2P(K_\\perp)}\\over{d^2K_\\perp}} = {{1}\\over{(2\\pi)^2}}\\int d^2 {\\vec b}\\ \ne^{iK_\\perp\\cdot b -\\int_0^{q_{max}} d^3{\\bar n}(k)[1-e^{-ik_t\\cdot b}]}\n\\end{equation}\nis the transverse momentum distribution \nof initial state soft gluons emitted in the  \nparton-parton collisions\nand where, for simplicity, \n$k_\\perp\\cdot b = {\\vec k}_\\perp\\cdot {\\vec b}$. In Equation (\\ref{dpdk})\n$q_{max}$ is the maximum transverse momentum allowed by kinematics to\nsingle soft gluon emission in a given hard collision, \naveraged over the parton \ndensities. \nAccording to the basic\nansatz of the Eikonal Minijet Model (EMM), \n\\begin{equation}\n\\label{sigmajet}\n\\sigma_{hard}\\equiv \\sigma^{AB}_{\\rm jet} (s) = \n\\int_{p_{tmin}}^{\\sqrt{s\/2}} d p_t \\int_{4\np_t^2\/s}^1 d x_1 \\int_{4 p_t^2\/(x_1 s)}^1 d x_2 \\sum_{i,j,k,l}\nf_{i|A}(x_1) f_{j|B}(x_2)~~\n \\frac { d \\hat{\\sigma}_{ij}^{ kl}(\\hat{s})} {d p_t}.\n\\end{equation}\n Here   $A$ and $B$ denote particles ($\\gamma, \\ p,\n\\dots$), $i, \\ j, \\ k, \\ l$ are parton types and $x_1,x_2$ the fractions\nof the parent particle momentum carried by the parton. $\\hat{s} =\nx_1 x_2 s$  and $\\hat{ \\sigma}$ are hard parton scattering\ncross--sections. As discussed in \\cite{corsetti}, kinematical considerations\nsuggest \\cite{greco}\n\\begin{equation}\n\\label{qmaxav}\nq_{max}(s)={{\\sqrt{s}} \n\\over{2}}{{ \\sum_{i,j}\\int {{dx_1}\\over{ x_1}}\nf_{i|A}(x_1)\\int {{dx_2}\\over{x_2}}f_{j|B}(x_2)\\sqrt{x_1x_2} \\int_{z_{min}}^1\n dz (1 - z)}\n\\over{\\sum_{i,j}\\int {dx_1\\over x_1}\nf_{i|A}(x_1)\\int {{dx_2}\\over{x_2}}f_{j|B}(x_2) \\int_{z_{min}}^1 (dz)}}\n\\end{equation}\nwith $z_{min}=4p_{tmin}^2\/(sx_1x_2)$ and $f_{i\/a}$ the valence \nquark densities used in the jet cross-section calculation.   \nThe steps we  follow to compare the model with data  are then the following:\n\\begin{enumerate}\n\\item choose the parameters for the hard scattering part, namely\n\\begin{description}\n\\item{(i)} parton densities (PDF), \n$p_{tmin}$ and $\\Lambda_{QCD}$ for the\nchosen PDF set in Equation (\\ref{sigmajet}) \n\\item{(ii)} model for $\\alpha_s$ in the infrared region and relevant \nparameters in Equation (\\ref{dpdk})\n\\end{description}\n\\item calculate $q_{max}(s,p_{tmin})$ for the given densities and\n  $p_{tmin}$\nusing Equation (\\ref{qmaxav})\n\\item calculate $n_{hard}(b,s)= A_{BN}^{hard}(b,s)\n  \\sigma_{jet}(s,p_{tmin})$\n \\item choose the parameters for the low energy part, namely\n\\begin{description}\n\\item{(i)}  the constant low energy cross-section $\\sigma_0$\n\\item {(ii)}  values for  $q_{max}^{soft}$\n\\end{description}\n\\item  calculate $n_{soft}(b,s)=\n A_{BN}^{soft}(b,s) \\sigma_0(1+\\epsilon {{2}\\over{\\sqrt{s}}})$ with\n $\\epsilon=0, 1$ depending upon the process being $p p$ or $p \n\\bar{p}$\n\\item calculate $n(b,s)$ and thus $\\sigma_{tot}$ \n\\item choose the parameter set which gives the best description of the\n  total cross-section up to the Tevatron data \\cite{e710,cdf,e811}\n\\end{enumerate}\nNotice that once a good set of parameters is found, one can \nuse $n(b,s)$ with fitted parameters to calculate survival \nprobabilities or diffractive Higgs production.\n\\subsection{Application to  total cross-section data}\nWe show in this section the application of the model to  the\ntotal \ncross-section for different PDFs. We find that our model is \nflexible enough to be able to reproduce the present data for \n$\\sigma_{tot}$ using all presently available PDFs. \nIn particular, all GRV \\cite{GRV,GRV94,GRV98} and MRST \\cite{MRST}\n densities give a good description  using \nthe singular $\\alpha_s$ model described in \\cite{corsetti}, \nwhile CTEQ densities \n\\cite{CTEQ} give an acceptable \nfit  up to Tevatron data, but  then fail to rise further. \n\nWe present these results by following the previously listed steps. We start\nby choosing the parameters for the hard scattering, and, following our\nprevious results \\cite{ggps}, we fix $p_{tmin}=1.15 \\ GeV$ in the jet\ncross-sections and calculate   $q_{max}$  for different PDF sets.\n We show the result in Fig.(1).\n\\begin{figure}\n\\label{qmax}\n\\includegraphics[width=\\textwidth,height=\\textwidth\n]{qmax_115_band.eps}\n\\caption{Average value of the maximum transverse momentum allowed to single\n  gluon emission, according to the model in {\\protect \\cite{ggps}}.}\n\\end{figure}\nWe notice the following:\\begin{itemize}\n\\item GRV densities are of two types, GRV98\\cite{GRV98}  \nfor which $q_{max}$ keeps on increasing \nlogarithmically and the older ones \n\\cite{GRV} for  which \n$q_{max}$ slows down past the TeV region, albeit still slowly increasing\n\\item CTEQ densities give values for $q_{max}$ which  increase more\n  rapidly than $\\ln {s}$ after typical Tevatron energies \n\\item MRST densities indicate a behaviour opposite to CTEQ, since they give\n$q_{max}$ values decreasing after the TeV cross mark.\n\\end{itemize} \nWe now turn to the jet cross-sections and examine\nthe  growth with energy of \n $\\sigma_{jet}$   for different PDFs. \n In Fig.(2)\nwe plot\nthese cross-sections for the same set of densities used to calculate\n$q_{max}$\nand for  $p_{tmin}=1.15 \\ GeV$.\n\\begin{figure}\n\\label{sigjets}\n\\includegraphics[width=\\textwidth,height=\\textwidth\n]{sigjet_115_band.eps}\n\\caption{Mini-jet QCD cross-sections for different PDFs as indicated in the\n  figure}\n\\end{figure}\nFrom this figure we notice that :\n\\begin{itemize}\n\\item the jet cross-sections  for GRV densities\n  increase faster than all the others\n\\item the jet cross-sections for CTEQ increase more or less similarly to\n  those for the MRST group\n\\end{itemize}\nThe implications are that $\\sigma_{jet}$ with GRV densities, \nwhich increase faster than $\\sigma_{jet}$ with MRST,\nneed more softening, $\\sigma_{jet}$ with CTEQ, which increase \nless than with GRV, should not be\nsmeared that much.\n\nTo proceed further, we now need to calculate the $b-distribution$ and \nfix the low-energy parameters, like $\\sigma_0$. The $b-distribution$ requires\nto input the behaviour of $\\alpha_s$ in the infrared region. We have shown\nin \\cite{ggps} the need to use a singular but integrable expression for \n$\\alpha_s$ in order to reproduce both the sudden rise and the subsequent\nsoftening of the total cross-section. Our choice is an expression like\n\\begin{equation}\n\\label{alphaRich}\n\\alpha_s(k_t)={{12 \\pi }\\over{(33-2N_f)}}{{p}\\over{\\ln[1+p({{k_t}\n\\over{\\Lambda}})^{2p}]}}\n\\end{equation}\nwhich depends on the singularity parameter $p$, in addition to the scale\n$\\Lambda$. In \\cite{ggps} we have chosen the value $p=0.75$ and \n$\\Lambda=100\\ MeV$, other choices\nare also possible \\cite{pramana}. Turning now to the low-energy part, $n_{soft}(b,s)$, we\nchoose $\\sigma_0=48\\ mb$ and use for $A_{BN}^{soft}$ a set of \n$q_{max}$ values which\nreproduce the low energy behaviour, and which appear in Fig. (1).  \n\\section{Comparison with data and expectations at LHC densities}\nWe can now input all the above in the eikonal representation for the total\ncross-section and obtain\nthe results shown in Fig.(\\ref{sigtotsing})\n\n\\begin{figure}[t]\n\\label{sigtotsing}\n\\includegraphics[width=\\textwidth,height=\\textwidth]{eiksig_115_band_pp.eps}\n\\caption{Data for total cross-sections for $pp$ and $p{\\bar p}$ \ncompared with model predictions at LHC for $p p$ scattering,\n using different PDFs.}\n\\end{figure}\n for the singular $\\alpha_s$ case\n and GRV, MRST and CTEQ \ndensities. For the sake of clarity, we only plot curves for \n$p p$ scattering, referring the reader to \n\\cite{ggps,pramana} for the curves for $p{\\bar p}$ and for a different\n parameter set,  or for predictions from \nother models \\cite{blockfroissart,dl,bghp,cudell,luna}. \nFrom the figure we see that the calculation with CTEQ densities \nappears very unlikely. The effect is due to the fact that the jet\ncross-sections in the CTEQ case do not rise as much as the others \nwhile the softening\neffect is stronger, as it is driven by $q_{max}$, which is strongly \nincreasing for these densities. As a\n result, the cross-section starts decreasing. \nNotice that while the behaviour of $\\sigma_{jet}$ is dominated by the \ngluon densities, that of $q_{max}$ is determined only by the valence \nquarks, as we assume this to be the leading order effect. \n\nThe curves shown in Fig.(\\ref{sigtotsing}) indicate that the coming\nmeasurement at $\\sqrt{s}=900\\ GeV$ will be very important in determining\nwhich of these curves  best describes the data. It   can then be used to \nselect  the parameter set, basically PDF's and $p_{tmin}$, \nfor a prediction at the  project LHC energy, $\\sqrt{s}=14\\ TeV$.   \nIf the UA5 \\cite{ua5} value  at $\\sqrt{s}=900\\ GeV$ \nis confirmed with a comparable error, \nthen, for the set of parameters discussed in this note,  at\n$\\sqrt{s}=14\\ TeV$ our model gives\n$\\sigma_{total}^{GRV98}=90.2\\  mb$, $\\sigma_{total}^{GRV}=100.2\\  mb$\nand $\\sigma_{total}^{MRST76}=103.4\\  mb$. As shown in \\cite{pramana}, \nchanging the parameter set, namely $\\sigma_0$, $p_{tmin}$ or the singularity \nindex $p$, give values in the range $88\\div 111\\ mb$.\n\n\n\n\n\n\n\n\n \n\n\n \n\n\n \n\n\n\n\n\n\\section{Conclusions}\nWe have presented a version of the eikonal minijet model which allows a\ngood description of total cross-sections at asymptotic energies and\ndiscussed its connection with the small x-behaviour of various types of\nparton densities.\nThis model is based on a softening of the\nmini-jet cross-sections due to  an s-dependent \nb-distribution in\nthe proton, which we calculate using a soft gluon resummation model down to\nzero energy of the soft gluons.\n\\section*{Acknowledgments}\nWe acknowledge  partial support from EU CT-2002-0311. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe breakthrough discovery of Witten-Kontsevich (\\cite{Witten,Kon}) established an intimate link between mathematical physics and enumerative geometry. From a general perspective, one aims at studying the Gromov-Witten invariants of a smooth projective variety $X$  in terms of suitable integrable hierarchies. From this point of view, the Witten-Kontsevich case corresponds to the situation when $X$ is a point and it was shown that the exponential of the generating function of intersection numbers on the moduli space of curves was a common solution of the Virasoro constraints and of the KdV hierarchy. Therefore, following the generalization for the case of the projective space proposed in \\cite{EHX}, on the one hand one wonders if the generating function fulfills a generalization of the Virasoro constraints. On the other hand, one also wants to know if the generating function is given by (the logarithm of) a particular tau-function of an integrable hierarchy. Nowadays, the case of varieties with semisimple quantum cohomology is well understood and the answer to both questions is affirmative (\\cite{Teleman} and \\cite{DubrovinZ2}; see also \\cite{DubrovinZ, Getzler,Givental, LiuTian, OK, OK-Toda}). \n\n\nIt is worth pointing out that, for each $X$, the explicit Virasoro operators as well as the relevant integrable hierarchy may vary; for instance, the $2$-Toda hierarchy appears when dealing with the equivariant GW invariants of ${\\mathbb P}^1$ (\\cite{OK-Toda}). Nevertheless, one recognizes some common features that arise among these results. Let us mention some of them. In \\cite{GiventalAn}, Givental studied a case in which the total descendent potential is a $\\tau$-function for the $n$KdV-hierarchy by using $n-1$ copies of the KdV. Thus,  the total descendent potential of a semisimple Frobenius manifold was defined in \\cite{Givental} as a product of $n$ copies of Witten-Kontsevich $\\tau$-functions. Dealing with a case of orbifold quantum cohomology, it has been proved in \\cite{JarvisKimura} that the Virasoro constraints decomposed as $n$ copies of (half of) the Virasoro algebra, that their solution was the product of Witten-Kontsevich $\\tau$-functions, and that the relevant integrable hierarchy consisted of $n$ commuting copies of the KdV hierarchy. Finally, in \\cite{DVV, KacSchwarz} it was shown that  the solution of the Virasoro constraints in the case of Witten-Kontsevich is unique  (up to a constant factor) and this uniqueness also holds in other setups (e.g. \\cite{LiuUnique}).\n\nThis paper, making use of the representation theoretic properties of the Virasoro algebra, \noffers new insights and results on these properties and provides evidences that the above mentioned properties rely heavily on the structure of the Virasoro algebra and its representations. Our study of explicit expressions for Virasoro representations (see  \\S2) is general enough to encode many of the known representations within the framework of Virasoro constraints. Further, it allows us to determine whether a representation is the tensor product of Lie algebra representations and if a solution factorizes as a product of solutions of those representations. An explicit realization of these ideas is carried out in \\S3 for the case of  smooth projective varieties with trivial odd cohomology and vanishing first Chern class. Thus, we think that our approach may help in determining the explicit expression of the Virasoro operators as well as the corresponding integrable hierarchies for other types of varieties $X$ (see \\S\\ref{subsec:final}).  Now, let us be more precise and explain the contents of the paper.\n\n\nWe begin by fixing a pair $(A,(\\, ,\\,))$ consisting of a finite dimensional vector space and a non-degenerate bilinear form. Associated to this data, we consider a Heisenberg algebra ${\\mathbb H}(A)$ and its universal enveloping algebra $\\uh[(A)]$. Let us denote by ${\\mathcal W}_>$ the positive half of the Virasoro algebra and recall that it contains $\\sl(2)$ canonically. Section 2 is entirely devoted to the study of Lie algebra maps ${\\mathcal W}_>\\to \\uh[(A)]$. To begin with, we show that, under some homogeneity condition, there is a canonical bijection between $\\Hom_{\\text{Lie-alg}}( {\\mathcal W}_>, \\uh[(A)])$ and $\\Hom_{\\text{Lie-alg}}( \\sl(2), \\uh[(A)]$ (Theorem~\\ref{thm:W^+-sl2}). This is highly non-trivial since, in general, the problem of extending a map defined on $\\sl(2)$ to ${\\mathcal W}_>$ involves  infinitely many conditions (see~\\cite{PT}). Accordingly, it is natural to expect that many properties of a map ${\\mathcal W}_> \\to \\uh[(A)]$ can be stated in terms of its restriction to $\\sl(2)$. Actually, we prove that such a map decomposes as tensor product of Lie algebra representations if and only if its restriction does. Moreover, this factorization is possible only if $A$ decomposes as the orthogonal sum of  two subspaces (Theorem~\\ref{thm:rhodecomp}). We finish this section by showing that the fact that ${\\mathcal W}_>$ admits no non-trivial finite dimensional representations has important consequences for the structure of the solutions of the equations $(\\rho_1\\otimes 1 + 1\\otimes \\rho_2)(L)(\\sum_i f_i\\otimes g_i)=0$, where $L\\in {\\mathcal W}_>$ (see Theorem~\\ref{thm:solutionrhoi}). That is, decompositions of the representation and of the solutions depend strongly on the structure of ${\\mathcal W}_>$ and of $(A,(\\, ,\\,))$. \n\nAlthough the previous results are interesting on their own, \\S3 explores their application to concrete situations; for instance, relations with integrable hierarchies (e.g. multicomponent KdV). The case we have chosen to illustrate this issue is that of the differential operators appearing in the Virasoro conjecture when $X$ has trivial odd cohomology (for instance, whenever $X$ has semisimple even quantum cohomology) and its first Chern class vanishes. Then, Theorem~\\ref{thm:Lbar=Lhat} shows explicitly how to obtain these operators as the images of the generators $L_k\\in{\\mathcal W}_>$ by:\n\t{\\small $$\n\t\\hat\\rho\\,:\\, {\\mathcal W}_>\\, \\overset{\\rho}\\longrightarrow\\,  \\uh[(A)] \\, \\overset{\\widehat{\\,}}\\longrightarrow\\, \\End\\big({\\mathbb C}[[\\{t_{i,\\alpha}\\vert 1\\leq \\alpha\\leq \\dim(A) , i=1,3,\\ldots\\}]]\\big)\n\t$$}for $A=H^*(X,{\\mathbb C})$ endowed with the Poincar\\'e pairing. Then, our results of \\S2 imply that $\\hat\\rho$ decomposes as the tensor product of Lie algebra representations associated to data $({\\mathbb C},(\\, , \\,))$; i.e. the $1$-dimensional case. The detailed study of the $1$-dimensional case carried out in \\S\\ref{subsec:solutions1dim} shows that, up to re-scaling the variables, the corresponding operators \\emph{always }come from a representation:\n\t$$\n\t\\sigma: {\\mathcal W}_> \\,\\longrightarrow \\, \\operatorname{Diff}^1({\\mathbb C}((z)))\n\t$$\nwhich means that we can profit from \\cite{KacSchwarz, Plaza-AlgSol} to build the unique solution in terms of a $\\tau$-function of the KdV hierarchy. Putting everything together, we have the main results of this section. First, in the case of $\\dim A=1$:\n\n\\begin{thm*}[see Theorem~\\ref{thm:solutionWscaleKdV}]\nLet $\\rho \\in \\homlie({\\mathcal W}_>,\\End({\\mathbb C}[[t_1,t_3,\\ldots]]))$ be such that $\\rho(L_k)$  is of type $k$ for $k\\geq -1$ and that all coefficients of $\\rho(L_{-1})$ are non zero. \n\nThen, there exists a unique $\\tau(t)\\in{\\mathbb C}[[t_1,t_3,\\ldots]]$, with $\\tau(0)=1$, such that:\n\t$$\n\t\\rho(L_k)(\\tau(t))\\,=\\,0\\qquad k\\geq -1\n\t$$\nFurther, the solution $\\tau(t)$ is a $\\tau$-function of the scaled KdV hierarchy.\n\\end{thm*}\\noindent and, for $\\dim A=N \\geq 2$:\n\n\n\\begin{thm*}[see Theorem~\\ref{thm:solutionproductKdV}]\nLet $\\rho:{\\mathcal W}_>\\to \\uh[(A)]$ be as in \\S\\ref{subsec:baby}.\n\nThere exist $S\\in \\Gl(A)$ and functions $\\tau_\\alpha(t_{1,\\alpha},t_{3,\\alpha},\\ldots)\\in {\\mathbb C}[[t_{1,\\alpha},t_{3,\\alpha},\\ldots]]$ such that:\n\t$$\n\t\\hat\\rho(L_k)\\big(S( \\prod_\\alpha \\tau_\\alpha( t_\\alpha))\\big)\\,=\\, 0\n\t$$\n\nFurther, $\\tau_\\alpha(t_{1,\\alpha},t_{3,\\alpha},\\ldots)$ are $\\tau$-functions of the scaled KdV hierarchy.\\end{thm*}\n\nWe hope that our methods shed some light on the explicit expressions of the Virasoro operators and of the relevant integrable hierarchies that appear when studying the Virasoro conjecture.  We also think that the techniques presented here can be applied to many instances of representations of ${\\mathcal W}_>$ which appear in a variety or problems such as recursion relations, Hurwitz numbers, knot theory, etc.~. We sketch some ideas in \\S\\ref{subsec:final} although all of them deserve further research.  \n\n\\emph{Acknowledgements.} We thank A. Givental and G. Borot for explaining us some facts on their papers. The first author  wish to express his gratitude to the Max Planck Institute f\\\"ur Mathematik (Bonn) for the invitation in fall 2014. \n\n\n\n\n\n\n\\section{Lie algebras}\\label{sec:ReprW>}\n\nLet ${\\mathcal W}$ be the Witt algebra; that is, the ${\\mathbb C}$-vector space\nwith basis $\\{L_k\\}_{k\\in{\\mathbb Z}}$ endowed with the Lie bracket\n$[L_i,L_j]=(i-j) L_{i+j}$, and let ${\\mathcal W}_>$ be the subalgebra\ngenerated by $\\{L_k\\}_{k\\geq -1}$. It contains a copy of $\\sl(2)$ via $\\sl(2)=<\\{L_{-1}, L_0, L_1\\}> \\subset {\\mathcal W}_>$. Recall that ${\\mathcal W}_>$ is also called the \\emph{positive half of the centereless Virasoro algebra}. \n\nIn this section, we study certain maps from  $\\sl(2)$ and their extensions to  ${\\mathcal W}_>$. These results will eventually allow us to relate the representation theories of ${\\mathcal W}_>$ and $\\sl(2)$. A further consequence is that, in order to construct the operators $L_0, L_1, L_2, \\ldots$ one only has to start with $L_{-1}$ and follow some simple procedures and choices. \n\nIt is worth mentioning that a study of the representation theory of ${\\mathcal W}_>$ in terms of the representation theory of its subalgebra $\\sl(2) \\subset {\\mathcal W}_>$ has been carried out in \\cite{PT} in full generality. \n\t\n\\subsection{Preliminaries}\\label{subsec:prelim-beta-Diff-to-C[[t]]}\n\n\nLet us be more precise. Let $(A, (\\, ,\\,))$ be given, where $A$ is a finite dimensional ${\\mathbb C}$-vector space and $(\\,,\\,)$ is a non-degenerated bilinear pairing. For a basis $\\{a_{\\alpha}\\vert \\alpha=1,\\ldots, n\\}$ of $A$, let $\\eta=(\\eta_{\\alpha\\beta})$ denote the matrix associated to the given bilinear product; that is, $\\eta_{\\alpha\\beta}:=(a_{\\alpha}, a_{\\beta})$. The inverse will be denoted with superindexes; i.e. $\\eta^{\\alpha\\beta}:=(\\eta^{-1})_{\\alpha\\beta}$. \n\nLet us consider unknowns $\\{p_i,q_i\\vert i\\geq 1\\}$ and introduce $p_{i,\\alpha}:=p_i\\otimes a_{\\alpha}$ and $q_{i,\\alpha}:=q_i\\otimes a_{\\alpha}$. Let ${\\mathbb H}(A)$ be the Heisenberg algebra generated by $\\{1, p_{i,\\alpha},q_{i,\\alpha}\\vert i\\geq 1, \\alpha=1,\\ldots, n\\}$, whose elements will be called \\emph{operators},  endowed with the Lie bracket: \n    \\begin{equation}\\label{eq:pqcommutation}\n    \\begin{gathered}\n    \\, [p_{i,\\alpha}, q_{j,\\beta}] \\,=\\, \\delta_{i,j} i \\eta^{\\alpha\\beta}\\cdot 1\n    \\\\\n   \\,  [p_{i,\\alpha}, p_{j,\\beta}] \\,=\\,[q_{i,\\alpha}, q_{j,\\beta}] \\,=\\, 0\n    \\\\\n   \\,  [p_{i,\\alpha}, 1] \\,=\\,[q_{i,\\alpha}, 1] \\,=\\, 0\n    \\end{gathered}\n    \\end{equation}\nWe define their degree by $\\deg(q_{i,\\alpha})=i$, $\\deg(p_{i,\\alpha})=-i$ and $\\deg(1)=0$. \n\nAlthough the definition of the Heisenberg algebra depends on the pair $(A, (\\, ,\\,))$, it will be simply denoted by ${\\mathbb H}$ if no confusion arises.\n\nFor ${\\mathbb H}$ as above, let us define $\\uh$ the universal enveloping algebra of ${\\mathbb H}$, which is the quotient of the tensor algebra of  ${\\mathbb H}$ by the two-sided ideal generated by the relations $u\\otimes v- v\\otimes u-[u,v]$. \n\nMotivated by the explicit forms of the Virasoro operators considered in the literature (\\cite{DVV, DubrovinZ,EO, Getzler, Givental,KSU, KazarianZograf}), we introduce the following notion. The ultimate meaning of this notion is unveiled in Lemma~\\ref{lem:Diff1scaleW}.\n \n\n\\begin{defn}\\label{defn:type}\nAn operator $T\\in \\uh$ is \\emph{of type $i\\geq -1$} if it is a linear combination of $p_{2i+3,\\alpha}$ and double products of degree $-2i$; i.e. $p_{j,\\alpha}p_{2i-j,\\beta}$, $q_{j,\\alpha}p_{2i+j,\\beta}$ and $q_{j,\\alpha}q_{-j-2i,\\beta}$. If $i=0$ we also allow a constant term.\n\nThe subset consisting of operators of type $i\\geq -1$ will be denoted by $\\uh[(A)]_i$ (or, simply, $\\uh_i$). \n\\end{defn}\n\nThis section deals with the study of homomorphisms of Lie algebras:\n\t$$\n\t\\rho:{\\mathcal W}_> \\, \\longrightarrow\\, \\uh \\qquad \\text{s.t. } \\rho(L_i)\\in \\uh_i\n\t$$\n\t\nLet us illustrate the previous definition. From now on, according to Einstein convention, summation over repeated indices will be understood. For instance, an operator of type $-1$ is of the form:\n    \\begin{equation}\\label{eq:type-1}\n    b_{-1}^{0,1} p_1 +  q_1 a_{-1}^{1,1}  q_1^T  + q_{i+2} b_{-1}^{i+2,i}  p_i \\, \\in\\, \\uh_{-1} \n    \\end{equation}\n(the sum  runs over the set of odd positive integers $i$),  $p_i$ is the column vector $(p_{i,1}\\ldots,p_{i,n})^T$, $q_i$ is the row vector $(q_{i,1}\\ldots,q_{i,n})$,  $b_{-1}^{0,1}$ is a row vector, $a_{-1}^{1,1}$  and $b_{-1}^{i+2,i}$ are $n\\times n$ matrices. For brevity, we set $a:=a_{-1}^{1,1} $. \n\nSimilarly, an type $0$ operator can be expressed as:\n    \\begin{equation}\\label{eq:type0}\n   b_{0}^{0,3} p_3 + b_0^{0,0}  + q_{i} b_{0}^{i,i}  p_i \\, \\in\\, \\uh_{0}\n    \\end{equation}\nwhile an operator of type $i\\geq 1$ is of the form:\n\t\\begin{equation}\\label{eq:type}\n\tb_i^{0,2i+3} p_{2i+3} + p_{j}^T c_i^{j,2i-j} p_{2i-j} +  q_j b_i^{j,2i+j} p_{2i+j}\n\t\\,\\in\\,\\uh_i \\qquad i\\geq 1\n\t\\end{equation}\nfor a row vector $b_i^{0,2i+3}$ and $n\\times n$-matrices $b_i^{j,2i+j}$ and $c_i^{j,2i-j}$, where  $c_i^{j,2i-j}=(c_i^{2i-j,j})^T$ and the sum runs over $j$ odd.\n\nIt is convenient to offer an interpretation of these matrices. Recall that $q_i$ is the row vector $(q_{i,1},\\ldots, q_{i,n})$, which can be thought as an ${\\mathbb H}$-valued vector of $A$. A similar argument holds for the column vector $p_i$. Thus, under a basis change in $A$, the matrix $b$ in $q_i\\cdot b\\cdot p_i$ behaves as a bilinear form on $A$. The same fact applies to all $a$, $b$ and $c$ matrices.  Similarly, column vectors $b_i^{0,2i+3}$ are understood as vectors on $A$ while row vectors are like linear forms. \n\nIt is worth noticing how these operators behave w.r.t. the Lie bracket. Indeed, the computations given in subsection~\\S\\ref{subsect:CR} and the linearity of the bracket show that it is compatible with the degree:\n\t\\begin{equation}\\label{eq:LieT}\n\t[\\,\\, , \\,\\, ] \\,:\\, \\uh_i \\times \\uh_j \\, \\longrightarrow\\, \\uh_{i+j}\n\t\\end{equation}\nwhere $i,j,i+j\\geq -1$. In particular, it follows that $\\oplus_{i\\geq -1} \\uh_i$ is a partial Lie algebra. \n\t\n\n\n\n\\subsection{Maps from $\\sl(2)$ to Heisenberg}\\label{subsect:sl2}\n\n\n\n\nLet $\\sl(2)$ be the Lie algebra of $\\operatorname{Sl}(2,{\\mathbb C})$. We fix a  basis $\\{e,f,h\\}$  of $\\sl(2)$  satisfying the relations:\n    $$\n    [e,f]=h\\; , \\qquad\n    [h,e]=2e \\; , \\qquad\n    [h,f]=-2f\\; .\n    $$\nIn particular, the previous choice yields a natural embedding:\n    \\begin{equation}\\label{eq:emb-sl2-W+}\n    \\iota:\\sl(2)\\,\\hookrightarrow\\, {\\mathcal W}_>\n    \\end{equation}\nby mapping $f$ to $L_{-1}$, $h$ to $-2L_0$, and $e$ to $-L_1$. \n\n\n\\begin{lem}\\label{lem:FyieldsH}\nLet $F\\in  \\uh_{-1}$ be as in \\eqref{eq:type-1}. \nAssume that\n$b_{-1}^{i+2,i}$ is invertible for all $i$. It holds that:\n    {\\small $$\n    \\left\\{\n    \\begin{gathered}\n    H\\in\\uh_0 \\text{ s.t.}\n    \\\\\n    [H,F]\\,=\\, - 2F\n    \\end{gathered}\n    \\right\\}\n    \\,\\simeq\\,\n    \\left\\{\n    \\begin{gathered}\n    (b,B)\\in {\\mathbb C}\\times \\operatorname{Mat}_{n\\times n}({\\mathbb C})\n    \\text{ s.t.}\n    \\\\ (B \\eta^{-1} + \\operatorname{Id}) (a +a^T) +  (a +a^T)(B \\eta^{-1} +\\operatorname{Id})^T = 0\n    \\end{gathered}\n    \\right\\}\n    $$}\n\\end{lem}\n\n\\begin{proof}\nOur task consists of computing the  bracket $[H,F]$ explicitly. Recall that, for simplicity, we have set $a=a_{-1}^{1,1}$. Since $H\\in\\uh_0$, it must be of the form $H:=  b_0^{0,3} p_3 +  b_{0}^{0,0}   +  q_{i}  b_{0}^{i,i} p_i $ where $b_0^{0,3}$ is a row vector, $b_{0}^{0,0}$ is an homothety, and $b_{0}^{i,i}$ are $n\\times n$ matrices.\n\nHaving in mind the commutation relations of \\S\\ref{subsect:CR}, the bracket $[H,F]$ is a linear combination of $p_{1}$, $q_{1\\alpha}q_{1\\beta}$ and $q_{i+2,\\alpha}p_{i,\\beta}$. Therefore, the expression $[H,F]=-2F$ is equivalent to the following identities:\n\t$$\n\t    \\begin{aligned}\n    \\big( 3 b_0^{0,3} \\eta^{-1}  b_{-1}^{3,1} - b_{-1}^{0,1}\\eta^{-1}  b_0^{1,1} \\big) p_1 \\, & =\\, -2 b_{-1}^{0,1} p_1\n    \\\\\n     q_1 b_0^{1,1}\\eta^{-1}  \\big( a  + a^T   \\big) q_1^T   \\, & =\\,  -2 q_1 a   q_1^T\n    \\\\\n    q_{i+2}\\big( (i+2) b_0^{i+2,i+2}\\eta^{-1}  b_{-1}^{i+2,i}  - i  b_{-1}^{i+2,i}\\eta^{-1}  b_0^{i,i}\\big) p_i\n    \\, & =\\,  - 2 q_{i+2} b_{-1}^{i+2,i} p_i\n    \\qquad\\forall i \\geq 1\n    \\end{aligned}\n\t$$\nObserve that $q_1 A q_1^T = q_1 B q_1^T$ if and only if $A+A^T=B+B^T$.  Hence, \nthe above system is equivalent to the following equations:\n        \\begin{subequations}\\label{eq:FyieldsH}\n    \\begin{align}\n    \\label{eq:FyieldsH:1}\n    3 b_0^{0,3} \\eta^{-1} b_{-1}^{3,1} - b_{-1}^{0,1}\\eta^{-1} b_0^{1,1} \\, & =\\, -2 b_{-1}^{0,1}\n    \\\\\n     \\label{eq:FyieldsH:2}\n    b_0^{1,1} \\eta^{-1} (a +a^T)+  (a +a^T)( b_0^{1,1}\\eta^{-1} )^T  \n     \\, & =\\,  - 2 (a +a^T)\n         \\\\\n    \\label{eq:FyieldsH:3}\n    (i+2) b_0^{i+2,i+2}\\eta^{-1} b_{-1}^{i+2,i}  - i b_{-1}^{i+2,i} \\eta^{-1}  b_0^{i,i}\n    \\, & =\\,  - 2 b_{-1}^{i+2,i}\n    \\qquad\\forall i\\geq 1 \n    \\end{align}\n        \\end{subequations}\n\nNote that, since $b_{-1}^{i+2,i}$ and $\\eta$ are invertible,  given a pair $(b,B)$ as in the statement, this system has a unique solution for $b_0^{0,0}=b$ and $b_0^{1,1}=B$; namely,\n\t\\begin{equation}\\label{eq:coefH}\n\t\\begin{aligned}\n\tb_0^{0,3}\\, &=\\, \\frac13 b_{-1}^{0,1}(\\eta^{-1} b_0^{1,1}-2) (\\eta^{-1} b_{-1}^{3,1})^{-1} \n\t\\\\\n\tb_0^{i+2,i+2} \\,&=\\, \\frac1{i+2}  b_{-1}^{i+2,i} (i \\eta^{-1} b_0^{i,i}-2 )(\\eta^{-1} b_{-1}^{i+2,i})^{-1} \\qquad\\forall i\\geq 1\n\t\\end{aligned}\n\t\\end{equation}\nThe converse is straightforward.\n\\end{proof}\n\n\n\\begin{exam} \nSet  $F= b_{-1}^{0,1}  p_1 +   q_1  a  q_1^T  +  \\frac{i+2}2 q_{i+2} p_i$ and $b_0^{1,1}=-\\frac12$, then  $H=-2 b_{-1}^{0,1} p_3 +   b_{0}^{0,0}   + i q_i p_i$.  Note that $i q_{i} p_i$ is the \\emph{degree operator}.\n\\end{exam}\n\n\\begin{exam} \nLet us consider the case where the chosen basis in $A$ is orthonormal; i.e. $\\eta$ is the identity matrix, and suppose that:\n    $$\n    F\\,=\\, b_{-1}^{0,1}  p_1 +   q_1  a  q_1^T  +  q_{i+2} p_i\\,\\in\\,\\uh_{-1}\n    $$\nThen, operators $H$ given by Lemma~(\\ref{lem:FyieldsH}) acquire the form:\n    $$\n    H \\,=\\,    \\frac13 b_{-1}^{0,1}( b_0^{1,1}-2)   p_3 \n    +   b_{0}^{0,0}   +  \\frac1{i} q_i  ( b_0^{1,1}-(i-1)) p_i,\\in\\,\\uh_0\n    $$\nwhere $ b_{0}^{0,0} \\in {\\mathbb C}$ and $b_0^{1,1}$ verifies $(b_0^{1,1} +\\operatorname{Id}) a  +  a (b_0^{1,1} +\\operatorname{Id})^T =0$. \n\\end{exam}\n\n\\begin{exam}\\label{exam:dim1}\nFinally, let $\\dim A=1$ , $a, \\eta\\in{\\mathbb C}^{\\ast}$ and $F = b_{-1}^{0,1}  p_1 +   q_1  a  q_1^T  +  q_{i+2} \\eta p_i$. Then, $ b_0^{i,i}=-\\eta$ for all $i$ and $H=-  b_{-1}^{0,1}  p_3     +   b_{0}^{0,0}   -   q_i  \\eta  p_i$. \n\\end{exam}\n\n\n\n\n\n\\begin{lem}\\label{lem:T'T'bracket}\nLet $H$ be as in equation~\\eqref{eq:type0} and $\\uh'_i$ be the subspace:\n\t$$\n\t\\uh'_i\\,:=\\, \\{ T\\in\\uh_i  \\text{ s.t. } [H,T]= 2i T \\}\n\t$$\nThen, it holds that $[ \\uh'_i ,  \\uh'_j ]\\subseteq \\uh'_{i+j} $.\n\\end{lem}\n\n\\begin{proof}\nThe claim follows easily from~\\eqref{eq:LieT} and the Jacobi identity.\n\\end{proof}\n\n\n\\begin{thm}\\label{thm:sl2-symmmatrices}\nLet $F$ and $H$ be as in equations \\eqref{eq:type-1} and \\eqref{eq:type0} respectively. \n\nThere is a surjective map:\n    {\\small\n    $$\n    \\left\\{\n        \\begin{gathered}\n            c \\in M_{n\\times n}({\\mathbb C})\\text{ such that }\n            \\\\\n             b_0^{0,0} =   \\operatorname{Tr}( c \\eta^{-1}  (a+a^T)  (\\eta^{-1})^T\n            \\\\\n            \\text{and equation~\\eqref{eq:HF-2F:2} below}\n        \\end{gathered}\n    \\right\\} \n    \\, \\longrightarrow\\,\n    \\left\\{\n        \\begin{gathered}\n        \\sigma \\in \\homlie(\\sl(2),\\uh) \\\\\n         \\text{such that } \\sigma(f)=F, \\\\\n         \\sigma(h)=H  \\text{ and }\\sigma(e)\\in\\uh_1\n        \\end{gathered}\n    \\right\\}\n    $$}\nMoreover, $c_1$ and $c_2$ have the same image iff $c_1+c_1^T= c_2+c_2^T$. Thus, the restriction of the above map to symmetric matrices yields a bijection. \n\\end{thm}\n\n\\begin{proof}\nGiving a map $\\sigma$ as in the r.h.s. is equivalent to set an operator $E\\in\\uh_1$, such that $[E,F] = H $ and $ [H,F]= -2F$. Consider:\n     \\begin{equation}\\label{eq:E=type1}\n     E\\,=\\,  b_1^{0,5} p_5 + p_1^T c_1^{1,1}p_1  + q_i  b_1^{i,i+2}  p_{i+2}\\,\\in\\, \\uh_1\n     \\end{equation}\nwhere, for simplicity, we will set $c= c_1^{1,1}$. The identity $[H,E]= 2E$, expressed in terms of the coefficients of the operators, is equivalent to the following equations (thanks to the computations of \\S\\ref{subsect:CR}):\n    \\begin{subequations}\n    \\label{eq:HF-2F}\n    \\begin{align}\n     3 b_{0}^{0,3}\\eta^{-1} b_1^{3,5} - 5 b_{1}^{0,5}\\eta^{-1} b_0^{5,5}\n     \\, & =\\,  2 b_1^{0,5}\\label{eq:HF-2F:1}\n    \\\\\n    - (c^T + c) \\eta^{-1}b_0^{1,1} - (\\eta^{-1}b_0^{1,1})^T  (c^T + c)  \n    \\, & =\\, 2 (c^T + c) \\label{eq:HF-2F:2}\n  \n    \\\\\n    r b_0^{r,r}  \\eta^{-1}  b_1^{r,r+2} - (r+2) b_1^{r,r+2} \\eta^{-1}  b_0^{r+2,r+2}\n    \\,&=\\, 2 b_1^{r,r+2} \\label{eq:HF-2F:3}\n    \\end{align}\n    \\end{subequations}\n\n\nAnalogously, expanding the relation $[E,F]=H$ with the help of \\S\\ref{subsect:CR} yields the system:\n    \\begin{subequations}\n    \\label{eq:EFH}\n    \\begin{align}\n    \\label{eq:EFH:1}\n    \\operatorname{Tr}( c \\eta^{-1} (a+a^T) (\\eta^{-1})^T ) \\, & = \\,  b_0^{0,0}\n    \\\\\n    \\label{eq:EFH:2}\n    - b_{-1}^{0,1}\\eta^{-1}  b_1^{1,3} + 5  b_1^{0,5} \\eta^{-1} b_{-1}^{5,3} \\, & = b_0^{0,3}\n    \\\\\n    \\label{eq:EFH:3}\n    3 b_1^{1,3}\\eta^{-1} b_{-1}^{3,1}   +  (a+a^T) (  \\eta^{-1})^T  (c+c^T )  \\, & = \\,  b_0^{1,1}\n    \\\\\n    \\label{eq:EFH:4}\n    (r+2) b_1^{r,r+2} \\eta^{-1} b_{-1}^{r+2,r} - (r-2) b_{-1}^{r,r-2} \\eta^{-1}   b_1^{r-2,r} \\, & = \\, b_0^{r,r} \\quad\\forall r>2\n    \\end{align}\n    \\end{subequations}\nHaving in mind the properties of the trace, one observe that these equations only depends on $c+c^T$.\n\n\nIt remains to show that equations \\eqref{eq:HF-2F} and \\eqref{eq:EFH} are equivalent to the conditions of the claim; that is, that they can be reduced to \\eqref{eq:HF-2F:2} and \\eqref{eq:EFH:1}. \n\nAssuming \\eqref{eq:HF-2F:2} and \\eqref{eq:EFH:1}, one gets $b_1^{1,3}$ from \\eqref{eq:EFH:3}; then, $b_1^{0,5}$ is determined by \\eqref{eq:EFH:2}; and, $b_1^{r,r+2}$ is obtained from \\eqref{eq:EFH:4}. We claim that \\eqref{eq:HF-2F:1} is fulfilled too. Indeed, a long but straightforward computation shows that \\eqref{eq:HF-2F:1} is derived from \\eqref{eq:coefH}, \\eqref{eq:HF-2F:2} together with  the case $r=3$ of \\eqref{eq:EFH:4}. Similarly,  \\eqref{eq:HF-2F:3} follows from  \\eqref{eq:coefH}, \\eqref{eq:FyieldsH:3}  and \\eqref{eq:EFH:4}.\n\\end{proof}\n\n\\subsection{Extending to ${\\mathcal W}_>$}\\label{subsect:W>}\n\nIn order to extend a map defined on $\\sl(2)$ to one on ${\\mathcal W}_>$, one should choose an endomorphism $T$, define $\\rho(L_i)$ by equations~\\eqref{eq:rho(L_2)def} and \\eqref{eq:rho(L_i)def} and check infinitely many constraints (see~\\cite{PT}). However, in our situation the following Lemma simplifies that approach drastically; there will exist a unique $T$ satisfying all the requirements. \n\n\\begin{lem}\\label{lem:adFinjective}\nLet $F,H$ be as in equations~\\eqref{eq:type-1}   and~\\eqref{eq:type0}. The map:\n\t$$\\ad(F):\\uh'_i \\overset{\\sim}\\longrightarrow \\uh'_{i-1}\n\t$$\nis an isomorphism for $i\\geq 2$.\n\\end{lem}\n\n\\begin{proof}\nFirst, one has to prove that given an operator:\n\t$$\n\tS\\,:=\\, b_{i-1}^{0,2i+1} p_{2i+1} \\, +\\, p_j^T c_{i-1}^{j,2i-j-2} p_{2i-j-2} \\, +\\, q_j b_{i-1}^{j,j+2i-2} p_{j+2i-2}\\,\\in\\,\\uh_{i-1}\n\t$$\nof type $i-1\\geq 1$, there is exactly one operator:\n\t$$\n\tT\\,:=\\, b_i^{0,2i+3} p_{2i+3} \\, +\\, p_j^T c_i^{j,2i-j} p_{2i-j} \\, +\\, q_j b_i^{j,j+2i} p_{j+2i}\\,\\in\\,\\uh_i\n\t$$\nof type $i$ satisfying $\\ad(F)(T)=S$ where  $\\ad$ denotes the adjoint representation and $F$ is given by equation~\\eqref{eq:type-1}.\n\nNow, one proceeds as in the proof of Lemma~\\ref{lem:FyieldsH} and shows that  $\\ad(F)(T)=[F,T]=S$ has exactly one solution. \n\n\nFinally, let us check that if $S\\in\\uh'_{i-1}$  and $\\ad(F)(T)=S$, then $T\\in\\uh'_{i}$. Using the injectivity of $\\ad(F)$ and the relation:\n    $$\n    \\begin{aligned}\n    \\ad(F)(\\ad(H)(T))\n    \\,& =\\,\\ad(H)(\\ad(F)(T)) + \\ad([F,H])(T)\n    \\,= \\\\ & =\\, \\ad(H)(S) + \\ad(2F)(T)\n    \\,=\\, 2(i-1)S + 2S\\,=\\, 2 i S\n    \\end{aligned}\n    $$\none obtains $\\ad(H)(T)=2iT$, as we wanted.\n\\end{proof}\n\n\n\\begin{thm}\\label{thm:W^+-sl2}\nLet $F$ be as in \\eqref{eq:type-1} where  $a $ is symmetric and $b_{-1}^{i,i-2}$ are invertible.\n\nThen, the map $\\iota^*$ of \\eqref{eq:emb-sl2-W+} yields a bijection:\n    {\\small\n    $$\n    \\left\\{\n        \\begin{gathered}\n        \\rho \\in \\homlie({\\mathcal W}_>,\\uh)\n        \\\\\n         \\text{such that }\n        \\rho(L_{-1})=F\n        \\\\\n        \\text{ and }\n       \n        \\rho(L_i) \\in\\uh_i \\text{ for }i\\geq 0\n        \\end{gathered}\n    \\right\\}\n    \\, \\overset{\\sim}\\longrightarrow\\,\n    \\left\\{\n        \\begin{gathered}\n        \\sigma \\in \\homlie(\\sl(2),\\uh) \\\\\n         \\text{such that }\n        \\sigma(f)=F\\, ,\n        \\\\\n        \\sigma(h) \\in\\uh_0 \\text{ and }\\sigma(e)\\in\\uh_1\n        \\end{gathered}\n    \\right\\}\n    $$}\n\\end{thm}\n\n\\begin{proof}\nGiven $\\rho$, we define $\\sigma:=\\iota^*(\\rho)$ and, therefore, $\\sigma(f)=\\rho(\\iota(f))=\\rho(L_{-1})$, $\\sigma(h)=\\rho(-2L_0)$ and $\\sigma(e)=\\rho(-L_1)$.\n\nFor the converse, one requires several steps and the previous Lemmas.\n\n\\emph{Step 1.} Let  $\\sigma$ be given. There exists a ${\\mathbb C}$-linear homomorphism $\\rho:{\\mathcal W}_>\\to \\uh$ such that $\\sigma=\\iota^*(\\rho)$. First, we set:\n    $$\n    \\rho(L_{-1}):= \\sigma(f)=F\n    \\; , \\quad\n    \\rho(L_0):= -\\frac12 \\sigma(h)\n    \\; , \\quad\n    \\rho(L_1):= -\\sigma(e)\n\t$$\nThe fact that $\\sigma$ is a map of Lie algebras and Lemma~\\ref{lem:FyieldsH} imply that:\n\t$$\n\t\\rho(L_0)\\,=\\, -\\frac12H\n\t$$\nwhere $H$ is as in equation~\\eqref{eq:type0}. Furthermore, it holds that $\\rho(L_{i})\\in\\uh'_i$ for $i=-1,1$. Having in mind Lemma~\\ref{lem:adFinjective} we obtain that there is a unique $T\\in\\uh'_2$ such that:\t\n\t$$\n\t\\operatorname{ad}(\\rho(L_{-1}))(T) \\,=\\, \\rho(L_1)\n\t$$\n\nThen, we define:\n\t\\begin{equation}\\label{eq:rho(L_2)def}\n\t\\rho(L_2)\\,:= \\, -3 T \\,\\in \\, \\uh'_2\n\t\\end{equation}\nand, recursively,\n    \\begin{equation}\\label{eq:rho(L_i)def}\n    \\rho(L_i):= \\frac1{i-2}[\\sigma(e),\\rho(L_{i-1})]\n    \\qquad \\text{for }i>2\n    \\;.\n    \\end{equation}\n\n\n\\emph{Step 2.} It holds that $ [\\rho(L_{0}),\\rho(L_j)] = -j\\rho(L_{j})$ for $j\\geq -1$. This is equivalent to show that\n$\\rho(L_j) \\in  \\uh'_j$ for all $j\\geq 1$.  Bearing in mind that $\\rho(L_1)\\in\\uh'_1$ and Lemma~\\ref{lem:T'T'bracket}, the conclusion follows.\n\n\\emph{Step 3.} It holds that $ [\\rho(L_{-1}),\\rho(L_j)] = -(1+j)\\rho(L_{j-1})$ for $j\\geq -1$. The cases $j\\leq 1$ follow from the fact that $\\sigma$ is a homomorphism of Lie algebras.  The choice of $T$ implies the case $j=2$. Let us proceed by induction on $j$.  For $j\\geq 3$, the definition of $\\rho(L_i)$, the Jacobi identity and the induction hypothesis yield:\n    {\\small $$\n    \\begin{aligned}\\;\n    [\\rho(L_{-1}), & \\rho(L_j)]\n    \\,  =\\,\n    [\\rho(L_{-1}),\\; -\\frac1{j-2}[\\rho(L_{1}),\\rho(L_{j-1})]]\n    \\, = \\\\ & = \\,\n    \\frac1{j-2}\\Big( [\\rho(L_{1}),\\; [\\rho(L_{j-1}), \\rho(L_{-1})]] \\;+\\;\n    [\\rho(L_{j-1}),\\; [\\rho(L_{-1}), \\rho(L_{1})]]\\Big)\n    \\, = \\\\ & = \\,\n    \\frac1{j-2}\\Big( [\\rho(L_{1}),\\; j \\rho(L_{j-2})] \\;+\\;\n     [\\rho(L_{j-1}),\\; (-2) \\rho(L_{0})]\\Big)\n     \\, = \\\\ & = \\,\n    \\frac1{j-2}\\big( j(3-j) \\rho(L_{j-1}) -2 (j-1)\\rho(L_{j-1})\\big)\n    \\,=\\, (-1-j)\\rho(L_{j-1})\n    \\end{aligned}\n    $$}\n\n\n\\emph{Step 4.} The identity:\n    \\begin{equation}\\label{eq:braket-hom}\n    [\\rho(L_i),\\rho(L_j)]\\,-\\,(i-j)\\rho(L_{i+j}) \\,=\\,0\n    \\end{equation}\nholds for $i,j\\geq 1$. We proceed by induction on $n=i+j$. The case $n=4$ (i.e. $i,j\\geq 1$ and $i+j=4$) holds by the very definition of $\\rho(L_4)$. Now, let us assume that it holds true up to  $n-1=i+j-1$ and let us prove the case $n=i+j>4$. Observe that, by Step 2, the l.h.s of the equation \\eqref{eq:braket-hom} lies in $\\uh_{i+j}'$. By Lemma~\\ref{lem:adFinjective}, it suffices to show that its image under $\\ad(F)=\\ad(\\rho(L_{-1}))$ vanishes. In fact, the Jacobi identity, the Step 3 and the induction hypothesis show that:\n\t{\\small $$\n\t\\begin{aligned}\n\t\\ad & (\\rho( L_{-1}))\\big(   [\\rho(L_i),\\rho(L_j)]\\,-\\,(i-j)\\rho(L_{i+j})\\big) \\, = \\\\ & = \\,\n\t [[\\rho(L_{-1}), \\rho(L_i)],\\rho(L_j)] \\,+\\,  [\\rho(L_i), [\\rho(L_{-1}),\\rho(L_j)]]\n\t \\,-\\, (i-j)[\\rho(L_{-1}),\\rho(L_{i+j})]\n\t \\, = \\\\ & = \\,\n\t [-(1+i)\\rho(L_{i-1}),\\rho(L_j)] \\,+\\, [\\rho(L_i), -(1+j)\\rho(L_{j-1})] \\,+\\, (i-j)(1+i+j) \\rho(L_{i+j-1})\n\t \\, = \\\\ & = \\,\n\t - \\big( (1+i)(i-j-1) \\,+\\, (1+j)(i-j+1)  \\,-\\, (i-j)(1+i+j) \\big) \\rho(L_{i+j-1})\n\t \\, = \\\\ & = \\,\n\t0\n\t \\end{aligned}\n\t $$}\n\n\\emph{Step 5.} $\\rho$ is a Lie algebra homomorphism. This follows from the properties of $\\sigma$ and Steps 2, 3, 4. \t\n\t\n\\end{proof}\n\n\n\\subsection{Factorization as a product}\n\nIt is remarkable that if the vector space $(A, (\\, ,\\,))$ decomposes as $A_1\\perp A_2$ (i.e. $A=A_1\\oplus A_2$ and $(a_1,a_2)=0$ for all $a_i\\in A_i$), then the very definition of the associated Heisenberg algebra implies that:\n\t$$\n\t{\\mathbb H}(A) \\,\\simeq\\, {\\mathbb H}(A_1)  \\widehat\\otimes_{{\\mathbb C}} {\\mathbb H}(A_2) \n\t$$\nas Lie algebras and ${\\mathbb H}(A_i)$ is a subalgebra of ${\\mathbb H}(A)$. So, we may wonder under which circumstances a morphism\n$\\rho :{\\mathcal W}_> \\to \\uh[(A)]$ would decompose accordingly. The following Theorem provides an answer in terms of the restriction $\\rho\\vert_{\\sl(2)}$. For this goal, recall that matrices $a$, $b$ and $c$'s behave as bilinear forms on $A$ (w.r.t. the action of $\\Gl(A)$).\n\n\\begin{thm}\\label{thm:rhodecomp}\nLet $F, H, E$ be as in~\\eqref{eq:type-1}, \\eqref{eq:type0} and \\eqref{eq:E=type1}. Let $\\rho :{\\mathcal W}_> \\to \\uh[(A)]$ satisfy $\\rho(L_{-1})=F$, $\\rho(L_0)=-\\frac12 H$ and $\\rho(L_1)=-E$. \n\nIf the vector space $A$ decomposes as $A_1\\perp A_2$ w.r.t. $\\eta$ and this decomposition is compatible with the action of $F$ and with the bilinear forms  $b_0^{1,1}$ and  $c_1^{1,1}$, then there are Lie algebra maps $\\rho_{i} :{\\mathcal W}_> \\to \\uh[(A_i)]$ for $i=1,2$ such that:\n\t$$\n\t\\rho \\,=\\, \\rho_1\\otimes 1 + 1\\otimes \\rho_2\n\t$$\n\nIf this is the case, and $\\rho(L_k)\\in \\uh[(A)]_k'$ for all $k\\geq -1$, then $\\rho_{i}(L_k)\\in \\uh[(A_{i})]_k'$ for all $k\\geq -1$ and $i=1,2$. \n\\end{thm}\n\n\n\\begin{proof}\n\\emph{Step 1}. The case of $\\rho(L_{-1})$. The hypothesis says that we can find $\\{a_{\\alpha}\\vert \\alpha=1,\\ldots,n\\}$, a basis of $A$, and an index $m$ such that, for $1\\leq i <m \\leq j\\leq n$, the vectors $a_i$ and $a_j$ are orthogonal w.r.t. to the bilinear form defined by $a $.  Equivalently, w.r.t. the splitting $A_1\\oplus A_2$ the matrix of this bilinear form acquires a block decomposition as follows:\n\t$$\n\ta  \\,=\\, \n\t\\begin{pmatrix} \\ast  & 0 \\\\ 0 & \\ast \\end{pmatrix}\n\t$$\nIt is now straightforward that the terms of the operator $F$ (as given in equation~\\eqref{eq:type-1}) can be grouped in two sets, the first one involving $p_{i,\\alpha}$ and $q_{i,\\alpha}$ for $i\\in{\\mathbb N}$ and $1\\leq \\alpha <m$, and the second one depending only on $p_{i,\\alpha}$ and $q_{i,\\alpha}$ for $i\\in{\\mathbb N}$ and $m\\leq \\alpha \\leq n$. Denote these operators as $\\bar L_{-1,1}$ and $\\bar L_{-1,2}$ respectively. One checks that:\n\t\\begin{equation}\\label{eq:barL_-1}\n\t\\begin{gathered}\n\t\\rho( L_{-1}) = \\bar L_{-1,1}\\otimes 1 + 1\\otimes  \\bar L_{-1,2}\n\t\\\\\n\t\\bar L_{-1,\\alpha}\\,\\in \\, \\uh[(A_{\\alpha})]_{-1}\\qquad\\alpha=1,2.\n\t\\end{gathered}\n\t\\end{equation}\n\t\n\\emph{Step 2}. The case of $\\rho(L_{0})$. Bearing in mind that it is defined as $-\\frac12H$ and that the coefficients of the latter fulfill the relations \\eqref{eq:FyieldsH}, one can proceed as in the previous case. More precisely, considering the following block decompositions:\n\t$$\n\t\t\\eta  \\,=\\, \n\t\\begin{pmatrix} \\eta_1  & 0 \\\\ 0 & \\eta_2 \\end{pmatrix}\n\t\\qquad\n\t\ta  \\,=\\, \n\t\\begin{pmatrix} a_1  & 0 \\\\ 0 & a_2 \\end{pmatrix}\n\t\\qquad\n\t\tc_{1}^{1,1} \\,=\\, \n\t\\begin{pmatrix} c_1  & 0 \\\\ 0 & c_2 \\end{pmatrix}\n\t$$\none may use the following identity as a defining relation for $\\bar L_{0,\\alpha} \\in \\uh[(A_{\\alpha})]_0$:\n\t$$\n\t\\begin{aligned} \n\t\\rho(L_0)-b_0^{0,0} \n\t\\, & =\\, \\big( \\bar L_{0,1}- 2\\Tr( c_{1}\\eta_1^{-1}( a_{1}+a_1^T)(\\eta_1^T)^{-1} )\\big) \n\t\\, + \\\\ & \\qquad +\\, \n\t \\big( \\bar L_{0,2}- 2\\Tr( c_{2}\\eta_2^{-1} (a_{2}+a_2^T)(\\eta_2^T)^{-1})\\big)  \n\t\\end{aligned}\n\t$$\n\n\\emph{Step 3}. The case of $\\rho(L_{k})$ for $k\\geq 1$.  Recall from the proof of Theorem~\\ref{thm:sl2-symmmatrices} that the coefficients $b_1^{r,r+2}$ of $\\rho(L_{1})$ can be expressed in terms of $a $, $b_0^{1,1}$ and $c_1^{1,1}$ and that a close look of these expressions shows that $b_1^{r,r+2}$ are compatible w.r.t. to the splitting of $A$. Thus, we can express $\\rho(L_1)$ as the sum of two factors; namely, $\\bar L_{1,\\alpha}$ for $\\alpha=1,2$ which consists of the terms of $\\rho(L_{1})$ in   $p_{i,\\alpha}, q_{i,\\alpha}$ for  $1\\leq \\alpha <m$ and  for $m\\leq \\alpha \\leq n$, respectively. Now, we proceed as above. \n\nFor the case of $\\rho(L_k)$ for $k\\geq 2$ one proceeds recursively (using the expressions of the proof of Lemma~\\ref{lem:adFinjective}). \n\n\n\n\\emph{Step 4}. \n$[ \\bar L_{k,\\alpha} ,  \\bar L_{l,\\beta}]=0$ for $k,l \\geq -1$ and  $\\alpha\\neq\\beta$, since these two operators involve disjoint sets of variables. \n\n\\emph{Step 5}. The maps $\\rho_{\\alpha}$. Consider:\n\t$$\n\t\\rho_{\\alpha}(L_k)\\,:=\\,\n\t\\bar L_{k,\\alpha} \n\t\\qquad\\text{for $k\\geq -1$ and $\\alpha=1,2$}\n\t$$\nThe previous steps show that $\\rho = \\rho_1\\otimes 1 + 1\\otimes \\rho_2$.\n\nIt remains to check that $\\rho_{\\alpha}$ are morphisms of Lie algebras. For this goal we will expand both sides of the identity $[\\rho(L_k),\\rho(L_l)]=(k-l)\\rho(L_{k+l})$ using the above facts. The l.h.s. is:\n\t{$$\n\t\\, [\\rho(L_k),\\rho(L_l)]\\,=\\, \n\t[\\bar L_{k,1}+\\bar L_{k,2} , \\bar L_{l,1}+\\bar L_{l,2}]\n\t\\,=\\, \n\t[\\bar L_{k,1} , \\bar L_{k,1}]  + [\\bar L_{k,2} , \\bar L_{l,2}]\n\t$$}\nwhile the r.h.s. reads:\n\t{$$\n\t(k-l)\\rho(L_{k+l}) \\,=\\, \n\t(k-l)( \\bar L_{k+l,1}+\\bar L_{k+l,2})\n\t$$}\nComparing both expressions and having in mind the separation of variables, it follows that:\n\t{$$\n\t[\\bar L_{k,\\alpha} , \\bar L_{k,\\alpha }]  \\,=\\, \n\t(k-l) \\bar L_{k+l,\\alpha}\n\t$$}\n\\noindent and we conclude that $\\rho_{\\alpha}$ is a map of Lie algebras ${\\mathcal W}_> \\to \\uh[(A_{\\alpha})]$.\n\n\\emph{Step 6}. Type of the operators. In order to show that $\\rho(L_k)\\in \\uh[(A)]_k'$ implies that  $\\rho_{\\alpha}(L_k)\\in \\uh[(A_{\\alpha})]_k'$, it suffices to expand the Lie bracket $[ \\rho(L_0), \\rho(L_k)]$ using  $ \\rho(L_k)= \\rho_1(L_k) \\otimes 1 + 1\\otimes \\rho_2(L_k)$. \n\\end{proof}\n\n\\begin{rem}\nIt is worth noticing that if a decomposition is compatible with $a $, it does not need to be compatible with $b_0^{1,1}$. Indeed, for $A={\\mathbb C}^2$, $\\eta=a  = \\tiny{\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} }$, the general form of $b_0^{1,1}$ is given by\n$\\tiny{\\begin{pmatrix} -1  & \\lambda \\\\ \\lambda  & -1 \\end{pmatrix} }$.\n\\end{rem}\n\n\nFor later use, the following general result will be required.\n\n\\begin{thm}\\label{thm:solutionrhoi}\nLet $\\rho_i:{\\mathcal W}_> \\to \\End V_i$, $i=1,2$, be two representations of the Lie algebra ${\\mathcal W}_>$. And let us consider the product representation:\n\t$$\n\t\\rho=\\rho_1\\otimes 1 +1\\otimes \\rho_2\\, : \\, {\\mathcal W}_> \\longrightarrow \\End (V_1\\otimes V_2)$$\n\nLet $\\sum_{i=1}^r f_{1,i}\\otimes f_{2,i} \\in V_1 \\otimes V_2$.  Assume that $f_{i,1},\\ldots,f_{i,r}$ are linearly independent (for $i=1,2$).\n\nIt then holds that:\n\t$$\n\t\\rho(L_k)(\\sum_{i}f_{1,i}\\otimes f_{2,i})\\,=\\, 0 \\quad \\forall k\\geq -1\n\t$$\nif and only if:\n\t$$\n\t\\rho_{i}(L_k)(f_{i,j})\\,=0\\, \\text{ for all $i,j$ and $k\\geq -1$ } \n\t$$\n\\end{thm}\n\n\\begin{proof}\nLet us prove the converse. Bearing in mind that $\\rho =\\rho_1\\otimes 1 + 1\\otimes \\rho_2$, one can do the following computation:\n\t{\\small $$\n\t\\begin{aligned}\n\t\\rho(L_k)(\\sum_i f_{1,i} & \\otimes f_{2,i} )\\, \n\t=\\, \n\t(\\rho_1\\otimes 1 + 1 \\otimes \\rho_2) (L_k)(\\sum_i f_{1,i}\\otimes f_{2,i} )\n\t\\, \\\\ & =\\, \n\t \\sum_i  \\rho_1(L_k)(f_{1,i}) \\otimes f_{2,i} \\, +\\, \n\t \\sum_i f_{1,i}\\otimes \\rho_2 (L_k)(f_{2,i}  )\n\t\\,=\\, 0\n\t\\end{aligned}\n\t$$}\nand the conclusion follows. \n\nThe direct implication is more subtle. The hypothesis and the decomposition of $\\rho$ yield:\n\t{\\small $$\n\t\\begin{aligned}\n\t0\\,=\\, \\rho(L_k)(\\sum_i f_{1,i} & \\otimes f_{2,i} )\\, \n\t=\\, \n\t(\\rho_1\\otimes 1 + 1 \\otimes \\rho_2) (L_k)(\\sum_i f_{1,i}\\otimes f_{2,i} )\n\t\\, \\\\ & =\\, \n\t \\sum_i  \\rho_1(L_k)(f_{1,i}) \\otimes f_{2,i} \\, +\\, \n\t \\sum_i f_{1,i}\\otimes \\rho_2 (L_k)(f_{2,i}  )\n\t\\end{aligned}\n\t$$}\nLet $E$ be  the vector space generated by $\\{f_{1,1},\\ldots, f_{1,r}\\}\\subset V_1$. Suppose that there exists $l$ such that $\\rho_1(f_{1,l})$ does not belong to $E$. Then, let $\\chi: V_1 \\to {\\mathbb C}$ be a linear form such that $\\chi(f_{1,i})=0$ for all $i$ and $\\chi( \\rho_1(f_{1,l}))\\neq 0$. Applying $\\chi$ to the above equation, one obtains:\n\t$$\n\t0\\,=\\, \\sum_i \\chi\\big( \\rho_1(L_k)(f_{1,i}) \\big) f_{2,i} \\,\\in\\, V_2\n\t$$\nwhich contradicts the fact that $f_{2,1},\\ldots,f_{2,r}$ are linearly independent. Therefore, it follows that $\\rho_1(f_{1,l})$ belongs to $E$ for all $l$ or, equivalently, \n\t$$\n\t\\begin{aligned}\n\t\\rho_{1,E}:{\\mathcal W}_> \\,&\\longrightarrow \\,\\End(E)\n\t\\\\\n\tL_k \\quad &\\longmapsto \\, \\rho_{1,E}(L_k):= \\rho_1(L_k)\\vert_E\n\t\\end{aligned}\n\t$$\nis a Lie algebra homomorphism. Recall that, being ${\\mathcal W}_>$ simple, the non-trivial representations of ${\\mathcal W}_>$ are faithful. Since $E$ is finite dimensional, $\\rho_{1,E}$ must be trivial; that is, $\\rho_{1,E}(L_k)=0$ for all $k$. In particular, \n\t$$\n\t0\\,=\\, \\rho_{1,E}(L_k)(f_{1,j}) \\,=\\,  \\rho_1(L_k)(f_{1,j}) \\qquad \\forall j\n\t$$\nThe identities $\\rho_2(L_k)(f_{2,j})=0$ are proven similarly. \n\\end{proof}\n\t\n\n\n\n\\subsection{Commutation Relations}\\label{subsect:CR}\n\nThis subsection only collects the explicit computations of some Lie brackets used  previously and, thus, the reader can skip it. From a formal point of view,  we are dealing with generators of $\\uh$, $ 1, q_{i,\\alpha}, p_{i,\\alpha}$,  with $i=1,2,\\ldots$ and $\\alpha=1,\\ldots, n$, that satisfy the following relations:\n    $$\n    \\begin{gathered}\n    \\, [p_{i,\\alpha}, q_{j,\\beta}] \\,=\\, \\delta_{i,j} i \\eta^{\\alpha\\beta}\\cdot 1 \n    \\\\\n   \\,  [p_{i,\\alpha}, p_{j,\\beta}] \\,=\\,[q_{i,\\alpha}, q_{j,\\beta}] \\,=\\, \n    [p_{i,\\alpha}, 1] \\,=\\,[q_{i,\\alpha}, 1] \\,=\\,  0\n    \\end{gathered}\n    $$\nand, because of the associativity of composition, we will also use:\n\t$$\n\t[a,bc]=\\, [a,b] c \\,+\\, b[a,c]\n\t$$\n\nWe will use the Einstein convention; that is, repeated subindices of the variables $p,q$'s imply the summation is to be done. Recall that $b_i^{0,2i+3}$  and $q_i:=(q_{i,1},\\ldots, q_{i,n})$ denote row vectors, $p_i:=(p_{i,1},\\ldots, p_{i,n})^T$ are column vectors (the superscript $T$ denotes the transpose), and $a , b_i^{j,2i+j}, c_i^{j,2i-j}$ are $n\\times n$ square matrices.\n\nLet us compute some Lie brackets. For instance,\n    $$\n    \\begin{aligned}\n    \\, & [ b_i^{0,2i+3} p_{2i+3} , b_{j}^{0,2j+3} p_{2j+3}]\n    \\,=\\\\ & \\qquad  =\\,\n    [ (b_i^{0,2i+3})_{\\alpha} (p_{2i+3})_{\\alpha}\t\\, ,\\, (b_{j}^{0,2j+3})_{\\beta} (p_{2j+3})_{\\beta} ]\n    \\,=\\\\ & \\qquad  =\\,\n    (b_i^{0,2i+3})_{\\alpha}  [ (p_{2i+3})_{\\alpha}\t\\, ,\\,(p_{2j+3})_{\\beta} ]  (b_{j}^{0,2j+3})_{\\beta} \\,=\\, 0\n    \\end{aligned}\n    $$\nwhere subindices $\\alpha,\\beta$ denote the corresponding entries of the vectors. Analogously, we have the following identities:\n    \\begin{subequations}\n    \\begin{align*}\n\t& [ q_1 a  q_1^T ,  b_i^{0,2i+3} p_{2i+3} ] \\,=\\, 0 \\qquad \\forall i\\geq 0\n\t\\\\ &\n    [ q_{r} b_{i}^{r,r+2i} p_{r+2i} ,  b_j^{0,2j+3} p_{2j+3}]\n    \\,=\\\\ & \\qquad  =\\,\n    - [b_j^{0,2j+3} p_{2j+3} ,  (q_{r})_{\\alpha}] (b_{i}^{r,r+2i} p_{r+2i})_{\\alpha}\n    \\\\ & \\qquad \\quad \n    -\\, (q_{r} b_{i}^{r,r+2i})_{\\alpha} [  b_j^{0,2j+3} p_{2j+3}, (p_{r+2i})_{\\alpha}]\n    \\,=\\\\ & \\qquad  =\\,\n    - (b_j^{0,2j+3})_{\\beta} [ (p_{2j+3})_{\\beta}, (q_{r})_{\\alpha}]\n    (b_{i}^{r,r+2i})_{\\alpha\\gamma} (p_{r+2i} )_{\\gamma}\n    \\,=\\\\ & \\qquad  =\\,\n    - (2j+3) (b_j^{0,2j+3})_{\\beta} \\eta^{\\beta\\alpha} (b_{i}^{2j+3,2j+3+2i})_{\\alpha\\gamma} (p_{2j+3+2i} )_{\\gamma}\n    \\,=\\\\ & \\qquad  =\\,\n    - (2j+3) b_j^{0,2j+3} \\eta^{-1} b_{i}^{2j+3,2j+3+2i} p_{2j+3+2i}\n     \\\\ & \n     [ p_{r} c_{i}^{r,2i-r} p_{2i-r} ,  b_j^{0,2j+3} p_{2j+3}]\n    \\,=\\, 0 \n    \\\\ & \n    [ q_1 a  q_1^T , q_{r} b_{j}^{r,r+2j} p_{r+2j} ]\n    \\,=\\, 0 \\qquad\\forall j\\geq 1\n       \\\\ &\n    [ q_1 a  q_1^T , q_{r} b_{0}^{r,r} p_{r} ]\n    \\,=\\\\ & \\qquad  =\\,\n   (  q_{1} b_{0}^{1,1} )_{\\alpha} [ q_1 a  q_1^T , ( p_{1})_{\\alpha} ]\n    \\,=\\\\ & \\qquad  =\\,\n\t- (  q_{1} b_{0}^{1,1} )_{\\alpha} \\big(\n\t[( p_{1})_{\\alpha} , ( q_{1})_{\\beta}] (a  q_1^T)_{\\beta} +\n\t(q_1 a )_{\\beta} [( p_{1})_{\\alpha} , ( q_{1})_{\\beta}]\\big)\n    \\,=\\\\ & \\qquad  =\\,\n    - q_{1} b_{0}^{1,1}\\eta^{-1}  \\big( a  +a^T \\big) q_1^T\n\\\\ &\n    [ q_1 a  q_1^T , p_{r}^T c_{j}^{r,2j-r} p_{2j-r} ]\n    \\,=\\\\ & \\qquad  =\\,\n    [ q_1 a  q_1^T , (p_{r})_{\\alpha}] (c_{j}^{r,2j-r} p_{2j-r})_{\\alpha}\n    \\,+\\,\n      (p_{r}^T c_{j}^{r,2j-r})_{\\alpha}  [ q_1 a  q_1^T ,  (p_{2j-r} )_{\\alpha}]\n    \\,=\\\\ & \\qquad  =\\,\n    -\\big( [(p_1)_{\\alpha}, (q_1)_{\\beta}] (a  q_1^T)_{\\beta} + (q_1 a )_{\\beta}[(p_1)_{\\alpha}, (q_1)_{\\beta}]\\big)\n    (c_{j}^{1,2j-1} p_{2j-1})_{\\alpha}\n    \\,- \\\\ & \\qquad \\qquad - \\, (p_{2j-1}^T c_{j}^{2j-1,1})_{\\alpha} \\big([(p_1)_{\\alpha},(q_1)_{\\beta}] (a  q_1^T)_{\\beta} + ( q_1 a )_{\\beta} [(p_1)_{\\alpha},(q_1)_{\\beta}]\\big)\n    \\,=\\\\ & \\qquad  =\\,\n    -q_1 \\big(  a    + ( a )^T\\big)(\\eta^{-1})^T  c_{j}^{1,2j-1} p_{2j-1}     \n    \\,- \\, p_{2j-1}^T c_{j}^{2j-1,1} \\eta^{-1} \\big(   a   + (a   )^T\\big) q_1^T\n    \\,=\\\\ & \\qquad  =\\,\n    -q_1 \\big(  a    + ( a )^T\\big)(\\eta^{-1})^T   \n    \\big(  c_{j}^{1,2j-1}  +    (c_{j}^{2j-1,1})^T \\big) p_{2j-1} -\n    \\\\  \n    & \\qquad \n    -  \\delta_{j1}  \\operatorname{Tr}\\big(c_{1}^{1,1} \\eta^{-1} ( a   + a^T)(\\eta^{-1})^T\\big)\n\\\\ &\n    [ q_{r} b_{i}^{r,r+2i} p_{r+2i}   , p_{s}^T c_{j}^{s,2j-s} p_{2j-s} ]\n    \\,=\\\\ & \\qquad  =\\,\n    [  q_{r} b_{i}^{r,r+2i} p_{r+2i} , (p_{s})_{\\alpha} ] (c_{j}^{s,2j-s} p_{2j-s})_{\\alpha}\n    \\,+\n    \\\\ & \\qquad\\quad + \\,\n      (p_{s}^T c_{j}^{s,2j-s} )_{\\alpha} [ q_{r} b_{i}^{r,r+2i} p_{r+2i} ,  (p_{2j-s})_{\\alpha} ]\n    \\,=\\\\ & \\qquad  =\\,\n    - [ (p_{s})_{\\alpha} ,  (q_{r})_{\\beta}  ] (b_{i}^{r,r+2i} p_{r+2i} )_{\\beta} (c_{j}^{s,2j-s} p_{2j-s})_{\\alpha}\n    \\,- \\\\ & \\qquad \\qquad -\\,\n      (p_{s}^T c_{j}^{s,2j-s} )_{\\alpha}  [ (p_{2j-s})_{\\alpha},  (q_{r})_{\\beta}   ](b_{i}^{r,r+2i} p_{r+2i})_{\\beta}\n    \\,=\\\\ & \\qquad  =\\,\n    - r p_{2j-r}^T (c_{j}^{r,2j-r})^T \\eta^{-1} b_{i}^{r,r+2i} p_{r+2i}\n    \\,-\\, r  p_{2j-r}^T c_{j}^{2j-r,r} \\eta^{-1} b_{i}^{r,r+2i} p_{r+2i}\n    \\,=\\\\ & \\qquad  =\\,\n    - r p_{2j-r}^T \\big( (c_{j}^{r,2j-r})^T + c_{j}^{2j-r,r}\\big)\\eta^{-1} b_{i}^{r,r+2i} p_{r+2i}\n\n  \n      \\\\ &\n    [ q_{r} b_{i}^{r,r+2i} p_{r+2i}   , q_{s} b_{j}^{s,s+2j} p_{s+2j} ]\n    \\,=\\\\ & \\qquad  =\\,\n    [  q_{r} b_{i}^{r,r+2i} p_{r+2i} , (q_{s})_{\\alpha} ] (b_{j}^{s,s+2j} p_{s+2j})_{\\alpha}\n    \\,+\\\n     \\\\ & \\qquad \\qquad +\\,\n      (q_{s} b_{j}^{s,s+2j} )_{\\alpha} [ q_{r} b_{i}^{r,r+2i} p_{r+2i} ,  (p_{s+2j})_{\\alpha} ]\n    \\,=\\\\ & \\qquad  =\\,\n    - (q_{r} b_{i}^{r,r+2i})_{\\beta}[    (q_{s})_{\\alpha}, (p_{r+2i})_{\\beta} ] (b_{j}^{s,s+2j} p_{s+2j})_{\\alpha}\n    \\,- \\\\ & \\qquad \\qquad -\\,\n      (q_{s} b_{j}^{s,s+2j} )_{\\alpha} [ (p_{s+2j})_{\\alpha}, (q_{r} )_{\\beta} ](b_{i}^{r,r+2i} p_{r+2i})_{\\beta}\n    \\,=\\\\ & \\qquad  =\\,\n    (r+2i) q_{r} b_{i}^{r,r+2i}  \\eta^{-1}   b_{j}^{r+2i,r+2i+2j} p_{r+2i+2j}\n    \\,-\\,\n      r q_{r-2j} b_{j}^{r-2j,r}\\eta^{-1} b_{i}^{r,r+2i} p_{r+2i}\n   \\end{align*}\n   \\end{subequations}\n\n\n\\section{An Application}\n\nAs an application of the previous sections, we offer here an example that illustrates how our results can be used for studying the representation of ${\\mathcal W}_>$ appearing in the study of the Virasoro conjecture. Regarding the Virasoro conjecture our main references are the works of  Dubrovin-Zhang, Eguchi-Hori-Xiong, Getzler, Givental and Liu-Tian (\\cite{DubrovinZ, EHX, Getzler, Givental, LiuTian}). \n\nIn our example will consider $(A, (,))$ to be the cohomology ring of a smooth projective variety $X$ with $c_1(X)=0$ and trivial odd cohomology groups. Recall that the hypothesis on the first Chern class is equivalent to the vanishing of the operator $R$ in \\cite{DubrovinZ, Getzler}; however, it does not seem difficult to extend the results of \\S\\ref{sec:ReprW>} to include this case. On the other hand, the hypothesis on the odd cohomology groups is fulfilled if $X$ has generically semisimple even quantum cohomology (\\cite{HMT}). It seems to be very hard to weaken this assumption.  \n\n\n\\subsection{Preliminaries}\n\nLet $A$ be a $n$ dimensional vector space over ${\\mathbb C}$ endowed with a bilinear form $(\\, , \\, )$. Let $\\{a_1,\\ldots,a_n\\}$ be a basis and $\\eta$ be the  matrix associated to the pairing,  $\\eta_{\\alpha\\beta}:=(a_{\\alpha},a_{\\beta})$. Let us  consider  the subspace ${\\mathbb C}[[t_1,t_3,t_5,\\ldots]]$ of the the boson Fock space ${\\mathbb C}[[t_1,t_2,\\ldots]]$ and the subalgebra of $ {\\mathbb C}[[t_1,\\ldots]] \\hat\\otimes_{{\\mathbb C}} S^{\\bullet}A$ generated by $t_{i,\\alpha}:= t_i\\otimes a_{\\alpha} $ with $i$ odd:\n\t{\\small\n\t\\begin{equation}\n\t\\label{eq:Vkdv}\n\t\\Vkdv(A)\\,:={\\mathbb C}[[\\{t_{i,\\alpha} \\vert 1\\leq \\alpha\\leq n, i \\text{ odd}\\} ]]\n\t\\,\\subseteq \\,  {\\mathbb C}[[t_1,\\ldots]] \\hat\\otimes_{{\\mathbb C}} S^{\\bullet}A\n\t\\end{equation}}\nIf no confusion arises, we will simply write $\\Vkdv$. \n\nNow we study a distinguished representation of ${\\mathcal W}_>$ in $\\Vkdv$; eventually, we will see that it is the representation coming from the action of the Heisenberg algebra via Givental's quantization (\\cite{Givental}). More precisely, we will combine the chain of inclusions of Lie algebras:\n\t$$\n\t\\sl(2)\\,\\hookrightarrow\\, {\\mathcal W}_> \\, \\hookrightarrow \\, \\uh\n\t$$\nwhich has been studied in the previous section, with a map:\n\t$$\n\t\\begin{aligned}\n\t\\widehat{\\,}\\,: \\uh[(A)] &\\, \\longrightarrow \\End_{{\\mathbb C}}(\\Vkdv(A))\n\t\\\\\n\tP &\\, \\longmapsto \\, \\hat P\n\t\\end{aligned}\n\t$$\nwhose obstruction to be compatible with the Lie brackets is governed by a cocycle. This map is defined following  the results of Dubrovin-Zhang, Givental and Kazarian (\\cite{DubrovinZ, Givental, Kaz}); namely, we set:\n\t\\begin{equation}\\label{eq:quant}\n\\hat 1 \\,=\\, 1 \\, , \\qquad \n\\hat p_{i,\\alpha} \\,=\\, \\eta^{\\alpha\\beta}\\frac{\\partial}{\\partial t_{i,\\beta}}   \\, ,\\qquad\n\\hat q_{i,\\alpha} \\,=\\,  i t_{i,\\alpha}\n\t\\end{equation}\n(recall that $i$ is a positive odd integer number).\n\n\\begin{rem}\nGivental has developed a beautiful formalism for this construction in terms of quantization of quadratic hamiltonians (\\cite{Givental}). An alternative approach, originated in the Japanese school and strongly linked to the Sato grassmannian, can be found in \\cite{KSU}. The forthcoming section (\\S\\ref{subsec:solutions1dim}) is deeply inspired by the latter. \n\\end{rem}\n\n\\subsection{The representation}\\label{subsec:representation}\n\nBearing in mind the results of \\S\\ref{subsect:sl2}, we know that the operator:\n\t$$\n\t    F\\,  :=\\, b_{-1}^{0,1} p_1 + q_1\\eta q_1^T + q_{i+2} \\eta p_i \n\t$$\ntogether with the data:\n\t\\begin{itemize}\n\t\\item  $b_{-1}^{0,1}$ arbitrary,\n\t\\item $b_0^{1,1}$ such that \\eqref{eq:FyieldsH:2} holds, and\n\t\\item  $c_1^{1,1} := \\frac1{16} \\eta^T b_0^{1,1} \\eta^{-1} (b_0^{1,1}\\eta^{-1}+2) $,\n\t\\end{itemize}\ndetermine a map  $\\sigma:\\sl(2)\\to \\uh$. Indeed, equations~\\eqref{eq:FyieldsH}, \\eqref{eq:EFH} and \\eqref{eq:HF-2F} allow us to obtain the explicit expressions for  $H$ and $F$:\n\t\\begin{equation}\\label{eq:HEparticular}\n\t\\begin{aligned}\n\tH\\, & = \\, \n\t\\frac13 b_{-1}^{0,1}(\\eta^{-1} b_0^{1,1}-2) p_3 +\n\t \\frac1i q_i (b_0^{1,1}-(i-1)\\eta) p_i +\n\t \\\\ & \\qquad + \n\t \\frac1{16} \\Tr(b_0^{1,1}\\eta^{-1} (b_0^{1,1}\\eta^{-1}+2) (1+\\eta^{-1}\\eta^T))\n\t\\\\\n\tE \\, &=\\, \\frac1{5!!} b_{-1}^{0,1} \n\t\\big(2\\eta^{-1}b_0^{1,1} - 2- (\\eta^{-1}+(\\eta^{-1})^T )(c_1^{1,1}+(c_1^{1,1})^T)\\big) p_5 \n\t+ \\\\\n\t & \\qquad +\n\t \\frac1{16} p_1^T \\eta^T b_0^{1,1} \\eta^{-1} (b_0^{1,1}\\eta^{-1}+2)  p_1 - \\\\\n\t & \\qquad - \\frac1{4 i (i+2) } q_{i} (  b_0^{1,1} \\eta^{-1}- (i-1) ) (  b_0^{1,1} \\eta^{-1}-(i+1) ) \\eta  p_{i+2}\n\t\\end{aligned}\n\t\\end{equation}\n\nNow, by  Theorem \\ref{thm:W^+-sl2}, the map $\\sigma$ extends uniquely to an homomorphism $\\rho:{\\mathcal W}_>\\to \\uh$. And one can now compute the induced action on $\\Vkdv$. Let us write down the first operators:\n    \\begin{subequations}\n    \\begin{align*}\n    \\hat{L}_{-1}\\, & :=\\, (\\rho(L_{-1}))^{\\hat\\,} \\, =\\, \\hat{F}\\, =\\, b_{-1}^{0,1} \\eta^{-1}\\frac{\\partial}{\\partial t_{1}}\n    \t + t_{1} \\eta t_{1}^T + (i+2) t_{i+2} \\frac{\\partial}{\\partial t_{i}}\n    \\\\\n     \\hat{L}_{0}\\, & :=\\, (\\rho(L_{0}))^{\\hat\\,} \\, =\\, -\\frac12 \\hat{H}\\, =\\, \n     - \\frac16 b_{-1}^{0,1}(\\eta^{-1} b_0^{1,1}-2) \\eta^{-1} \\frac{\\partial}{\\partial t_{3}}  -\n      \\\\ & \\qquad \t-\\frac12 t_i (b_0^{1,1}\\eta^{-1} -(i-1)) \\frac{\\partial}{\\partial t_{i}}\n      -\\frac1{32} \\Tr(b_0^{1,1}\\eta^{-1} (b_0^{1,1}\\eta^{-1}+2) (1+\\eta^{-1}\\eta^T))\n     \\\\\n     \\hat{L}_{1}\\, & :=\\, (\\rho(L_{1}))^{\\hat\\,} \\, =\\, - \\hat{E}\\, = \\\\ & \\qquad\n     - \\frac1{5!!} b_{-1}^{0,1} \n\t\\big(2\\eta^{-1}b_0^{1,1} - 2- (\\eta^{-1}+(\\eta^{-1})^T )(c_1^{1,1}+(c_1^{1,1})^T)\\big) \\eta^{-1}\\frac{\\partial}{\\partial t_{5}} - \\\\ & \\qquad\n     -  \\frac1{16} (\\frac{\\partial}{\\partial t_{1}})^T \\eta^T  b_0^{1,1} \\eta^{-1} (b_0^{1,1}\\eta^{-1}+2)   \\frac{\\partial}{\\partial t_{1}} + \\\\\n\t & \\qquad + \\frac1{4 (i+2)} t_{i} (  b_0^{1,1} \\eta^{-1}- (i-1) ) (  b_0^{1,1} \\eta^{-1}-(i+1) )  \\frac{\\partial}{\\partial t_{i+2}}\n     \\end{align*}\n     \\end{subequations}\nwhere, as usual, we write $t_i$ for the row vector $(t_{i,1},\\ldots, t_{i,n})$ and $\\frac{\\partial}{\\partial t_{i}}$ for the column vector $(\\frac{\\partial}{\\partial t_{i,1}},\\ldots, \\frac{\\partial}{\\partial t_{i,n}})^T$. \n     \n     \n\n\\subsection{The operators of the Virasoro Conjecture: a baby model}\\label{subsec:baby}\n\nNow, we are ready to show how the operators appearing in the Virasoro conjecture agree with our approach for the case of manifold with trivial odd cohomology and whose first Chern class vanishes. \n\nFrom now on, we suppose we are given $X$, whose first Chern class is zero, and with trivial odd cohomology. Under this hypothesis, the Poincar\\'e pairing defines on $A:= H^{\\bullet}(X,{\\mathbb C})$  a symmetric non-degenerated bilinear form:\n\t$$\n\t( a, b ) \\,=\\, \\int_X a\\cup b \\quad \\text{ for }a,b\\in A ,\n\t$$\nLet $r:=\\operatorname{dim}(X)$ and fix a basis $\\{a_{\\alpha}\\vert \\alpha=1,\\ldots, n \\}$ of $A$,  with $a_1=1\\in H^0(X,{\\mathbb C})$, such that it is homogeneous w.r.t. the Hodge decomposition; that is, $a_{\\alpha}\\in H^{p_{\\alpha},q_{\\alpha}}(X)$ for certain $p_{\\alpha}, q_{\\alpha}$.  Let $\\bar \\eta$ the matrix associated to the Poincar\\'e pairing w.r.t. the chosen basis and let us define  $\\mu_{\\alpha}:= p_{\\alpha}-\\frac{r}2$ and $\\mu$ the matrix with $\\mu_1,\\ldots, \\mu_r$ along its diagonal and $0$ elsewhere. Observe that the compatibility of the Poincar\\'e pairing w.r.t. the Hodge decomposition yields:\n\t\\begin{equation}\\label{eq:mua+mb=0}\n\t\\bar\\eta_{\\alpha\\beta}\\neq 0 \\qquad \\implies \\qquad \\mu_{\\alpha} + \\mu_{\\beta}=0\n\t\\end{equation}\n\n\nThe operators appearing in the Virasoro conjecture when the first Chern class vanishes (\\cite[Equation (1.2)]{Getzler}) are as follows:\n\t{\\small\n\t{\\begin{equation}\\label{eq:Getzler-101}\n\t\\begin{aligned}\n\t\\bar L_{-1}\\,& :=\\,\n\t- \\frac{\\partial}{\\partial \\bar t_{0,1}}  + \\frac{1}{2\\hbar} \\bar t_{0} \\bar \\eta \\bar t_0^T + \\bar t_{i+1} \\frac{\\partial}{\\partial \\bar t_{i}} \n\t\\\\\n\t\\bar L_0 \\,& :=\\, -\\frac{3-r}2\\frac{\\partial}{\\partial  \\bar t_{1,1}}  + (\\mu_{\\alpha}+i+ \\frac{1}2) \\bar t_{i,\\alpha}  \\frac{\\partial}{\\partial \\bar t_{i,\\alpha}}  +  \n\t\\frac{1}{48}(3-r)\\int_X c_r(X)\n\t\\end{aligned}\n\t\\end{equation}}}\nand, for $k\\geq 1$, as:\n\t{\\small\n\t\\begin{equation}\\label{eq:GetzlerLk}\n\t\\begin{aligned}\n\t\\bar L_k \\,& :=\\, -\\frac{\\Gamma(k+\\frac{5-r}2)}{\\Gamma(\\frac{3-r}2)} \n\t\\frac{\\partial}{\\partial \\bar t_{k+1,1}} +\n\t\t \\frac{\\Gamma(\\mu_{\\alpha}+i+k+\\frac{3}2)}{\\Gamma(\\mu_{\\alpha}+i+\\frac{1}2)} \\bar t_{i,\\alpha }   \\frac{\\partial}{\\partial \\bar t_{k+i,\\alpha}} \n\t\t\\\\ & \\qquad\n\t\t+ \\frac{\\hbar}2 (-1)^{i} \\frac{\\Gamma(\\mu_{\\alpha}+i+k+\\frac{3}2)}{\\Gamma(\\mu_{\\alpha}+i+\\frac{1}2)} \\bar \\eta^{\\alpha\\beta} \\frac{\\partial}{\\partial \\bar t_{-1-i,\\alpha}}  \\frac{\\partial}{\\partial \\bar t_{k+i,\\beta}} \n\t\\end{aligned}\n\t\\end{equation}}\nwhere $c_r(X)$ is the $r$-th Chern class and we have used variables $\\bar t_{i,\\alpha}$ with $\\alpha=1,\\ldots, n$ and $i=0,1,2,\\ldots$.\n\nSimilarly to the case of $\\uh$, we say that a second order differential operator in $\\{\\bar t_{i,\\alpha}\\}$ is of type $i$ if it is a linear combination of $\\frac{\\partial}{\\partial \\bar t_{i+1,\\alpha}}$ and the following terms \n$\\frac{\\partial}{\\partial \\bar t_{j-1,\\alpha}}\\frac{\\partial}{\\partial \\bar t_{i-j,\\beta}}$, $\\bar t_{j,\\alpha} \\frac{\\partial}{\\partial \\bar t_{j+i,\\beta}}$ and $\\bar t_{j-1,\\alpha}\\bar t_{-i-j,\\beta}$ and, if $i=0$, a constant term. Observe that $\\bar L_k$ is of type $k$. Now, we offer a simple proof of a folk statement.\n\n\\begin{prop}\nThe operators $\\{\\bar L_k\\vert k\\geq 2\\}$ are uniquely determined by $\\{\\bar L_{-1},\\bar L_0, \\bar L_1\\}$ and the condition that $\\bar L_k$ is of type $k$ for all $k\\geq -1$. \n\\end{prop}\n\n\\begin{proof}\nUnder the change of variables $\\bar t_i:= \\sqrt{2\\hbar} (2i+1)!! t_{2i+1}$, it is clear that a second order differential operator in $\\bar t_i$'s  is of type $k$ if and only if is equal to $\\hat T$ for $T\\in \\uh_{k}$. Now, it is easy to check that the hypothesis of Theorem \\ref{thm:W^+-sl2} hold; namely, $\\hat F=\\bar L_{-1}$ and $\\bar L_k$ are of type $k$ for $k=0,1$. The conclusion follows. \n\\end{proof}\n\n\\begin{thm}\\label{thm:Lbar=Lhat}\nIt holds:\n\t$$\n\t\\bar L_i \\,=\\, \\hat L_i \\qquad i=-1,0,1,\\ldots\n\t$$\nfor the choice $\\bar t_i:= \\sqrt{2\\hbar} (2i+1)!! t_{2i+1}$, $\\eta=\\bar \\eta$, $b_{-1}^{0,1}= (0,\\ldots, 0, \\frac{-1}{\\sqrt{2\\hbar}})$ and:\n\t$$\n\tb_0^{1,1}\\,:=\\, -(2 \\mu+1) \\eta \\,=\\, -{\\small \\begin{pmatrix} 0 & & 2\\mu_1+1 \\\\ & \\iddots & \\\\ 2\\mu_n+1 & & 0\\end{pmatrix}}$$\n\\end{thm}\n\n\\begin{proof}\nTheorem \\ref{thm:W^+-sl2} implies that it suffices to show that $\\bar L_i \\,=\\, \\hat L_i$ for $i=-1,0,1$. Indeed, this fact follows from the explicit substitution of $t_i$, $\\eta$, etc. as in the statement in the operators $ \\hat L_i$. The only identity which is not obvious is the one corresponding to the constant term of $ \\hat L_0$. Bearing in mind the definitions and the fact that $\\eta$ is symmetric, this term is:\n\t{\\small $$\n\t\\begin{aligned}\n\t -\\frac1{32} & \\Tr (b_0^{1,1}\\eta^{-1} (b_0^{1,1}\\eta^{-1}+2) (1+\\eta^{-1}\\eta^T))\n\t \\,= \\\\ & = \\,\n\t -\\frac1{16}\n\t \\sum_{\\alpha=1}^n (2 p_{\\alpha} -r+1) (2 p_{\\alpha}-r -1)  \n\t  \\,=\\,\n\t \\frac14 \\sum_{p,q} h^{p,q} (\\frac{r+1}2 - p ) ( p-\\frac{r-1}2)\n\t \\end{aligned}\n\t $$}\nwhere $h^{p,q}=\\dim H^p(X,\\Omega^q)$.\n\nNow, observe that the Libgober-Wood identity (\\cite[Proposition~2.3]{LW}) can be stated as:\n\t{\\small $$\n\t\\sum_{p,q} (-1)^{p+q}h^{p,q}  (\\frac{r+1}2 - p ) ( p-\\frac{r-1}2) \n\t\\,=\\, \\frac16\\int_X \\big(\\frac{3-r}2 c_r(X)- c_1(X)c_{r-1}(X)\\big)\n\t$$}\nRecalling that we are assuming that that it has trivial odd cohomology, the constant term equals:\n\t$$\n\t \\frac1{48}\\int_X \\big( (3-r) c_r(X)- 2 c_1(X)c_{r-1}(X)\\big)\n\t$$\nwhich agrees with the free term of $\\bar L_0$ (see \\eqref{eq:Getzler-101}) since $c_1(X)=0$. \n\\end{proof}\n\n\\begin{rem}\nIt is worth noticing that up to rescaling the variables and a Dilaton shift, these operators coincide with those of \\cite[Equation (3.5)]{DVV} and \\cite[\\S3]{Givental} (for $b_1=0$) and with those of \\cite[Equation (7.33)]{Dijkgraaf} and \\cite[Equation (2.59)]{Witten} (for $b_1=-\\frac13\\sqrt{\\frac{\\eta}{2\\hbar}}$). \n\\end{rem}\n\n\nNow, we will go one step further in the study of the above representation. Recall that in \\S\\ref{subsec:prelim-beta-Diff-to-C[[t]]} it was stated that matrices $a$, $b_{i}^{j-2i,j}$ and $c_i^{j,2i-j}$ behave as bilinear forms under the action of $\\Gl(A)$. \\emph{A fundamental observation is that all results and equations above are invariant under the action of the general linear group} (acting as base changes on the given basis $\\{a_1,\\ldots, a_n\\}$). Let us briefly discuss this statement. For instance, let $S\\in \\Gl(A)$, then the row vector $q_i=(q_{i,1},\\ldots, q_{i,n})$ is transformed to $q_i S^T$, accordingly the column vector $p_i$ goes to $S p_i$. The action of $S$ sends the bilinear form $\\eta$  to $(S^{-1})^T \\eta S^{-1}$ and analogously with $a$, etc.~. Note that, since $\\eta$ and $b_0^{1,1}$ behave as bilinear forms, $\\eta^{-1}b_0^{1,1}$ defines an endomorphism of $A$.  Finally, the Heisenberg algebra is also affected. \n\n\n\\begin{defn}\nLet ${\\mathbb H}^{\\eta}$ be the Heisenberg algebra defined in \\eqref{eq:pqcommutation}. Given a map of Lie algebras $\\rho:{\\mathcal W}_>\\to {\\mathcal U}({\\mathbb H}^{\\eta})$ and  $S\\in\\Gl(A)$,  we denote by $\\rho^S$ the map of Lie algebras:\n\t$$\n\t{\\mathcal W}_>\\, \\overset{\\rho}\\longrightarrow \\,  {\\mathcal U}({\\mathbb H}^{\\eta})\n\t\\,\\overset{\\sim}\\longrightarrow\\,  {\\mathcal U}({\\mathbb H}^{(S^{-1})^T \\eta S^{-1}})\n\t$$\nwhere the last map sends \t$q_{i}$ to $q_{i} S^T$ and $p_i$ to $S p_i$. \n\\end{defn}\n\nWith the hypothesis and choices of above, we have the following,\n\n\n\\begin{thm}\\label{thm:Virasorodecomp}\nLet $\\rho:{\\mathcal W}_>\\to {\\mathcal U}({\\mathbb H}^{\\eta})$ be as above; i.e. $\\hat\\rho$ defines  the Virasoro constraints,  \\eqref{eq:Getzler-101} and \\eqref{eq:GetzlerLk},  of a smooth projective variety with trivial odd cohomology and vanishing first Chern class. \n\nThen  there exists $S\\in \\Gl(A)$ such that  $\\rho^S$ decomposes as the product of $n$ representations of dimension $1$; that is, there exist $\\rho_i: {\\mathcal W}_>\\to \\uh[({\\mathbb C})]$ such that:\n\t\\begin{equation}\\label{eq:rho=sumrhoalpha}\n\t\\rho^S \\,=\\, \n\t\\rho_1 \\otimes 1\\otimes\\ldots \\otimes 1 \\, +\\, \\ldots\\, +\\,\n\t1\\otimes\\ldots \\otimes 1 \\otimes \\rho_n\n\t\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nLet us consider a basis  which is orthonormal for $\\eta$. Let $S\\in\\Gl(A)$ be the matrix associated to this change of basis. Due to the choices of $a$, $b_{-1}^{i+2,i}, b_0^{1,1}$ and $c_1^{1,1}$, it is trivial that $S$ also bring them into diagonal form; or, equivalently, there is a common orthogonal basis for all these bilinear pairings. Applying Theorem \\ref{thm:rhodecomp}, one concludes.\n\\end{proof}\n\nIn this situation, for each $\\alpha=1,\\ldots,n$, one obtains a one dimensional representation $\\rho_{\\alpha}$ or, what is tantamount, our study essentially reduces to the case of Example~\\ref{exam:dim1}. That is,  $\\dim A=1$ , $a= \\eta\\in{\\mathbb C}^{\\ast}$ and, thus, $b_0^{1,1}=-\\eta$. Setting $b_0:= b_{0}^{0,0}$, one has that \\eqref{eq:HEparticular} gives:\n\t$$\n\t\\begin{aligned}\n\tF \\,& =\\,  b_{-1}  p_1 +   q_1  \\eta  q_1  +  q_{i+2}\\eta p_i\n\t\\\\\n\tH\\, &=\\,  -  b_{-1}   p_3   -  q_i  \\eta  p_i  - \\frac18 \n    \t\\\\\n\tE\\,&=\\, - \\frac1{4} b_{-1}  p_5 - \\frac1{4} p_1 \\eta  p_1 -  \\frac1{4}  q_{i}  \\eta  p_{i+2}\n\t\\end{aligned}\n\t$$\nwhere $b_{-1} $ and $\\eta$ are computed from the $n$-dimensional setup \\eqref{eq:Getzler-101}.\n\nThese three operators determine $\\rho$ completely and, according to the map \\eqref{eq:quant} and Theorem~\\ref{thm:Lbar=Lhat}, one has:\n\t{\\small\n\t{\\begin{equation}\\label{eq:barL}\n\t\t\\begin{aligned}\n\t\\bar L_{-1}\\,& :=\\, \n\tb_{-1} \\sqrt{2\\hbar} \\eta^{-1} \\frac{\\partial}{\\partial \\bar t_{0}}  \n\t+ \\frac{1}{2\\hbar} \\eta \\bar t_{0}^2\n\t+ \\bar t_{i+1} \\frac{\\partial}{\\partial \\bar t_{i}} \n\t\\\\\n\t\\bar L_0 \\,& :=\\, \\frac32  b_{-1} \\sqrt{2\\hbar} \\eta^{-1} \\frac{\\partial}{\\partial \\bar t_{1}}  \n\t+ (i+\\frac12) \\bar t_{i}  \\frac{\\partial}{\\partial \\bar t_{i}}  + \\frac{1}{16} \n\t\\\\\n\t\\bar L_1 \\,& :=\\, \\frac{5!!}4  b_{-1} \\sqrt{2\\hbar} \\eta^{-1} \\frac{\\partial}{\\partial \\bar t_{2}}  \n\t+ \\frac{\\hbar}2\\eta^{-1} \\frac{\\partial}{\\partial \\bar t_0} \\frac{\\partial}{\\partial \\bar t_0} \n\t+ (i+\\frac12)(i+\\frac32) \\bar t_{i } \\frac{\\partial}{\\partial \\bar t_{i+1}} \n\t\\end{aligned}\n\t\\end{equation}}}\n\n\n\n\\subsection{On the solutions for the $1$-dimensional situation}\\label{subsec:solutions1dim}\n Once the representation has been decomposed in terms of $1$-dimensional parts, we wonder if one could deduce some properties of the solutions of the Virasoro constraints. Our approach follows closely our previous work \\cite{Plaza-AlgSol} which is inspired in \\cite{KacSchwarz}. Briefly, the idea is to show that each of our representations $\\rho_i$ come from an action of ${\\mathcal W}_>$ on the Sato Grassmannian and that that they admit exactly one solution $\\tau_i$, which are $\\tau$-functions for the KP hierarchy and, then, conclude that the product $\\tau_1\\cdot\\ldots\\cdot \\tau_n$ is a solution for $\\rho^S$. \n\nLet us begin recalling that the Sato Grassmannian is the set of subspaces $U\\subset {\\mathbb C}((z))$ such that the kernel and cokernel of $\\pi_U:U\\to {\\mathbb C}((z))\/{\\mathbb C}[[z]]$ are finite dimensional (\\cite{Sato, SW}). Actually, it is an infinite dimensional scheme (\\cite{AMP}) and carries a distinguished line bundle, the determinant line bundle ${\\mathbb D}$. Each integer $n$ correspondes to a connected component, $\\operatorname{Gr}^n$; namely, those subspaces $U$ such that $\\dim\\ker\\pi_U-\\dim\\coker\\pi_U=n$. Sato-Sato's achievement was to show that there was a bijection between the set of those $U$ s.t. $\\pi_U$ is an isomorphism and the set of functions $\\tau(t)\\in{\\mathbb C}[t_1,t_2,\\ldots]]$ with $\\tau(0)=1$ and fulfilling the KP hierarchy (thus, each $U$ has a $\\tau$-function; see \\cite{Sato,SW,AMP} for details). The same holds for the Sato grassmannian of ${\\mathbb C}((z))^{\\oplus n}$ and the $n$-multicomponent KP hierarchy.\n\nThe fact that the space of global section of ${\\mathbb D}^*$ is isomorphic to the semi-infinite wedge product or Fermion Fock space:\n\t{\\small $$\n\tH^0(\\operatorname{Gr}^n,{\\mathbb D}^*)\\simeq \\wedge^{\\frac{\\infty}2}{\\mathbb C}((z)) \\, =\\,\n\t<\\left\\{ \n\t\\begin{gathered}\n\tz^{i_1}\\wedge z^{i_2}\\wedge\\ldots \\text{ s.t. } i_1<i_2<\\ldots \\\\\n\t \\text{ and } i_k=k+n \\,\\,  \\forall k>>0\n\t \\end{gathered}\\right\\}>\n\t $$}\nhave allowed its extensive use in CFT's (in particular, by the Japanese school, see \\cite{KSU} and references therein). Recall that the boson-fermion correspondence is the isomorphism (we restrict us to $\\operatorname{Gr}^0$; that is, the charge $0$ sector):\n\t$$\n\t \\wedge^{\\frac{\\infty}2}{\\mathbb C}((z)) \\, \\simeq \\, {\\mathbb C}[[t_1,t_2,\\ldots]]\n\t $$\nthat maps $z^{i_1}\\wedge z^{i_2}\\wedge\\ldots$ to the Schur polynomial associated with the partition $1-i_1\\geq 2-i_2\\geq \\ldots$. Similarly, the space of global sections of ${\\mathbb D}^*$ over the Sato grassmannian of ${\\mathbb C}((z))^{\\oplus n}$ is isomorphic to ${\\mathbb C}[[\\{t_{i,\\alpha} \\vert \\alpha=1,\\ldots, n , i=1,2,\\ldots \\}]]$. \n\n\t\nGiven a subgroup of the restricted linear group of ${\\mathbb C}((z))$ (see \\cite{SW}), one has an induced action on $\\operatorname{Gr}^n({\\mathbb C}((z)))$. Moreover, if the action preserves the determinant bundle, it will yield a projective action on the space of global sections. In fact, an analogous statement holds for the case of Lie algebras. Let us illustrate this issue with the case of the Lie algebra $\\operatorname{Diff}^1({\\mathbb C}((z)))$ of first order differential operators on ${\\mathbb C}((z))$. An operator $D\\in\\operatorname{Diff}^1({\\mathbb C}((z)))$ acts on sections as follows. If the matrix $(d_{ij})$ corresponding to $D$ w.r.t. the basis $\\{z^i\\}$ has no non-trivial diagonal elements, then:\n    $$\n    D ( z^{i_1}\\wedge z^{i_2}\\wedge\\ldots)\n    \\,:=\\,\n     D ( z^{i_1})\\wedge z^{i_2}\\wedge\\ldots +\n     z^{i_1}\\wedge  D ( z^{i_2})\\wedge\\ldots + \\ldots\n     $$\nIf the matrix $(d_{ij})$ is diagonal, then:\n    $$\n    D ( z^{i_1}\\wedge z^{i_2}\\wedge\\ldots)\n    \\,:=\\,\n     \\sum_{j=1}^{\\infty} (d_{i_j i_j}- d_{jj})  z^{i_1}\\wedge z^{i_2}\\wedge\\ldots\n     $$\nHaving in mind the boson-fermion correspondence, the above construction gives rise to a linear map:\n    \\begin{equation}\\label{eq:betaDiffEnd}\n    \\begin{aligned}\n    \\operatorname{Diff}^1({\\mathbb C}((z))) \\,&\n    \\overset{\\beta}\\longrightarrow\n    \\End({\\mathbb C}[[t_1,t_2,\\ldots]])\n    \\\\\n    D\\,& \\longmapsto \\, \\beta( D)\n    \\end{aligned}\n    \\end{equation}\nwhich defines a projective representation. Note, nevertheless, that if we are given a map of Lie algebras $\\sigma:{\\mathcal W}_>\\to\\operatorname{Diff}^1({\\mathbb C}((z)))$, then, $\\beta\\circ\\sigma$ can be canonically promoted to a linear representation since ${\\mathcal W}_>$ has no non-trivial central extensions. Indeed, for this goal, if suffices to add a constant to $\\beta\\circ\\sigma(L_0)$.\n\nThe following results will show that the operators of \\S\\ref{subsec:baby} arise from the previous setup. \n\n\\begin{lem}\\label{lem:DinDiffistypei}\nLet $D\\in\\operatorname{Diff}^1({\\mathbb C}((z)))$. Then, $\\beta(D)$  is of type $i$ if and only if $D$ is a linear combination of $ 1$, $ z^{-(2i+3)} $ and $  z^{-2i}(z\\partial_z+\\frac{1-2i}2)$. \n\\end{lem}\n\n \\begin{proof}\nRecall that $\\operatorname{Diff}^1({\\mathbb C}((z)))$ is generated as ${\\mathbb C}$-vector space by $1$, $z^m$ for $m\\in{\\mathbb Z}$ acting as an homothety and $ z^{m}(z\\partial_z+\\frac{m+1}2)$ for $m\\in{\\mathbb Z}$. Let us recall from~\\cite[Table 1]{Kaz} the description of  the operators induced by them via the boson-fermion correspondence: \n    $$\n    \\beta(z^m) \\,=\\,\n        \\begin{cases}\n        m t_m & \\text{ for } m >0 \\\\\n        0 & \\text{ for } m=0 \\\\\n        \\frac{\\partial}{\\partial t_{-m}} &\\text{ for } m<0\n        \\end{cases}\n    $$\nand, for $m>0$, \n    {\\small $$\n    \\beta\\big(z^m(z\\partial_z+\\frac{1+m}2)\\big)\n    \\,  = \\,\n    \\frac12 \\sum_{j=1}^{m-1} j(m-j) t_j t_{m-j} +\n    \\sum_{j=1}^{\\infty} (j+m) t_{m+j} \\frac{\\partial}{\\partial t_{j}}\n    $$}\nAnalogously, the\naction of $z^{-m}(z\\partial_z+\\frac{1-m}2)$ on ${\\mathbb C}((z))$\ncorresponds to the action of:\n    {\\small $$\n\t\\beta\\big(z^{-m}(z\\partial_z+\\frac{1-m}2))\\big)\n    \\, = \\,\n    \\sum_{j=1}^{\\infty} j t_j  \\frac{\\partial}{\\partial t_{m+j}}  +\n    \\frac12\\sum_{j=1}^{m-1} \\frac{\\partial}{\\partial t_{j}} \\frac{\\partial}{\\partial t_{m-j}}\n    $$}\nFinally, recall that the case $m=0$ is regularized as follows:\n    {\\small $$\n    \\beta\\big(z^{-m}(z\\partial_z+\\frac{1-m}2))\\big)    \\, := \\,  \\sum_{j=1}^{\\infty} j t_j \\frac{\\partial}{\\partial t_{-j}}\n    $$}\nChecking the degrees, the conclusion follows.\n\\end{proof}\n\n\\begin{lem}\\label{lem:sigma}\nLet $\\sigma\\in\\homlie({\\mathcal W}_>,\\operatorname{Diff}^1({\\mathbb C}((z))))$. Recall that $\\Vkdv = {\\mathbb C}[[t_1,t_3,\\ldots]]$. \n\n$\\beta({\\sigma(L_i)}) \\vert_{\\Vkdv}$ takes values in $\\Vkdv$ and it is of type $i$ for all $i$, if and only if there exists $s,  t\\in {\\mathbb C}$ such that:\n    $$\n    \\sigma(L_i)\\,=\\,\n    t^i\\Big(\n        \\frac12 z^{-2i}(z\\partial_z + \\frac{1-2i}{2})+ s z^{-2i-3}\\Big)\n        \\qquad \\forall i\\geq -1\n    $$\n\\end{lem}\n\n\\begin{proof}\nThe ``if'' part follows from Lemma \\ref{lem:DinDiffistypei} and the fact that $\\sigma$ as in the statement defines a map of Lie algebras. Let us now deal with the ``only if'' part.\n\n\nWe know from~\\cite[\\S2]{Plaza-AlgSol} (see also~\\cite{Plaza-Opers}) that there is a 1-1-correspondence:\n    {\\footnotesize\n    $$\n    \\left\\{\\begin{gathered}\n    \\sigma\\in\\Hom_{\\text{Lie-alg}}({\\mathcal\n    W}_>,\\operatorname{Diff}^1({\\mathbb C}((z))))\n   \\\\\n    \\text{such that }\\sigma\\neq 0\n    \\end{gathered}\\right\\}\n    \\, \\overset{1-1}\\longleftrightarrow \\,\n    \\left\\{\\begin{gathered}\n    \\text{triples }(h(z),c,b(z))\\text{ such that}\n    \\\\\n    h'(z)\\in{\\mathbb C}((z))^*,\\; c\\in{\\mathbb C},\\; b(z)\\in{\\mathbb C}((z))\n    \\end{gathered}\\right\\}\n    $$\n    }\nwhich is explicitly given by:\n    \\begin{equation}\\label{eq:explicitexpreL_idiffC((z))}\n    \\sigma(L_i) \\,=\\,\n    \\frac{-h(z)^{i+1}}{h'(z)} \\partial_z  -\n    (i+1) c \\cdot h(z)^i + \\frac{h(z)^{i+1}}{h'(z)}b(z)\n    \\end{equation}\n\nOn the other hand, due to Lemma~\\ref{lem:DinDiffistypei}, the fact that $\\sigma(L_i)$ is of type $i$ implies that there exist $r_i,s_i,t_i\\in{\\mathbb C}$ satisfying:\n    \\begin{equation}\\label{eq:explicitexpreL_ilinearcomb}\n    \\sigma(L_i) \\,=\\,\n    r_i\\cdot 1 + s_i\\cdot z^{-(2i+3)} + t_i\\cdot z^{-2i}(z\\partial_z+\\frac{1-2i}2)\n    \\end{equation}\n\nComparing the coefficients of $\\partial_z$ in the previous identities, it follows that $h(z)=\\frac{t_i}{t_{i-1}} z^{-2}$. Hence, the quotients $\\frac{t_i}{t_{i-1}}$ are all equal to a constant, say $t$. Hence $t_i=t^i t_0$  and $h(z)=t z^{-2}$. Further, the case $i=0$ yields $t_0=\\frac12$.\n\nPluging this in equations~\\eqref{eq:explicitexpreL_idiffC((z))} and~\\eqref{eq:explicitexpreL_ilinearcomb}, one gets:\n    {\\small\n    $$\n    -(i+1)c(tz^{-2})^i\n    -\\frac{(tz^{-2})^{i+1}}{2 t z^{-3}} b(z)\n    \\,=\\,\n    r_i + s_i z^{-(2i+3)} + \\frac12 t^i z^{-2i}(\\frac{1-2i}2)\n    $$}\nand, thus:\n    $$\n    b(z)\\,=\\,\n    -2(i+1)c z^{-1}\n    -  2 t^{-i} r_i z^{2i-1} - 2 s_i t^{-i} z^{-4} - \\frac12  z^{-1}(1-2i)\n    $$\nObserve that the l.h.s. does not depend on $i$, one gets many conditions. First, for $i\\neq 0$ the term $z^{2i-1}$ is an odd power of $z$ different from $z^{-1}$ that can not be cancelled with any other term; consequently, $r_i=0$ for $i\\neq 0$. Further, since the coefficient of $z^{-4}$ in $b(z)$ has to be independent of $i$, it follows that $t^{-i} s_i$ is a constant independent of $i$; and, thus, equal to $s_0$. Finally, the coefficient of $z^{-1}$ in $b(z)$ is:\n    $$\n    -2(i+1)c - 2 r_0 \\delta_{i,0} - \\frac12 (1-2i)\n    $$\nSince it has to be independent of $i$, it follows that $c=\\frac12 $, $r_0=0$ and, thus:\n    $$\n    b(z)\\,=\\, -\\frac32  z^{-1} - 2 s_0 z^{-4}\n    $$\nSubstituting $h(z), c, b(z)$ into expression~\\eqref{eq:explicitexpreL_idiffC((z))} and setting $s=s_0$, one obtains the result.\n\\end{proof}\n\nLet us recall that the rescaling of the variables yields an action on the boson Fock space. More precisely, $\\lambda =\\{\\lambda_i \\} \\in\\prod_{i \\text{ odd}}{\\mathbb C}^*$ maps $t_i$ to $\\lambda_i t_i$. Accordingly, it acts on $ \\homlie({\\mathcal W}_>,\\End(\\Vkdv))$ and sends $\\rho$ to $\\rho^{\\lambda}:=\\lambda\\circ\\rho\\circ \\lambda^{-1}$. \n\n\n\\begin{defn}\nThe $\\lambda$-scaled KP hierarchy is the hierarchy obtained by replacing $t_i$ by $\\lambda_i t_i$ in the KP hierarchy (for given  $\\lambda=(\\lambda_i ) \\in\\prod_{i \\in{\\mathbb N}}{\\mathbb C}^*$).\nA function $\\tau_1(t)\\in {\\mathbb C}[[t_1, t_2,\\ldots]]$ is called $\\tau$-function of the $\\lambda$-scaled KP hierarchy if $\\tau(\\lambda^{-1}t):=\\tau(\\lambda_1^{-1}t_1, \\lambda_2^{-1} t_2,\\ldots)$ is a $\\tau$-function of the KP hierarchy. For brevity, we simply say scaled KP. We do similarly for KdV, multicomponent KP. \n\\end{defn}\n\nNote that the $\\lambda$-scaled KP hierarchy for $\\lambda=(\\mu^i)$ for $\\mu\\in{\\mathbb C}^*$ coincides with the KP hierarchy. However, this does not happen in general. \n\nThe following Lemma is the key point to go from Virasoro to KdV. \n\n\\begin{lem}\\label{lem:Diff1scaleW}\nThe map $\\beta$ of \\eqref{eq:betaDiffEnd} induces a bijection between:\n\\begin{itemize}\n\t\\item the set of $ \\sigma\\in\\Hom_{\\text{Lie-alg}}({\\mathcal  W}_>,\\operatorname{Diff}^1({\\mathbb C}((z))))$ such that there exist $s\\in {\\mathbb C}$\n\tsatisfying:\n\t$$\n\t \\sigma(L_i)= \\frac12 z^{-2i}\\big(z \\partial_z + \\frac{1-2i}{2}\\big)+ s z^{-2i-3}  \n\t     $$\n\t\\item the set of scale equivalence classes of $\\rho \\in \\homlie({\\mathcal W}_>,\\End(\\Vkdv))$ whose coefficients of quadratic terms in $\\rho(L_{-1})$ do not vanish and such that  $\\rho(L_i)$  is of type $i$ for $i\\geq -1$.\n\t\\end{itemize}\n\\end{lem}\n\n\\begin{proof}\nFirst, we prove the statement with no reference to $r(z)$ on the first item and with no mention to a linear function on the second item. Under these circumstances,  given $\\sigma$ as in the statement, Lemma~\\ref{lem:DinDiffistypei} shows that $(\\beta\\circ\\sigma)(L_i) \\vert_{\\Vkdv}$ takes values in $\\Vkdv$ and it is of type $i$ for all $i$. An explicit computation yields:\n\t$$\n\t\\begin{aligned}\n    (\\beta\\circ\\sigma)(L_{-1})  \\,& = \\,\n    s \\frac{\\partial}{\\partial t_{1}}+\n    \\frac14 t_1^2 +\n    \\frac12 \\sum_{j=1}^{\\infty} j t_{j+2} \\frac{\\partial}{\\partial t_{j}}\n\\\\\n    (\\beta\\circ\\sigma)(L_{0}) & \\,= \\,\n     s \\frac{\\partial}{\\partial t_{3}} +\n    \\frac12 \\sum_{j=1}^{\\infty} j t_{j} \\frac{\\partial}{\\partial t_{j}}\n    \\\\\n    (\\beta\\circ\\sigma)(L_{i}) & \\,= \\,\n    s \\frac{\\partial}{\\partial t_{2i+3}} +\n    \\frac14  \\sum_{j=1}^{2i-1} \\frac{\\partial}{\\partial t_{j}} \\frac{\\partial}{\\partial t_{2i-j}}+\n    \\frac12 \\sum_{j=1}^{\\infty} j t_{j} \\frac{\\partial}{\\partial t_{2i-j}}\n        \\end{aligned}\n    $$\n(where $j$, as usual, is odd) and thus:\n    $$\n    [(\\beta\\circ\\sigma)(L_{-1}), (\\beta\\circ\\sigma)(L_{1}) ]\n    \\; - \\beta({[\\sigma(L_{-1}), \\sigma(L_{1})]})\n    \\,=\\,\n    -\\frac18\n    $$\nwhich implies that we have a map of Lie algebras defined by:\n    \\begin{equation}\\label{eq:rhoWK}\n    \\rho(L_i)\\,:=\\, (\\beta\\circ\\sigma)(L_i)+ \\frac1{16}\\delta_{i,0}\n    \\end{equation}\n\nConversely, let us start with $\\rho$ as in the second set of the statement. The assumptions yield the following expression:\n\t$$\n\t\\rho(L_{-1})\\, = \\,   b_{-1}^{0,1} \\frac{\\partial}{\\partial t_{1}} +  a  t_1^2   + b_{-1}^{i+2,i}  t_{i+2} \\frac{\\partial}{\\partial t_{i}}\n\t$$\nwith $ a, b_{-1}^{i+2,i}  \\neq 0$. Considering the action of $\\prod_{i \\text{ odd}}{\\mathbb C}^*$ by conjugation, one finds $\\lambda = \\{\\lambda_i \\in{\\mathbb C}^* \\vert i\\text{ odd}\\}$ and $s\\in {\\mathbb C}$ such that:\n\t$$\n\t\\rho^{\\lambda}(L_{-1}) \\, =\\, \\lambda \\circ \\rho(L_{-1}) \\circ \\lambda^{-1}\n\t\\,=\\,\n\ts \\frac{\\partial}{\\partial t_{1}} +  \\frac14 t_1^2   + (\\frac{i+2}2) t_{i+2} \\frac{\\partial}{\\partial t_{i}}\n\t$$\n\nLemma \\ref{lem:sigma} and the previous discussion show that $\\rho^{\\lambda}$ is the representation associated to the map  $\\sigma : {\\mathcal W}_>\\to \\operatorname{Diff}^1({\\mathbb C}((z)))$ defined by:\n\t$$\n\t\\sigma(L_i)= \\frac12 z^{-2i}(z\\partial_z + \\frac{1-2i}{2})+ s z^{-2i-3}\n\t\\qquad \\forall i\\geq -1\n\t$$\n\\end{proof}\n\n\\begin{rem}\\label{rem:VirtoKdV}\nThe statement can be generalized. On the one hand, we may consider the conjugation of $\\sigma$ by an operator of the type  $\\exp(r(z))$ while, on the other hand, we replace $\\rho$ by its conjugate by $\\exp(\\beta(r(z)))$. For instance, for $r(z)\\in {\\mathbb C}[[z^2]]$, one has that $\\beta(r(z))$ is a linear function on $t_1,t_3,\\ldots$. Thus, the first representation is:\n \t$$\n\t \\sigma(L_i)= \\frac12 z^{-2i}\\big(z(- r(z)+\\partial_z) + \\frac{1-2i}{2}\\big)+ s z^{-2i-3}  \n\t     $$\nwhile  $\\rho$ is as in the statement up to a linear function on $t_i$'s. \n\\end{rem}\n\n\\begin{rem}\nIt is worth noticing that the Virasoro operators studied by Witten (\\cite{Witten}) correspond to the case $s=-\\frac12$, $r(z)=0$. Kac-Schwarz (\\cite{KacSchwarz}), using the fact the these operators come from a representation in $\\operatorname{Diff}^1({\\mathbb C}((z)))$, proved that there is a point in the Sato Grassmannian whose $\\tau$-function is a solution of these equations and, hence, is a solution of KdV hierarchy too.  A study of common solutions of Virasoro-like constraints and KdV has been carried out in \\cite{Plaza-AlgSol}. \n\\end{rem}\n\n\\begin{lem}\\label{lem:uniquenesTauVirasoro}\nLet $\\rho$ be as in Lemma~\\ref{lem:Diff1scaleW} and let $\\tau(t)\\in \\Vkdv={\\mathbb C}[[t_1,t_3,\\ldots]]$. Then, the Virasoro constraints:\n\t$$\n\t\\rho(L_k)(\\tau(t)) \\,=\\, 0\\qquad k\\geq -1\n\t$$\nwith the initial condition $\\tau(0)=1$ admits no solution for $s=0$ and at most one solution for $s\\neq 0$. \n\\end{lem}\n\n\\begin{proof}\nSince $\\tau(0)=1$, let us consider the problem in terms of a formal function $F(t)\\in \\Vkdv= {\\mathbb C}[[t_1,t_3,\\ldots]]$ with $F(0)=0$ and $\\tau(t)=\\exp(F(t))$. The function $F(t)$ has a series expansion:\n    \\begin{equation}\\label{eq:F=generating}\n    F(t)\\,=\\, \\sum_{\\bf n} f_{\\bf n} {\\bf t}^{\\bf n}\n        \\end{equation}\nwhere ${\\bf n}:=\\{n_1,n_3,\\ldots\\}$ is a sequence of non-negative integers such that $n_i=0$ for all $i\\gg 0$, $f_{\\bf n}\\in {\\mathbb C}$ and ${\\bf t}^{\\bf n}:=\\prod_{i\\geq 1} t_i^{n_i}$. Further, the topology of $\\Vkdv={\\mathbb C}[[t_1,t_3,\\ldots]]$ comes from the definition $\\deg(t_i)=i$. In particular, the degree of ${\\bf t}^{\\bf n}$ is given by $\\vert {\\bf n}\\vert:= \\sum_{i\\geq 0} i n_i$.\n\nFor the sake of brevity, let us denote by  $f_{n_1 n_3\\ldots n_k} = f_{\\bf n}$ for ${\\bf n}=\\{n_1,n_3,\\ldots\\}$ with $n_k\\neq 0$ and $n_i=0$ for all $i>k$ and we set $f_0=F(0)=0$. As a brief summary, let us write down the monomials and their coefficients up to degree $5$:\n    {\\small\n    $$\n    \\begin{array}{ccccccc}\n    \\text{degree} & 0 & 1 & 2 & 3 & 4 & 5\n    \\\\\n    \\text{monomials} & 1 & t_1 & t_1^2 & t_1^3 , t_3 & t_1^4 , t_1t_3 &\n    t_1^5 , t_1^2 t_3 , t_5\n    \\\\\n    \\text{coefficient} & f_0 & f_1 & f_2 & f_3, f_{01} & f_4, f_{11} &\n    f_5, f_{21}, f_{001}\n    \\end{array}\n    $$}\n\nAfter rescaling $t_i$'s and conjugation by an exponential, if needed, we may assume that $\\rho$ is given by \\eqref{eq:rhoWK}. The hypothesis $\\rho(L_k)(\\tau(t)) =0$ is equivalent to the vanishing of the corresponding homogeneous parts of degree $i$ for $i=0,1,2,\\ldots$. An explicit computation for low values of $k$ and $i$ yields:\n\t    $$\n    \\begin{array}{ccc}\n    k &\\quad  i \\quad& \\text{part of degree $i$ in }\\rho(L_k)(\\tau(t))\n    \\\\\n    -1 & 0 & s f_1 \n    \\\\\n    -1 & 1 & 2s f_2 t_1 \n    \\\\\n    -1 & 2 & 3s f_3 t_1^2 + \\frac12 t_1^2 \n    \\\\\n    -1 & 3 & s\\big(4 f_4 t_1^3 +f_{13} t_3\\big) + t_3 f_1  \n    \\\\\n    0 & 0 & s f_{01}+\\frac1{16} \n    \\\\\n    0 & 1 &  s f_{11} t_1 + t_1 f_1  \n    \\\\\n    0 & 2 & s f_{21} t_1^2 +2 f_2 t_1^2  \n    \\\\\n    1 & 0 & s f_{001} + \\frac12 f_2 +\\frac14 f_1^2  \n    \\\\\n    1 & 1 & s f_{101} + \\frac32 f_3 t_1 + f_{11} t_1  \n    \\end{array}\n    $$\nThus, it is clear that if a solution $F$ does exist, then $s\\neq 0$. In this case,  the vanishing of the above polynomials implies that  $f_1=0$, $f_2=0$, $f_3=-\\frac1{3! s}$, $f_{01}=-\\frac1{16 s}$, $f_{11}= 0$, $f_4=0$, $f_{11}= 0$, etc.~. Writing down the general expression for the homogeneous part of degree $i$ of $\\rho(L_k)(\\tau(t))$, one observes that it allows us to determine $f_{\\bf n}$ with $\\vert n\\vert =i$ and $n_k\\neq 0$ in terms of $f_{\\bf n}$ with $\\vert n\\vert \\leq i-2$. Thus, if a solution $F$ exists, the coefficients $f_{\\bf n}$ can be recursively determined.\n\n\n\\end{proof}\n\n\n\n\n\n\n\\begin{thm}\\label{thm:solutionWscaleKdV}\nLet $\\rho \\in \\homlie({\\mathcal W}_>,\\End({\\mathbb C}[[t_1,t_3,\\ldots]]))$ be such that $\\rho(L_k)$  is of type $k$ for $k\\geq -1$ and that all coefficients of $\\rho(L_{-1})$ are non zero. \n\nThen, there exists a unique $\\tau(t)\\in{\\mathbb C}[[t_1,t_3,\\ldots]]$, with $\\tau(0)=1$, such that:\n\t$$\n\t\\rho(L_k)(\\tau(t))\\,=\\,0\\qquad k\\geq -1\n\t$$\nFurther, the solution $\\tau(t)$ is a $\\tau$-function of the scaled KdV hierarchy.\n\\end{thm}\n\n\\begin{proof}\nLemma~\\ref{lem:Diff1scaleW} implies that there is $\\lambda$ and $\\sigma:{\\mathcal W}_>\\to \\operatorname{Diff}^1({\\mathbb C}((z)))$ such that $\\rho^{\\lambda}=\\beta_*(\\sigma)$.  Recalling  Theorem~3.12  of~\\cite{Plaza-AlgSol}, one knows that there is a function $\\tau_0(t)$ which satisfies that $\\rho^{\\lambda}(L_n)(\\tau_0(t))=0$ and that it is a $\\tau$-function of the KdV hierarchy. Then, $\\tau(t):=\\tau_0(\\lambda t)$ fulfills the requirements. Since Lemma~\\ref{lem:uniquenesTauVirasoro} implies the uniqueness of the solution, the conclusion follows. \n\n\\end{proof}\n\n\n\\begin{rem}\nLet us make two comments on the solutions. First, an instance of the notion of scaled KdV appears already in Kontsevich's Theorem when it is claimed that the exponential of the generating function in variables $T_{2i+1}:=t_i\/(2i+1)!!$ is a $\\tau$-function for the KdV hierarchy (\\cite[Theorem 1.2]{Kon}). On the other hand, although the  dilation shift $\\bar t_{i}\\mapsto \\bar t_{i}-\\delta_{i,0}$ transforms the operators $\\rho(L_k)$, it should be noted that it does not induce an automorphism of the algebra ${\\mathbb C}[[\\bar t_0,\\bar t_1,\\ldots]]$.\n\\end{rem}\n\n\n\n\n\n\\subsection{On the solutions for the $n$-dimensional situation}\n\nLet us now focus in the $n$-dimensional situation. That is, we aim at studying the interplay between Virasoro representations and multicomponent KP hierarchy. Special attention will be paid at their common solutions. \n\nRecall that $\\Vkdv(A)$ is the subalgebra of ${\\mathbb C}[[t_1,t_3,\\ldots]]\\widehat\\otimes_{{\\mathbb C}} S^\\bullet A$ generated by $t_i\\otimes a$. Then, $S\\in\\Gl(A)$ acts on it by the automorphism of algebras $t_i\\otimes a\\mapsto t_i\\otimes S(a)$. \n\n\\begin{thm}\\label{thm:solutionproductKdV}\nLet $\\rho:{\\mathcal W}_>\\to \\uh[(A)]$ be as in \\S\\ref{subsec:baby}.\n\nThere exist $S\\in \\Gl(A)$ and functions $\\tau_\\alpha(t_{1,\\alpha},t_{3,\\alpha},\\ldots)\\in {\\mathbb C}[[t_{1,\\alpha},t_{3,\\alpha},\\ldots]]$ such that:\n\t\\begin{equation}\\label{eq:scaleequivmultiKP}\n\t\\hat\\rho(L_k)\\big(S( \\prod_\\alpha \\tau_\\alpha( t_\\alpha))\\big)\\,=\\, 0\n\t\\end{equation}\n\nFurther, $\\tau_\\alpha(t_{1,\\alpha},t_{3,\\alpha},\\ldots)$ are $\\tau$-functions of the scaled KdV hierarchy.\n \\end{thm}\n\n\\begin{proof}\nTheorem~\\ref{thm:Virasorodecomp} shows that there is $S\\in\\Gl(A)$ such that $\\rho^S$ decomposes as the tensor product of $n$ $1$-dimensional Lie algebra representations of ${\\mathcal W}_>$. More precisely,  if $\\{a_\\alpha\\}$ is the chosen basis for $A$, then $\\{S(a_\\alpha)\\}$ is a orthogonal basis for $\\eta$. Consequently, there are $\\rho_\\alpha: {\\mathcal W}_>\\to \\uh[(<S(a_\\alpha)>)]$ such that \\eqref{eq:rho=sumrhoalpha} holds. \n\nNow, apply the results of \\S\\ref{subsec:solutions1dim} on the $1$-dimensional case. Indeed, since $\\eta$ is non-degenerated and $\\{S(a_\\alpha)\\}$ is a orthogonal basis,  from Theorem~\\ref{thm:solutionWscaleKdV} one obtains functions $\\tau_\\alpha(t_\\alpha)$, such that $\\tau_\\alpha(0)=1$,  $\\rho_\\alpha(L_k)(\\tau_\\alpha)=0$ for all $\\alpha,k$ and they are  $\\tau$-functions for the scaled KdV hierarchy. \n\nObserve that \\eqref{eq:scaleequivmultiKP} holds if and only if $\\hat\\rho^S(L_k)( \\prod_\\alpha \\tau_\\alpha( t_\\alpha))$ vanishes. Applying the converse of Theorem~\\ref{thm:solutionrhoi} one concludes. \n\\end{proof} \n\n\n\n\\begin{rem}\nThe previous Theorem means that, assuming the uniqueness of the solution (\\cite[Theorem 3.10.20]{DubrovinZ}), the solution of the Virasoro constraints has to be of the above form; that is, an operator acting on a product of Witten-Kontsevich $\\tau$-functions. Thus, it agrees with the results of Givental (\\cite{Givental}) for the total descendent potential. It would be interesting to relate both expressions explicitly (see also \\cite{FvLS, GiventalAn,Lee}). Alternatively, one could combine Teleman's classification of semi simple cohomological field theories (\\cite{Teleman}) with Givental's results to deduce that this is the right expression for the solution. Nevertheless our result can be applied on other frameworks, as it will mentioned in \\S\\ref{subsec:final}).\n\\end{rem}\n\n\n\n\n\n\\begin{cor}\nLet $\\rho$ be as in the Theorem~\\ref{thm:solutionproductKdV}. \n\nIf $S, \\tau_\\alpha$ satisfy \\eqref{eq:scaleequivmultiKP}, then $\\rho^S=\\rho_1+\\ldots+\\rho_n$ and $\\hat\\rho_\\alpha(L_k)(\\tau_\\alpha)=0$. \n\nThe matrix $S$ is unique up to an ortogonal matrix. \n\\end{cor}\n\n\\begin{proof}\nIf $S$ and $\\tau_\\alpha$ are such that \\eqref{eq:scaleequivmultiKP} vanishes, then the following expression also vanishes:\n\t$$\n\t0\\,=\\, \n\t{\\exp(-\\sum_\\alpha \\tilde\\tau_\\alpha(t_\\alpha))} S^{-1} \\hat\\rho(L_k)\\big(S( \\prod_\\alpha \\tau_\\alpha( t_\\alpha))\\big) \n\t\\,=\\, \n\t\\frac{ \\hat \\rho^S(L_k)(\\exp(\\sum_\\alpha \\tilde\\tau_\\alpha(t_\\alpha)))} {\\exp(\\sum_\\alpha \\tilde\\tau_\\alpha(t_\\alpha))}\n\t$$\nRecall that an operator $\\rho^S(L_k)$ of type \\eqref{eq:type} is the same as $\\rho(L_k)$  where the matrix $a$  has been replaced by $(S^{-1})^T a S^{-1}$ (and, accordingly, $b_k$, $c_k$, etc.). Expanding the case $k=-1$ of the last identity, one obtains that $(S^{-1})^T \\eta S^{-1}$ is diagonal. Then, Theorem~\\ref{thm:rhodecomp}, implies that $\\rho^S$ decomposes as a sum $\\rho_1+\\ldots+\\rho_n$ and Theorem~\\ref{thm:solutionrhoi} implies that $\\hat\\rho_\\alpha(L_k)(\\tau_\\alpha(t_\\alpha))=0$. \n\nIt is straightforward that $S$ is unique up to an ortogonal matrix.\n\\end{proof} \n\n\n\n\\begin{cor}\nLet $\\rho$ be as in the Theorem~\\ref{thm:solutionproductKdV}.  If either $S$ is diagonal or $\\tau_\\alpha$ are $\\tau$-functions of the same scaled hierarchy, then the solution is a $\\tau$-function for the scaled multicomponent KP hierarchy. \n\\end{cor}\n\n\\begin{proof}\nIn particular, the product $S( \\prod_\\alpha \\tau_\\alpha( t_\\alpha))$ is uniquely determined by $\\rho$. Each function $\\tau_\\alpha( t_\\alpha)$ satisfies the scaled KdV and, thus, there are $\\lambda_\\alpha:=(\\lambda_{i,\\alpha})\\in\\prod_{i\\text{ odd}}{\\mathbb C}^*$ such that $\\tau_\\alpha( \\lambda_\\alpha^{-1} t_\\alpha)$ defines a point $U_\\alpha\\in \\operatorname{Gr}({\\mathbb C}((z)))$. If $\\rho$ is expressed w.r.t. a basis $\\{a_1,\\ldots , a_n\\}$, then $S$ determines a second basis $\\{S(a_1),\\ldots, S(a_n)\\}$ or, equivalently, an isomorphism  ${\\mathbb C}\\oplus\\ldots \\oplus{\\mathbb C}\\simeq A$. This isomorphism induces:\n\t{\\small $$\n\t\\operatorname{Gr}({\\mathbb C}((z)))\\times \\ldots\\times \\operatorname{Gr}({\\mathbb C}((z))) \\hookrightarrow \n\t\\operatorname{Gr}({\\mathbb C}((z))\\oplus\\ldots\\oplus{\\mathbb C}((z))) \n\t\\simeq \\operatorname{Gr}(A\\otimes {\\mathbb C}((z)))\n\t$$}\nSince $\\tau$-function of the image of $(U_1,\\ldots, U_n)$, which is $U_1\\oplus\\ldots\\oplus U_n$, is given by $\\prod_\\alpha \\tau_\\alpha(\\lambda_\\alpha^{-1} t_\\alpha)$ it follows that $S \\prod_\\alpha \\tau_\\alpha(t_\\alpha)$ is a $\\tau$-function of the scaled multicomponent KP in the two cases of the statement.\n\\end{proof}\n\n\n\\begin{rem}\nRecalling Remark~\\ref{rem:VirtoKdV}, we observe that Theorems~\\ref{thm:solutionWscaleKdV} and~\\ref{thm:solutionproductKdV} could be weaken and stated for representations satisfying the hypothesis up to a linear function on $t_i$'s.\n\\end{rem}\n\n\n\n\\subsection{Final Comments}\\label{subsec:final}\n\nLet us finish with some brief comments. From a general perspective, we hope that our methods shed some light on the explicit expressions of the Virasoro operators and of the relevant integrable hierarchies that appear in   the Virasoro conjecture.  Furthermore, they can also be applied to many instances of representations of ${\\mathcal W}_>$ such as recursion relations, Hurwitz numbers, knot theory, etc.~. \n\nAs an illustration, let us point out the results of \\cite{AmbjornChekhov,KazarianZograf} on Hurwitz numbers. In both cases, the authors study the generating functions of the number of coverings of ${\\mathbb P}_1\\setminus\\{0,1,\\infty\\}$ with some properties. It is shown that these functions satisfy Virasoro constraints, KP hierarchy and topological recursion (of the Eynard-Orantin type \\cite{EO}). It is remarkable that the Virasoro constraints are explicitly expressed as differential operators of the form considered in \\S\\ref{sec:ReprW>} for the case $A={\\mathbb C}$. Thus, the results of \\S\\ref{subsec:solutions1dim} can be directly applied to conclude that Virasoro constraints imply the scaled KP hierarchy. \n\n\nOur results could also be of interest within the context of Eynard-Orantin topological recursion (\\cite{EO}). Indeed,  we learn from \\cite{MulaseSafnuk} that Mirzakhani's recursion formula for the Weil-Petersson volumes (\\cite{Mir}) is indeed a Virasoro constraint imposed on a generating function of these volumes and that this function satisfies the KdV hierarchy. It is worth pointing out some recent results on the relation of topological recursion and Virasoro constraints (\\cite{Eynard, Milanov}). On the one hand, it has been shown in \\cite{Eynard} that these Virasoro constraints are actually equivalent to Eynard-Orantin topological recursion for some spectral curve. On the other hand, one knows from \\cite{Milanov}  that the correlation functions of a semisimple cohomological field theory satisfy the Eynard-Orantin topological recursion and that these recursion formulas are equivalent to $n$ copies of the Virasoro constraints for the ancestor potential. Therefore, two problems can be faced with our techniques. First, we think that Theorem~\\ref{thm:solutionproductKdV} should imply  some bilinear relations of Hirota type for the solution of the Eynard-Orantin topological recursion. Second, due to the uniqueness of the solution and the fact that the solution satisfies the KP hierarchy, there must be a relation of the Eynard-Orantin spectral curve and the Krichever construction.  \n\nSimilarly, it would be interesting to interpret the recent papers \\cite{Da,DOSS} from our perspective. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\paragraph{}We have proposed to augment the standard model by a ``knot algebra\" that will define new ``elementary particles\".\n\nDirac \\cite{Dirac1} and Schwinger \\cite{Schwinger1} have both proposed adding hypothetical magnetic poles to the Quantized Maxwell Theory. We have shown how magnetic poles might be expected already in the context of the ``Knot Algebra\", where the field operators of the standard model acquire the factors $D^j_{mm'}(q,z)$, which are the irreducible representations of the quantum group $SL_q(2)$. This extension of the standard model is topological and could unlock magnetic poles. At the same time, because of advances in observational astronomy, the quantization of the gravitational field itself has become a topological challenge, ie, in most of the already explored universe, the influence of the gravitational field has been believed until recently to be too weak to be topologically significant. There is now, however, a growing opinion that different topologies may describe the consequences of earlier and major astronomical events, including gravitational collapse and nuclear explosions, as illustrated by the Gamow ``Big Bang\" \\cite{Hawking}. With the recent discovery of gravitational waves, physical spacetime may have become a field of speculation about its topological structure.\n\nSome of these topological possibilities are discussed by van der Bij\\cite{Bij1} in his review of recent experimental results at CERN.  It is also there pointed out that physical spacetime at earlier times may have been 3-dimensional\\cite{carlip} before entering its present phase of physical expansion. Both the new topologies and the new possibilities offered by a hypothetical earlier 2-d space were discussed by us in earlier works describing the knot model\\cite{Fink1}. \n\n\n\\section{Knot Model \\cite{Fink1} \\cite{finkelstein14a} }\n\\paragraph{} The original purpose of the knot algebra was to provide additional degrees of freedom to the Standard Model. Since the Knot Model associates a classical knot (N, w, r) with every quantum ``knot state\" $(j, m, m')$ of the standard model, it is necessary to add an odd number, o, to a defining equation of the ``knot algebra\" as follows:\n\\begin{equation}\n    \\boxed{(j, m, m')_q = \\frac{1}{2}(N, w, r+o)}\n    \\label{knotAssociation}\n\\end{equation}\nwhere w and r are of opposite parity while $m$ and $m'$ are of the same parity. Here q is the deformation parameter of $SL_q(2)$\n\nWe postulate that the quantum states of the ``new standard model\" $(j, m, m')_q$ are restricted by \\ref{knotAssociation}.\n\nThe Noether charges of the associated simplest classical knots are then\n\\begin{empheq}{align}\n    Q_w \\equiv -k_wm \\equiv -k_w\\frac{w}{2} \\label{qwdef} \\\\\n    Q_r \\equiv -k_rm' \\equiv -k_r\\frac{r+1}{2}  \\label{qrdef}\n\\end{empheq}\nwhere we set o = 1 for the simplest knots. The $k_w$ and $k_r$ are themselves undetermined constants that determine the writhe and rotation charges. The total Noether charge of the simplest knots satisfying (2.2) and (2.3) is then\n\\begin{equation}\n    Q = -k_w\\frac{w}{2}-k_r\\frac{r+1}{2}\n\\end{equation}\n\nThe association of classical knots with leptons and quarks is somewhat similar to the first association of classical knots by Bohr with the limiting states of the quantized Hydrogen atom. \n\nThere is in addition an empirical relation between the isotopic spin $(t, t_3, t_0)$ of elementary fermions and the knot characterization of the trefoils $(N, w, r)$, as shown in Table 1.\n\n\\renewcommand{\\arraystretch}{2}\n\n\\begin{figure}[H]\n\\begin{tabular}{r c r r r | p{14mm} c r r c | r}\n\\multicolumn{10}{c}{\\textbf{Table 1:} Empirical Support for $6(t,-t_3,-t_0) = (N,w,r+1)$} \\\\ \n\\hline \\hline\n & Elementary Fermions & $t$ & $t_3$ & $t_0$ & Classical Trefoil & $N$ & $w$ & $r$ & $r+1$ & $D^{N\/2}_{\\frac{w}{2}\\frac{r+1}{2}}$ \\\\[0.2cm]\n\\hline\n\\multirow{2}{*}{\\hspace{-4pt}leptons $\\Bigg \\{$} & $(e, \\mu, \\tau)_L$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ & & 3 & 3 & 2 & 3  & $D^{3\/2}_{\\frac{3}{2}\\frac{3}{2}}$\\\\\n& $(\\nu_e, \\nu_{\\mu}, \\nu_{\\tau})_L$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & & 3 & $-3$ & 2 & 3 & $D^{3\/2}_{-\\frac{3}{2}\\frac{3}{2}}$\\\\\n\\multirow{2}{*}{quarks $\\Bigg \\{$} & $(d, s, b)_L$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{6}$ & & 3 & 3 & $-2$ & $-1$ & $D^{3\/2}_{\\frac{3}{2}-\\frac{1}{2}}$\\\\\n& $(\\Bar{u}, \\Bar{c}, \\Bar{t})_L$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{6}$ & & 3 & $-3$ & $-2$ & $-1$ & $D^{3\/2}_{-\\frac{3}{2}-\\frac{1}{2}}$\\\\ [0.2cm]\n\\hline\n\\end{tabular}\n\n\\caption*{{The symbols $(\\quad)_L$ designate the left chiral states in the usual notation. The topological labels $(N,w,r)$ on the right side of the table provide a natural way to label the same chiral states. Note that $D^{3\/2}_{\\frac{w}{2}\\frac{r+1}{2}}$, which labels the chiral states, correlates with the elementary fermions on the left.}}\n\\end{figure}\n\nThe chiral states $(N, w, r+1)$ which may be described as 2d projections of 3d trefoils, while the corresponding $(t, -t_3, -t_0)$ states, which may be described as 2d projections of quark states of spin, together suggest the possible relevance of an early two dimensional space (for which there may be independent evidence\\cite{carlip}).\n\nThere is also the empirical relation \\ref{empirical}, which is shown in Tables 1 and 2.\n\n\\begin{equation}\n    (j, m, m')_q = 3(t, -t_3, -t_0) \\label{empirical}\n\\end{equation}\n\nOnly for the particular row-to-row correspondences shown in Table 1 does \\ref{empirical} hold, i.e., \\emph{each of the four families of fermions labelled by $(t_3, t_0)$ is uniquely correlated with a specific $(w,r)$ classical knot, and therefore with a specific state $D^{3\/2}_{\\frac{w}{2}\\frac{r+1}{2}}(q)$ of the quantum knot.}\n\nNote in Table 1 that the $t_3$ doublets of the standard model now correspond to the writhe doublets ($w = \\pm 3$). Note also that with this same correspondence the leptons and quarks form a knot rotation doublet ($r = \\pm 2$); the lepton-quark relation depends on $(w,r)$.\n\nRetaining the row to row correspondence described in the Tables, it is then possible to compare in Table 2 the electroweak charges, $Q_e$, \\emph{of the most elementary fermions with the total Noether charges, $Q_w + Q_r$, of the simplest quantum knots, which are the quantum trefoils.}\n\n\\begin{center}\n\\begin{tabular} {c r r r c | c l c c c}\n\\multicolumn{10}{c}{{\\textbf{Table 2:} Electric Charges of Leptons, Quarks, and Quantum Trefoils}} \\\\\n\\hline \\hline\n\\multicolumn{5}{c |}{{Standard Model}} & \\multicolumn{5}{c}{{Quantum Trefoil Model}} \\\\\n\\hline\n{$(f_1, f_2, f_3)$} & {$t$} & {$t_3$} &{$t_0$} & {$Q_e$} & {$(N,w,r)$} & {$D^{N\/2}_{\\frac{w}{2}\\frac{r+1}{2}}$} & {$Q_w$} & {$Q_r$} & {$Q_w + Q_r$} \\\\ [0.2cm]\n\\hline\n$(e, \\mu, \\tau)_L$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ & $-e$ & $(3,3,2)$ & $D^{3\/2}_{\\frac{3}{2} \\frac{3}{2}}$ & $-k_w \\left( \\frac{3}{2} \\right)$ & $-k_r \\left( \\frac{3}{2} \\right)$ & $-\\frac{3}{2}(k_r + k_w)$ \\\\\n$(\\nu_e, \\nu_{\\mu}, \\nu_{\\tau})_L \\hspace{-10pt}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $0$ & $(3,-3,2)$ & $D^{3\/2}_{-\\frac{3}{2} \\frac{3}{2}}$ & $-k_w \\left( -\\frac{3}{2} \\right)$ & $-k_r \\left( \\frac{3}{2} \\right)$ & $\\frac{3}{2}(k_w - k_r) $ \\\\\n$(d,s,b)_L$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{6}$ & $-\\frac{1}{3}e$ & $(3,3,-2)$ & $D^{3\/2}_{\\frac{3}{2} -\\frac{1}{2}}$ & $-k_w \\left( \\frac{3}{2} \\right)$ & $-k_r \\left( -\\frac{1}{2} \\right)$ & $\\frac{1}{2}(k_r - 3k_w)$ \\\\\n$(u,c,t)_L$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{6}$ & $\\frac{2}{3}e$ & $(3,-3, -2)$ & $D^{3\/2}_{-\\frac{3}{2} -\\frac{1}{2}} \\hspace{-5pt}$ & $-k_w \\left( -\\frac{3}{2} \\right)$ & $-k_r \\left( -\\frac{1}{2} \\right)$ & $\\frac{1}{2}(k_r + 3k_w)$ \\\\\n& & \\multicolumn{3}{c |}{$ \\hspace{-8pt} Q_e = e(t_3+t_0) \\hspace{-10pt}$} & \\multicolumn{2}{c}{\\normalsize $(j, m, m') = \\frac{1}{2}(N, w, r + 1)$} & $Q_w = -k_w \\frac{w}{2} \\hspace{-5pt}$ & $Q_r = -k_r \\frac{r+1}{2} \\hspace{-15pt}$ & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\n\\emph{One sees that \\boxed{Q_w + Q_r = Q_e} is satisfied for charged leptons, neutrinos and for both up and down quarks with only a single value of k as follows:}\n\\begin{equation}\n\\boxed{k_r = k_w (= k)=\\frac{e}{3}} \\label{3k}\n\\end{equation}\nand also that $t_3$ isospin and $t_0$ hypercharge then respectively measure the writhe and rotation charges of the associated classical knot:\n\\begin{equation}\nQ_w = et_3 \\label{qwet3}\n\\end{equation}\n\\begin{equation}\nQ_r = e t_0 \\label{qret0}\n\\end{equation}\nThen $Q_w + Q_r = Q_e$ becomes by \\eqref{qwet3} and \\eqref{qret0} an alternative statement of\n\\begin{equation}\nQ_e = e(t_3 + t_0)\n\\end{equation}\nof the standard model. It is important to note that this classification of fundamental particles in Tables 1 and 2 is made by \\emph{charge and not by mass}\n\nIn SLq(2) measure $Q_e = Q_w + Q_r$ is by \\eqref{qwdef} and \\eqref{qrdef}:\n\\begin{equation}\n\\boxed{Q_e = -\\frac{e}{3}(m+m'), \\label{qeem}}\n\\end{equation}\nor by (2.3)\n\\begin{equation}\nQ_e = - \\frac{e}{6}(w+r+1). \\label{qeewr}\n\\end{equation}\nfor the quantum trefoils, that represent the elementary fermions.\n\n\nWe can denote the fundamental two dimensional representation of $(j, m, m')_q$ by\n\n\\begin{equation}\nD^{\\frac{1}{2}}_{mm'}(q) = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\label{jhalfrep}\n\\end{equation}\n\nIn the physical model with the \\eqref{jhalfrep} coupling we interpret $(a,b,c,d)$ as creation operators for $(a,b,c,d)$ particles, which we term preons. Then we assume that $D^j_{mm'}(a,b,c,d)$ is the creation operator for the state representing the superposition of $(n_a, n_b, n_c, n_d)$ preons. Since we shall regard the preons as fermions, they will also carry an anti-symmetrizing index to satisfy the Pauli principle. \n\nHere we may additionally generalize by describing some aspects of a generic field theory where the field quanta have two couplings that may be expressed in the coupling matrix\n\\begin{equation}\n\\boxed{\n\\varepsilon_q = \\begin{pmatrix}\n0 & \\alpha_2 \\\\\n-\\alpha_1 & 0\n\\end{pmatrix}. \\cite{FinkSchwing}} \\label{couplingmatrix}\n\\end{equation}\nwhere the couplings $\\alpha_1$ and $\\alpha_2$ are assumed to be dimensionless and real and may be written as\n\\begin{equation}\n(\\alpha_1, \\alpha_2) \\text{ or } (\\alpha_2, \\alpha_1) = \\left( \\frac{e}{\\sqrt{\\hbar c}}, \\frac{g}{\\sqrt{\\hbar c}} \\right) \\label{couplingconstants}\n\\end{equation}\nwhere $e$ and $g$ refer to a specific two charge model and have dimensions of an electric charge. We assume that e and g may be energy dependent and normalized at relevant energies. The reference charge is the universal constant $\\sqrt{\\hbar c}$. \n\nThe fundamental assumption that we make on this coupling matrix is that it is invariant under SLq(2) as follows\n\\begin{equation}\nT \\varepsilon_q T^t = T^t \\varepsilon_q T = \\varepsilon_q \\label{slq2invariant}\n\\end{equation}\nwhere $t$ means transpose and $T$ is a two dimensional representation of SLq(2):\n\\begin{equation}\nT = \\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} . \\label{2drepslq2}\n\\end{equation}\nwhere the elements of $T$ obey the knot algebra:\n\\begin{equation}\n\\begin{split}\nab = qba \\qquad bd = qdb \\qquad ad-qbc = 1 \\qquad bc &= cb \\\\\nac = qca \\qquad cd = qdc \\qquad da -q_1cb = 1 \\qquad q_1 &\\equiv q^{-1}\n\\end{split}\n\\tag{A}\n\\end{equation}\nand\n\\begin{equation}\n\\boxed{ q = \\frac{\\alpha_1}{\\alpha_2}}\n\\end{equation}\nso that the two physical couplings fix the algebra through their ratio.\n\nIf also\n\\begin{equation}\n\\text{det} \\hspace{2pt} \\varepsilon_q = 1 \\label{couplingdet}\n\\end{equation}\none has\n\\begin{equation}\n\\alpha_1 \\alpha_2 = 1\n\\end{equation}\nIf the two couplings $(\\alpha_1, \\alpha_2)$ are given by \\eqref{couplingconstants}, where $e$ and $g$ are the electroweak and ``gluon''-like couplings, or electric and magnetic couplings respectively, then\n\\begin{equation}\n\\boxed{eg = \\hbar c} \\label{quantize}\n\\end{equation}\n\nSince $g$ represents magnetic charge, \\eqref{quantize} copies the Dirac proposal according to which the magnetic charge is very much stronger than the electric charge. If the magnetic pole is very much heavier as well, it may be observable only at early and not at current cosmological temperatures or at currently achievable accelerator energies.\n\nWe may alternatively assume that q-magnetic poles do exist and shall study the possible extension of q-knot symmetry to magnetic charges, in particular as sources of the gravitational field, with \\emph{strength} q.\n\\begin{equation}\n    \\boxed{g = qe}    \n\\end{equation}\nby (2.17), where q is the deformation parameter of $SL_q(2)$, so that one may assume either (2.20) or (2.21), and $g\/e$ may be either $\\hbar c \/ e^2$ or $q$ or both: then $q = \\hbar c \/ e^2$. This new expression for q would directly relate Gamow cosmology to Bohr quantum mechanics, and would be required to be experimentally realized.\n\n\\section{Graphical Representation of Corresponding Classical Structures}\nThe representation of the four classical trefoils as composed of three overlapping preon loops is shown in Figure 1. In interpreting Figure 1, note that the two lobes of all the preon loops make opposite contributions to the rotation, $r$, so that the total rotation of each preon loop vanishes. When the three $a$-preons and $c$-preons are combined to form charged leptons and neutrinos, respectively, as suggested by Harari\\cite{harari79}, Shupe\\cite{Schupe1}, and Raitio\\cite{Raitio1} each of the three labelled circuits is counterclockwise and contributes $+1$ to the rotation while the single unlabeled and shared (overlapping) circuit is clockwise and contributes $-1$ to the rotation so that the total $r$ for both charged leptons and neutrinos is $+2$. In this way the leptons, neutrinos, up and down quarks may be considered composite massive particles that are \\emph{sources of the gravitational field}, and are here abreviated by a, b, c, d, in accord with the fundamental representation of $SL_q(2)$. \\vspace{-0.3em}\n\n\\newgeometry{bottom=2.5cm}\n\\begin{center}\n\\normalsize\n\\begin{tabular}{c | c} \n         \\multicolumn{2}{c}{} \\\\ [-0.54cm]\n\t\\multicolumn{2}{c}{\\textbf{Figure 1:} Preonic Structure of Elementary Fermions} \\\\ [-0.49cm]\n\t\\multicolumn{2}{c}{$Q = -\\frac{e}{6}(w+r+o)$, and $(j, m, m') = \\frac{1}{2} (N, w, r+o)$} \\\\ [-0.25cm]\n\t\\multicolumn{2}{c}{$D^j_{mm'} = D^{\\frac{N}{2}}_{\\frac{w}{2} \\frac{r+o}{2}}$}\\\\\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{r}{ \\ul{$(w, r, o)$}} \\\\ [-0.3cm]\n\t\t\\multicolumn{3}{l}{Charged Leptons, $D^{3\/2}_{\\frac{3}{2} \\frac{3}{2}} \\sim a^3$} \\\\ [0.2cm]\n\t\t \\Large{$\\varepsilon^{ijk}$} & $\\hspace{17pt} a_j$ & \\\\ [0.4cm]\n\t\t $\\hspace{28pt} a_i$ & & $\\hspace{-58pt} a_k$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{ \\xoverh }\n[u(1)] [l(0.5)]\n!{\\color{black} \\xbendu }\n[l(2)]\n!{\\vcap[2] =>}\n!{\\xbendd- \\color{black}}\n[d(1)] [r(0.5)]\n!{\\xunderh }\n[d(0.5)]\n!{ \\xbendr}\n[u(2)]\n!{\\hcap[2]=>}\n[l(1)]\n!{\\xbendl- }\n[l(1.5)] [d(0.25)]\n!{\\xbendl[0.5]}\n[u(1)] [l(0.5)]\n!{\\xbendr[0.5] =<}\n[u(1.75)] [l(2)]\n!{ \\xoverv}\n[u(1.5)] [l(1)]\n!{\\color{black} \\xbendr-}\n[u(1)] [l(1)]\n!{ \\hcap[-2]=>}\n[d(1)]\n!{ \\xbendl \\color{black}}}} \\\\ [-1.6cm]\n\\multicolumn{3}{l}{ \\hspace{33pt} \\textcolor{red}{\n\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(2.25)] [r(3.25)]\n!{\\xcapv[1] =>}\n[l(2.75)] [u(.75)]\n!{\\xbendr[-1] =>}\n[d(1)] [r(1.25)]\n!{\\xcaph[-1] =>}\n}}}\n \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{150pt}$(3,2,1)$}\n\t\t \n\t\\end{tabular}\n\t&\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{r}{\\ul{$(w, r, o)$}} \\\\ [-0.3cm]\n\t\t\\multicolumn{3}{l}{$a$-preons, $D^{1\/2}_{\\frac{1}{2} \\frac{1}{2}}$} \\\\ [1.58cm]\n\t\t  & &\\hspace{-63pt} $a_i$ \\\\ [-1.3cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverv=>}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}}} \\\\ [-1.2cm]\n\\multicolumn{3}{l}{\\hspace{39pt} \\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[d(0.75)] [l(0.75)]\n!{\\xcaph[-1]=>}\n}\n}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{150pt}$(1,0,1)$}\n\t\t \n\t\\end{tabular} \\\\ [-0.1cm] \\hline\n\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{Neutrinos, $D^{3\/2}_{-\\frac{3}{2} \\frac{3}{2}} \\sim c^3$} \\\\ [0.2cm]\n\t\t \\Large{$\\varepsilon^{ijk}$}& $\\hspace{17pt} c_j$ & \\\\ [0.4cm]\n\t\t $\\hspace{28pt} c_i$ & & $\\hspace{-58pt} c_k$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderh }\n[u(1)] [l(0.5)]\n!{\\color{black} \\xbendu}\n[l(2)]\n!{ \\vcap[2]=>}\n!{\\xbendd- \\color{black}}\n[d(1)] [r(0.5)]\n!{\\xoverh}\n[d(0.5)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]=>}\n[l(1)]\n!{\\xbendl-}\n[l(1.5)] [d(0.25)]\n!{ \\xbendl[0.5]}\n[u(1)] [l(0.5)]\n!{\\xbendr[0.5]=<}\n[u(1.75)] [l(2)]\n!{\\xunderv}\n[u(1.5)] [l(1)]\n!{\\color{black} \\xbendr-}\n[u(1)] [l(1)]\n!{\\hcap[-2]=>}\n[d(1)]\n!{\\xbendl \\color{black}}}} \\\\ [-1.5cm]\n\\multicolumn{3}{l}{\\textcolor{red} {\\hspace{32pt} \\xygraph{\n!{0;\/r1.3pc\/:}\n[u(2.25)] [r(3.25)]\n!{\\xcapv[1] =<}\n[l(2.75)] [u(.75)]\n!{\\xbendr[-1] =<}\n[d(1)] [r(1.25)]\n!{\\xcaph[-1] =<}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{145pt}$(-3,2,1)$}\n\t\t \n\t\\end{tabular}\n\t&\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$c$-preons, $D^{1\/2}_{-\\frac{1}{2} \\frac{1}{2}}$} \\\\ [1.58cm]\n\t\t  & &\\hspace{-63pt} $c_i$ \\\\ [-1.3cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderv=>}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}}} \\\\ [-1.2cm]\n\\multicolumn{3}{l}{\\textcolor{red}{\\hspace{39pt} \\xygraph{\n!{0;\/r1.3pc\/:}\n[d(0.75)] [l(0.75)]\n!{\\xcaph[-1]=<}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{145pt}$(-1,0,1)$}\n\t\t \n\t\\end{tabular} \\\\ [-0.1cm] \\hline\n\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$d$-quarks, $D^{3\/2}_{\\frac{3}{2} -\\frac{1}{2}} \\sim ab^2$} \\\\ [0.2cm]\n\t\t & $\\hspace{19pt} b$ & \\\\ [0.4cm]\n\t\t $\\hspace{28pt} a_i$ & & $\\hspace{-56pt} b$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverh }\n[u(1)] [l(0.5)]\n!{\\color{black} \\xbendu}\n[l(2)]\n!{\\vcap[2]=<}\n!{\\xbendd- \\color{black}}\n[d(1)] [r(0.5)]\n!{ \\xoverv}\n[u(0.5)] [r(1)]\n!{ \\xbendr }\n[u(2)]\n!{\\hcap[2]=<}\n[l(1)]\n!{\\xbendl- }\n[u(0.5)] [l(3)]\n!{\\xoverv}\n[u(1.5)] [l(1)]\n!{\\color{black} \\xbendr- }\n[l(1)] [u(1)]\n!{\\hcap[-2]=<}\n[d(1)]\n!{\\xbendl  \\color{black}}\n[r(2)] [u(0.5)]\n!{\\xcaph- =>}}} \\\\[-1.6cm]\n\\multicolumn{3}{l}{\\hspace{42pt} \\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(1.75)] [l(1.5)]\n!{\\xbendd[-1]=<}\n[u(0.75)] [r(1)]\n!{\\xbendl[-1]=<}\n[d(1)] [l(2.25)]\n!{\\xcaph[-1]=>}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{140pt}$(3,-2,1)$}\n\t\t \n\t\\end{tabular}\n\t&\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$b$-preons, $D^{1\/2}_{\\frac{1}{2} -\\frac{1}{2}}$} \\\\ [1.58cm]\n\t\t  & &\\hspace{-63pt} $b$ \\\\ [-1.3cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverv=<}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}}} \\\\ [-1.5cm]\n\\multicolumn{3}{l}{\\hspace{40pt} \\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(0.75)] [l(0.75)]\n!{\\xcaph[1]=>}\n}}} \\\\ [-0.1cm]\n\t\t \\multicolumn{3}{r}{\\hspace{145pt}$(1,0,-1)$}\n\t\t \n\t\\end{tabular} \\\\ [-0.1cm] \\hline\n\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$u$-quarks, $D^{3\/2}_{-\\frac{3}{2} -\\frac{1}{2}} \\sim cd^2$} \\\\ [0.2cm]\n\t\t & $\\hspace{19pt} d$ & \\\\ [0.4cm]\n\t\t $\\hspace{28pt} c_i$ & & $\\hspace{-56pt} d$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderh }\n[u(1)] [l(0.5)]\n!{\\color{black} \\xbendu}\n[l(2)]\n!{\\vcap[2] =<}\n!{\\xbendd- \\color{black}}\n[d(1)] [r(0.5)]\n!{\\xunderv}\n[u(0.5)] [r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2] =<}\n[l(1)]\n!{\\xbendl-}\n[u(0.5)] [l(3)]\n!{\\xunderv}\n[u(1.5)] [l(1)]\n!{\\color{black} \\xbendr-}\n[l(1)] [u(1)]\n!{\\hcap[-2]=<}\n[d(1)]\n!{\\xbendl \\color{black}}\n[r(2)] [u(0.5)]\n!{\\xcaph- =>}}} \\\\[-1.75cm]\n\\multicolumn{3}{l}{\\hspace{44pt}\\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(1.75)] [l(1.5)]\n!{\\xbendd[-1]=>}\n[u(0.75)] [r(1)]\n!{\\xbendl[-1]=>}\n[d(1)] [l(2.25)]\n!{\\xcaph[-1]=<}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{140pt}$(-3,-2,1)$}\n\t\t \n\t\\end{tabular}\n\t&\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$d$-preons, $D^{1\/2}_{-\\frac{1}{2} -\\frac{1}{2}}$} \\\\ [1.58cm]\n\t\t  & &\\hspace{-63pt} $d$ \\\\ [-1.3cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderv=<}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}}} \\\\ [-1.5cm]\n\\multicolumn{3}{l}{\\hspace{39pt} \\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(0.75)] [l(0.75)]\n!{\\xcaph[1]=<}\n}}} \\\\ [-0.1cm]\n\t\t \\multicolumn{3}{r}{\\hspace{140pt}$(-1,0,-1)$} \\\\\t\t \n\t\\end{tabular} \\\\ [-0.43cm]\n\t\\multicolumn{2}{c}{The clockwise and counterclockwise arrows are given opposite weights $(\\mp 1)$ respectively.} \\\\ [-0.51cm]\n\t\\multicolumn{2}{c}{The (rotation\/writhe charge) is measured by the sum of the weighted (black\/red) arrows.} \\\\\n\t[-0.51cm]\n\t\\multicolumn{2}{c}{The central loops of the trefoils contribute oppositely to the rotation of the complete trefoil.}\n\n\t\\end{tabular}\n\\end{center}\n\n\\restoregeometry\n\n\\setlength{\\baselineskip}{1.6\\baselineskip}\n\nFor quarks the three labelled loops contribute $-1$ and the shared loop $+1$ so that $r=-2$, as required.\n\nIn each case the three preons that form a lepton trefoil contribute their three negative rotation charges. The geometric and charge profile of the lepton trefoil is thus similar to the geometric and charge profile of a triatomic molecule composed of neutral atoms since the valence electronic charges of the atoms, which cancel the nuclear electronic charges of the atoms, are shared among the atoms to create the chemical binding of the molecule just as the negative rotation charges which cancel the positive rotation charges of the preons are shared among the preons to create the preon binding of the trefoils. There is a similar correspondence between quarks and antimolecules.\n\nWe next display the general representation $D^j_{m m'}(q)$ of the algebra $SL_q(2)$ in the Weyl monomial basis just as in the current standard model.\n\\begin{equation}\n\\boxed{ {D^j_{mm'}(q) = \\sum_{n_a, n_b, n_c, n_d} A(q | n_a, n_b, n_c, n_d) a^{n_a} b^{n_b} c^{n_c} d^{n_d} \\label{long}}}\n\\end{equation}\n\nwhere $a, b, c, d$ satisfy the algebra (A) and the sum  on $n_a, n_b, n_c, n_d$ is over all positive integers and zero that satisfy the following equations: \\cite{finkelstein14a}\n\\begin{empheq}[box=\\fbox]{align}\n    n_a + n_b + n_c + n_d &= 2j \\label{n2j}\\\\\n    n_a + n_b - n_c - n_d &= 2m \\label{n2m}\\\\\n    n_a - n_b + n_c - n_d &= 2m' \\label{n2m'}\n\\end{empheq}\n\nand\n\n\\begin{equation}\nA (q \\vert n_a n_b n_c n_d) = \\left [ \\frac{ \\langle n_+ ' \\rangle_1 ! \\langle n_- ' \\rangle_1 ! }{\\langle n_+ \\rangle_1 ! \\langle n_- \\rangle_1 !} \\right ]^{\\frac{1}{2}} \\frac{ \\langle n_+ \\rangle_1 !}{\\langle n_a \\rangle_1 ! \\langle n_b \\rangle_1 !} \\frac{\\langle n_- \\rangle_1 !}{\\langle n_c \\rangle_1 ! \\langle n_d \\rangle_1 !}.\n\\end{equation}\nwhere $n_\\pm = j \\pm m$, $n ' _\\pm = j \\pm m '$, and $\\langle n \\rangle _1 = \\frac{q_1 ^n - 1}{q_1 ^1 - 1}$ and $q_1 = q^{-1}$\n\nThe two dimensional representation, $T$, introduced by \\eqref{2drepslq2} now reappears as the $j = \\frac{1}{2}$ fundamental representation\n\\begin{equation}\n\\boxed{D^{\\frac{1}{2}}_{mm'}(q) = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\label{}}\n\\end{equation}\nIn the physical model with the \\eqref{couplingmatrix} coupling we interpret $(a,b,c,d)$ in \\eqref{long} as creation operators for $(a,b,c,d)$ particles, which we have termed preons. Then $D^j_{mm'}(a,b,c,d)$ is the creation operator for the state representing the superposition of $(n_a, n_b, n_c, n_d)$ gravitational preons. Since we shall regard the preons as fermions, they will also carry an anti-symmetrizing index to satisfy the Pauli principle.\n\n\\emph{In proceeding to the quantization of the gravitational field we represent its sources as Standard Model x Knot Model sources ($SL_q(2)$) where the particles of the Knot Model are the preons (a, b, c, d) described in Fig. 1 as $D^{3\/2}_{m, m'}(q)$ and also represented as twisted loops as the auxiliary knot particles}. The physical particles are here graphically represented by closed loops of energy-momentum with $n \\frac{e}{3}$ charge where n is the number of net positive turns of the tangent in one transit of the loop.\n\n\n\\section{Presentation of the Model in the Preon Representation\\cite{fink2018}\\cite{FinkSchwing}} \n\nThe particles $(a,b,c,d)$ described in the following sections are assumed to be either e or g preons and carry both e and g charges. The knot representation (3.1) of $D^j_{mm'}$ as a function of $(a,b,c,d)$ and $(n_a,n_b,n_c,n_d)$ implies the following constraints on the exponents:\n\n\\begin{IEEEeqnarray*}{+rCl+x*}\nn_a+n_b+n_c+n_d & = & 2j & (4.1) \\\\\nn_a+n_b - n_c - n_d & =  & 2m & (4.2) \\\\\nn_a-n_b+n_c-n_d & = & 2m' . & (4.3)\n\\end{IEEEeqnarray*}\nThe two relations defining the quantum kinematics and giving physical meaning to $D^j_{mm'}$, namely the postulated \\eqref{knotAssociation}:\n\\begin{equation*}\n(j,m,m')_q = \\frac{1}{2}(N,w, r+o) \\quad \\text{field (flux loop) description} \\tag{4.4}\n\\end{equation*}\nand the semi-empirical \\eqref{empirical} shown in the tables.\n\\begin{equation*}\n(j,m,m')_q = 3(t, -t_3, -t_0)_L \\quad \\text{particle description} \\tag{4.5}\n\\end{equation*}\nimply two complementary interpretations of the relations (4.1) -- (4.3). By (4.4) and these equations, one has a \\textit{field} description $(N, w, \\tilde{r})$ of the quantum state $(j, m, m')_q$ as follows\n\\begin{equation}\n\\left.\n{\\arraycolsep=1.2pt\n\\begin{array}{rl}\nN &= n_a + n_b + n_c + n_d \\\\\nw &= n_a + n_b - n_c - n_d  \\\\\n\\tilde{r} \\equiv r+o &= n_a - n_b + n_c - n_d\n\\end{array}\n}\n\\quad \\right\\} \\text{field (flux loop) (N, w, $\\tilde{r}$) description} \\label{field (flux loop) description}\n\\tag{4.6}\n\\end{equation}\nIn the last line of \\eqref{field (flux loop) description}, where $\\tilde{r} \\equiv r+o$ and $o$ is a parity index, $\\tilde{r}$ has been termed ``the quantum rotation,\" and $o$ the ``zero-point rotation.\" \n\nBy \\eqref{empirical} one has a ``\\textit{particle} description\" $(t, t_3, t_0)$ of the same quantum state $(j, m, m')$.\n\\begin{comment}\n\\begin{align}\nt &= \\frac{1}{6} (n_a + n_b +n_c +n_d) \\\\\nt_3 &= -\\frac{1}{6}(n_a + n_b - n_c - n_d) \\quad \\text{particle description} \\multirow{3}{*}{\\}} \\\\ \nt_0 &= -\\frac{1}{6}(n_a - n_b + n_c - n_d)\n\\end{align}\n\\end{comment}\n\\begin{equation}\n\\left.\n{\\arraycolsep=1.2pt\n\\begin{array}{rl}\nt &= \\frac{1}{6} (n_a + n_b +n_c +n_d)  \\\\\nt_3 &= -\\frac{1}{6}(n_a + n_b - n_c - n_d)  \\\\\nt_0 &= -\\frac{1}{6}(n_a - n_b + n_c - n_d) \n\\end{array}\n}\n\\quad \\right\\} \\text{particle ($t$, $t_3$, $t_0$) description} \\label{particle_description}\n\\tag{4.7}\n\\end{equation}\n\\emph{In \\eqref{particle_description}, $(t, t_3, t_0)$ are to be read as SLq(2) preon indices agreeing with standard SU(2) $\\times$ U(1) notation} only \\emph{at $j = \\frac{3}{2}$. In general, however, we do not assume $SU(2)\\times U(1)$ and instead may assume that $t_3$ measures writhe charge, $t_0$ measures rotation hypercharge and $t$ measures the total preon population or the total number of crossings of the associated classical knot.}\n\nThe attempt to invert (4.4) as (4.6) is quite successful but the corresponding attempt to rewrite the semi-empirical (2.5) as $t, t_3, t_0$ is not. We interpret this difference in favor of (4.4) or as a weakness of (4.5).\n\n\n\\section{Interpretation of the Complementary Equations}\nThere is also an alternative particle interpretation of the flux loop equations \\eqref{field (flux loop) description}\n\\begin{align*}\nN = n_a + n_b + n_c + n_d \\tag{5.1$N$} \\\\\nw = n_a + n_b - n_c - n_d \\tag{5.1$w$} \\\\\n\\tilde{r} = n_a - n_b + n_c - n_d \\tag{5.1$\\tilde{r}$}\n\\end{align*}\n\nHere the left-hand side with coordinates $(N,w,\\tilde{r})$ label a 2d-projected knot, and the right-hand side describes the preon population of the corresponding quantum state. \n\n\\emph{Equation (5.1$N$) states that the number of crossings, $N$, equals the total number of preons, $N'$, as given by the right side of this equation. Since we assume that the preons are fermions, the knot describes a fermion or a boson depending on whether the number of crossings is odd or even.} Viewed as a knot, a fermion becomes a boson when the number of crossings is changed by attaching or removing a geometric curl\n\\begin{turn}{90}\n\\xygraph{\n    !{0;\/r0.75pc\/:}\n    !{\\xunderh}\n    [u l(0.75)]!{\\xbendu}\n    [l (1.5)]!{\\vcap[1.5]}\n    !{\\xbendd-}}\n\\end{turn}\n. This picture is consistent with the view of a geometric curl as an opened preon loop, in turn viewed as a twisted loop\n\\begin{turn}{90}\n\\xygraph{\n    !{0;\/r0.75pc\/:}\n    !{\\xunderh}\n    [u l(0.75)]!{\\xbendu}\n    [l (1.5)]!{\\vcap[1.5]}\n    !{\\xbendd-}\n    [l] [d(0.5)]!{\\xbendu-}\n    [ld]!{\\vcap[-1.5]}\n    [u] [r(0.5)]!{\\xbendd}\n}\n\\end{turn}. Each counterclockwise or clockwise classical curl corresponds to a preon creation operator or antipreon creation operator respectively.\n\\newline\n\n\\subsection{\\emph{Gravitational} Preon Numbers}\n\nSince $a$ and $d$ are creation operators for antiparticles with opposite charge and hypercharge, while $b$ and $c$ are neutral antiparticles with opposite values of the hypercharge, we may introduce the \\emph{gravitational preon numbers}\n\\begin{align}\n\\nu_a &= n_a -n_d \\\\\n\\nu_b &= n_b - n_c\n\\end{align}\nThen (5.1$w$) and (5.1$\\tilde{r}$) may be rewritten in terms of gravitational preon numbers as\n\\begin{align}\n&\\nu_a + \\nu_b = w \\hspace{3pt} (= -6t_3) \\\\\n&\\nu_a - \\nu_b = \\tilde{r} \\hspace{3pt} (=-6t_0)\n\\end{align}\nBy (5.3) and (5.4) the conservation of the preon numbers and of the charge and hypercharge is equivalent to the conservation of the writhe and rotation, which are topologically conserved at the 2d-classical level. In this respect, these quantum conservation laws for preon numbers correspond to the classical conservation laws for writhe and rotation.\n\nEqns. $(5.1N) - (5.1\\tilde{r})$may also be interpreted directly in terms of Fig. 2 by describing the right-hand side of these equations as the possible populations of the conjectured preons at these crossings of Fig. 2 and interpreting the left-hand side as parameters of the binding field that links the 3 conjectured preons.\n\\newline\n\n\n\\section{Summary on the Measure of Charge by SU(2) $\\times$ U(1) and by SLq(2)} \nThe SU(2)$\\times$U(1) measure of charge requires the assumption of fractional charges for the quarks. The SLq(2) measure requires the replacement of the fundamental charge $(e)$ for charged leptons by a new fundamental charge $(e\/3)$ or $(g\/3)$ for charged preons but then does not require fractional charges for quarks\\cite{Schupe1} if quark charge is expressed in terms of the new ``fundamental or minimal charges\", i.e. $(e\/3)$ or $q(e\/3)$.\n\nThe SLq(2), or $(j,m,m')_q$ measure, has a direct preon interpretation since $2j$ is the total number of preonic sources, while $2m$ and $2m'$ respectively measure the numbers of writhe and rotation sources of preonic charge.\\cite{Fink2} Since $N$, $w$, and $r$ all measure the handedness of the source, charge is measured by the chirality of the source. \\emph{The electric charge of the resultant trefoil or of any composite of preons would then be a measure of the chirality generated by the knotting of an original unknotted flux loop of energy-momentum.}\n\nIf neutral unknotted flux tubes predated the particles, and the particles were initially formed by the knotting of unknotted flux tubes of energy-momentum, then the simplest fermions that could have formed with topological stability must have had three 2d crossings and therefore three preons, but the topological stability of this trefoil could be protected only as long as spacetime remains 2-dimensional as suggested by the comments on Tables 1 and 2.\n\nThe total SLq(2) charge sums the signed two dimensional clockwise and counterclockwise turns that any energy-momentum current makes both at the crossings and in making a single circuit of the 2d-projected knot. This measure of charge, ``knot charge\", which is suggested by the leptons and quarks, appears more fundamental than the electroweak isotopic measure that originated in the neutron-proton system, since it reduces the concept of charge to the chirality of the corresponding energy-momentum curve which may also be described as a SLq(2) chirality and in this way reduces charge to a topological concept similar to the way energy-momentum is geometrized by the curvature of spacetime in the Einstein-Hilbert equations or in lattice methods of Regge for solving the differential equations. In this way the energy-momentum conservation may also be formulated graphically.\\cite{Regge}\n\n\n\\section{Other Possible Physical Interpretations of Corresponding Quantum States}\n\n\\begin{comment}\nThe point particle $(N', \\nu_a, \\nu_b)$ representation and the flux loop $(N, w, \\tilde{r})$ complementary representation are related by the $\\delta$-function transform:\n\\begin{align}\n\\tilde{D}^{N'}_{\\nu_a, \\nu_b}(q, a, b, c, d) = \\sum_{N,w,r} \\delta(N',N)\\delta(\\nu_a+\\nu_b, w)\\delta(\\nu_a - \\nu_b, \\tilde{r}) D^{N\/2}_{\\frac{w}{2} \\frac{\\tilde{r}}{2}} (q, a, b, c, d)\n\\end{align}\n\\end{comment}\n\nSince one may interpret the elements $(a,b,c,d)$ of the SLq(2) algebra as creation operators for either preonic particles or current loops, the $D^j_{mp}(q)$ may be interpreted as a creation operator for a composite quantum particle composed of either preonic particles $(N', \\nu_a, \\nu_b)$ or current loops $(N, w, \\tilde{r})$ . \\emph{These two complementary views of the same particle may be reconciled as describing $N$-preon systems bound by a knotted field having $N$-crossings with the preons at the crossings as illustrated in Figure 2 for $N=3$.} In the limit where the three outside lobes become small or infinitesimal compared to the central circuit, the resultant structure will resemble a three particle system tied together by a string.\n\\begin{figure}[!tb]\n\\begin{center}\n\\normalsize\n\\begin{tabular}{c | c}\n\t\\multicolumn{2}{c}{\\textbf{Figure 2:} Leptons and Quarks Pictured as Three Preons Bound by a Trefoil Field} \\\\ [-0.0cm]\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{r}{ \\ul{$(w, r, o)$}} \\\\ [-0.3cm]\n\t\t\\multicolumn{3}{l}{Neutrinos, $D^{3\/2}_{-\\frac{3}{2} \\frac{3}{2}} \\sim c^3$} \\\\ [0.2cm]\n\t\t & $\\hspace{17pt} c_j$ & \\\\ [-.22 cm]\n\t\t \\multicolumn{3}{c}{\\hspace{-1.655 cm}\\vspace{0.22 cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\\\ [-0.6cm]\n\t\t $\\hspace{28pt} c_i$ {\\hspace{.33cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{-1cm} & & {\\hspace{-2.65cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{2.15cm} $\\hspace{-56pt} c_k$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderh }\n[u(1)] [l(0.5)]\n!{\\xbendu}\n[l(2)]\n!{\\vcap[2]=>}\n!{\\xbendd-}\n[d(1)] [r(0.5)]\n!{\\xoverh}\n[d(0.5)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]=>}\n[l(1)]\n!{\\xbendl-}\n[l(1.5)] [d(0.25)]\n!{\\xbendl[0.5]}\n[u(1)] [l(0.5)]\n!{\\xbendr[0.5]=<}\n[u(1.75)] [l(2)]\n!{\\xunderv}\n[u(1.5)] [l(1)]\n!{\\xbendr-}\n[u(1)] [l(1)]\n!{\\hcap[-2]=>}\n[d(1)]\n!{\\xbendl}}} \\\\[-1.5cm]\n\\multicolumn{3}{l}{\\textcolor{red} {\\hspace{32pt} \\xygraph{\n!{0;\/r1.3pc\/:}\n[u(2.25)] [r(3.25)]\n!{\\xcapv[1] =<}\n[l(2.75)] [u(.75)]\n!{\\xbendr[-1] =<}\n[d(1)] [r(1.25)]\n!{\\xcaph[-1] =<}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{145pt}$(-3,2,1)$}\n\t\t \n\t\\end{tabular}\n&\n\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{r}{ \\ul{$(w, r, o)$}} \\\\ [-0.3cm]\n\t\t\\multicolumn{3}{l}{Charged Leptons, $D^{3\/2}_{\\frac{3}{2} \\frac{3}{2}} \\sim a^3$} \\\\ [0.2cm]\n\t\t & $\\hspace{17pt} a_j$ & \\\\ [-.22 cm]\n\t\t \\multicolumn{3}{c}{\\hspace{-1.58 cm}\\vspace{0.22 cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\\\ [-0.6cm]\n\t\t $\\hspace{28pt} a_i$ {\\hspace{.33cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{-1cm}  & & {\\hspace{-2.65cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{2.15cm} $\\hspace{-58pt} a_k$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverh }\n[u(1)] [l(0.5)]\n!{\\xbendu}\n[l(2)]\n!{\\vcap[2]=>}\n!{\\xbendd-}\n[d(1)] [r(0.5)]\n!{\\xunderh}\n[d(0.5)]\n!{\\xbendr }\n[u(2)]\n!{\\hcap[2]=>}\n[l(1)]\n!{\\xbendl-}\n[l(1.5)] [d(0.25)]\n!{\\xbendl[0.5]}\n[u(1)] [l(0.5)]\n!{\\xbendr[0.5]=<}\n[u(1.75)] [l(2)]\n!{\\xoverv}\n[u(1.5)] [l(1)]\n!{\\xbendr-}\n[u(1)] [l(1)]\n!{\\hcap[-2]=>}\n[d(1)]\n!{\\xbendl}}} \\\\ [-1.6cm]\n\\multicolumn{3}{l}{\\hspace{33pt} \\textcolor{red}{\n\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(2.25)] [r(3.25)]\n!{\\xcapv[1] =>}\n[l(2.75)] [u(.75)]\n!{\\xbendr[-1] =>}\n[d(1)] [r(1.25)]\n!{\\xcaph[-1] =>}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{150pt}$(3,2,1)$}\n\t\t \n\t\\end{tabular} \\\\ [-0.1cm] \\hline\n\t\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$d$-quarks, $D^{3\/2}_{\\frac{3}{2} -\\frac{1}{2}} \\sim ab^2$} \\\\ [0.2cm]\n\t\t & $\\hspace{19pt} b$ & \\\\ [-.22 cm]\n\t\t  \\multicolumn{3}{c}{\\hspace{-1.52 cm}\\vspace{0.22 cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\\\ [-0.6cm]\n\t\t $\\hspace{28pt} a_i$ {\\hspace{.33cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{-1cm}  & & {\\hspace{-2.65cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{2.15cm} $\\hspace{-56pt} b$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverh }\n[u(1)] [l(0.5)]\n!{\\xbendu}\n[l(2)]\n!{\\vcap[2]=<}\n!{\\xbendd-}\n[d(1)] [r(0.5)]\n!{\\xoverv}\n[u(0.5)] [r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2] =<}\n[l(1)]\n!{\\xbendl-}\n[u(0.5)] [l(3)]\n!{\\xoverv}\n[u(1.5)] [l(1)]\n!{\\xbendr-}\n[l(1)] [u(1)]\n!{\\hcap[-2]=<}\n[d(1)]\n!{\\xbendl}\n[r(2)] [u(0.5)]\n!{\\xcaph- =>}}} \\\\ [-1.6cm]\n\\multicolumn{3}{l}{\\hspace{42pt} \\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(1.75)] [l(1.5)]\n!{\\xbendd[-1]=<}\n[u(0.75)] [r(1)]\n!{\\xbendl[-1]=<}\n[d(1)] [l(2.25)]\n!{\\xcaph[-1]=>}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{140pt}$(3,-2,1)$}\n\t\t \n\t\\end{tabular}\n\t&\n\\begin{tabular}{c c c}\n\t\t\\multicolumn{3}{l}{$u$-quarks, $D^{3\/2}_{-\\frac{3}{2} -\\frac{1}{2}} \\sim cd^2$} \\\\ [0.2cm]\n\t\t & $\\hspace{19pt} d$ & \\\\ [-.22 cm]\n\t\t  \\multicolumn{3}{c}{\\hspace{-1.75 cm}\\vspace{0.22 cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\\\ [-0.6cm]\n\t\t $\\hspace{28pt} c_i$ {\\hspace{.33cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{-1cm}  & & {\\hspace{-2.9cm}\\textcolor{blue} {\\scalebox{2}{\\Huge .}}} \\hspace{2.15cm} $\\hspace{-56pt} d$ \\\\ [-2.8cm]\n\t\t \\multicolumn{3}{l}{\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderh }\n[u(1)] [l(0.5)]\n!{\\xbendu}\n[l(2)]\n!{\\vcap[2]=<}\n!{\\xbendd-}\n[d(1)] [r(0.5)]\n!{\\xunderv}\n[u(0.5)] [r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]=<}\n[l(1)]\n!{\\xbendl-}\n[u(0.5)] [l(3)]\n!{\\xunderv}\n[u(1.5)] [l(1)]\n!{\\xbendr-}\n[l(1)] [u(1)]\n!{\\hcap[-2]=<}\n[d(1)]\n!{\\xbendl}\n[r(2)] [u(0.5)]\n!{\\xcaph- =>}}} \\\\ [-1.75cm]\n\\multicolumn{3}{l}{\\hspace{44pt}\\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(1.75)] [l(1.5)]\n!{\\xbendd[-1]=>}\n[u(0.75)] [r(1)]\n!{\\xbendl[-1]=>}\n[d(1)] [l(2.25)]\n!{\\xcaph[-1]=<}\n}}} \\\\ [-0.7cm]\n\t\t \\multicolumn{3}{r}{\\hspace{140pt}$(-3,-2,1)$}\n\t\t \n\t\\end{tabular} \\\\ [-0.2cm]\n\\multicolumn{2}{c}{The preons conjectured to be present at the crossings are suggested by the blue dots at the crossings} \\\\ [-0.4cm]\n\\multicolumn{2}{c}{ of the lepton-quark diagrams, or at the crossings of any diagram with more crossings.}\n\\end{tabular}\n\\end{center}\n\\end{figure}\n\\emph{The physical models suggested by Fig. 2 may be further studied with the aid of preon Lagrangians similar to that given in reference 3}. The Hamiltonians of these three body systems may be parametrized by degrees of freedom characterizing both the preons and the binding field that come from the \\emph{form factors required by SLq(2) invariance}. The masses of the leptons, quarks, and binding quanta are determined by the eigenvalues of this Hamiltonian in terms of the parameters describing the constituent preons and energy-momentum flux loops. There is currently no experimental guidance at these conjectured energies. These three body systems are, however, familiar in different contexts, namely\n\\begin{center}\nH$^3$ composed of one proton and two neutrons: $PN^2$ \\\\\n$P$ composed of one down and two up quarks: $DU^2$ \\\\\n$N$ composed of one up and two down quarks: $UD^2$ \\\\\n\\end{center}\nwhich are similar to\n$U$ as $cd^2$ and $D$ as $ab^2$,\nwhere $U$ and $D$ are up and down quarks, presented as three binding preon states. These different realizations of energy-momentum and charge represent different expressions of curvature and chirality, and in particular as displayed by a closed loop of energy-momentum.\n\n\\section{Alternate Interpretation}\nIn the model suggested by Fig. 2 the parameters of the preons and the parameters of the current loops are to be understood as codetermined. On the other hand, in an alternative interpretation of complementarity, the hypothetical preons conjectured to be present in Figure 2 may carry no independent degrees of freedom and may simply describe \\emph{concentrations of energy and momentum at the crossings of the energy-momentum tube}. In this interpretation of complementarity, $(t, t_3, t_0)$ and $(N,w, \\tilde{r})$ are just two ways of describing the same quantum trefoil of field. \\emph{In this picture the preons are bound}, i.e. they do not appear as free particles. This view of the elementary particles as either non-singular lumps of field or as solitons has also been described as a unitary field theory\\cite{Raitio1}.\n\n\\section{Lower Representations}\nWe have so far considered the states $j = 3, \\frac{3}{2}, \\frac{1}{2}$ representing electroweak vectors, leptons and quarks, and preons, respectively. We finally consider the states $j=1$ and $j=0$. Here we shall not examine the higher $j$ states.\n\nIn the adjoint representation $j=1$, the particles are the vector bosons by which the $j=\\frac{1}{2}$ preons interact and there are two crossings. These vectors are different from the $j=3$ vectors by which the $j=\\frac{3}{2}$ leptons and the $j=\\frac{3}{2}$ quarks interact.\n\\bigskip\n\nIf $j=0$, the indices of the quantum knot are\n\\begin{align}\n(j,m,m')_q = (0,0,0)\n\\end{align}\nand by the basic rule \\eqref{knotAssociation} for interpreting the knot indices on the left chiral fields\n\\begin{align}\n\\frac{1}{2}(N,w,\\tilde{r}) = (j, m, m')_q &= (0, 0, 0) \n\\end{align}\nThen the $j=0$ quantum states correspond to classical loops with no crossings $(N=0)$ just as preon states correspond to classical twisted loops with one crossing. Since $N=0$, the $j=0$ states also have no preonic sources of charge and therefore no electroweak interaction.  \\emph{It is possible that these }$j=0$ \\emph{hypothetical quantum states are realized as (electroweak non-interacting) loops of field flux with} $w=0$, $\\tilde{r}= r+o = 0$\\emph{, and }$r = \\pm1$, $o =\\mp1$ \\emph{ i.e. with the topological rotation }$r=\\pm1$. The two states $(r, o) = (+1, -1)$ and $(-1, +1)$ are to be understood as quantum mechanically coupled.\n\nIf, as we are assuming, the leptons and quarks with $j= \\frac{3}{2}$ correspond to 2d projections of knots with three crossings, and if the heavier preons with $j= \\frac{1}{2}$ correspond to 2d projections of twisted loops with one crossing, then if the $j=0$ states correspond to 2d projections of simple loops with no crossings, one might ask if these particles with no electroweak interactions and which are smaller and heavier than the preons, are among the candidates for ``dark matter.\" If these $j=0$ particles predate the $j=\\frac{1}{2}$ preons, one may refer to them as ``yons\" as suggested by the term ``ylem\" for primordial matter.\n\n\\section{Speculations about an earlier universe and dark matter} \\vspace{-0.8em}\nOne may speculate about an earlier universe before leptons and quarks had appeared, when there was no charge, and when energy and momentum existed only in the SLq(2) $j=0$ neutral state as simple loop currents of gravitational energy-momentum. Then the gravitational attraction would bring some pairs of opposing loops close enough to permit the transition from two $j=0$ loops into two opposing $j=\\frac{1}{2}$ twisted loops. A possible geometric scenario for the transformation of two simple loops of current (yons) with opposite rotations into two $j=\\frac{1}{2}$ twisted loops of current (preons) is suggested in Fig. 3. Without attempting to formally implement this scenario, one notes according to Fig. 3 that the fusion of two yons may result in a doublet of preons as twisted loops, which might also qualify as Higgs particles.\n\n\\newpage\n\n\\begin{center}\n\\normalsize\n\\begin{tabular}{c  c  c}\n\\multicolumn{3}{c}{ \\textbf{Figure 3:} Creation of Preons as Twisted Loops} \\\\\n\\vspace{-3pt}\n\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\hcap[2]=>}\n!{\\hcap[-2]}} \\hspace{3pt} \n\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\hcap[2]}\n!{ \\hcap[-2]=< }} &\n\\xygraph{\n!{0;\/r1.3pc\/:}\n[d(1.25)]\n!{\\xcaph[3]=<@(0)}} & \\hspace{-4pt}\n\\xygraph{\n!{0;\/r1.3pc\/:}\n[d(1)]\n!{\\color{red} \\vcap[2]=> \\color{blue}}\n!{\\vcap[-2]}} \\hspace{-7pt} \n\\xygraph{\n!{0;\/r1.3pc\/:}\n[d(1)]\n!{\\color{blue} \\vcap[2]=<}\n!{\\color{red} \\vcap[-2]=< \\color{black}}}\\\\ [-1.7cm]\n\\hspace{4pt}\\scriptsize{$r=1 \\hspace{20pt}+ \\hspace{16pt}r=-1$} & & \\scriptsize{$\\hspace{3pt}r=1 \\hspace{8pt} r=-1$} \\\\ [-0.7cm]\n\\hspace{0pt}\\scriptsize{$\\tilde{r}=0 \\hspace{32pt} \\hspace{16pt}\\tilde{r}=0$} & & \\scriptsize{$\\hspace{0pt}\\tilde{r}=0 \\hspace{10pt} \\tilde{r}=0$} \\\\\nTwo $j=0$ neutral loops & gravitational attraction & interaction causing the crossing or  \\\\ [-0.55cm]\n with opposite topological & & redirection of neutral current flux \\\\ [-0.55cm]\n  rotation & & shown below \\\\ [-0.55cm]\n\\end{tabular}\n\\end{center}\n\\begin{center}\n\\begin{tabular}{c c}\n\\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(0.75)] [l(0.75)]\n!{\\xcaph[1]=>}\n}} &\n\\textcolor{red}{\\xygraph{\n!{0;\/r1.3pc\/:}\n[u(0.75)] [l(0.75)]\n!{\\xcaph[1]=<}\n}} \\\\ [-1cm]\n\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xoverv=<}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}} &\n\\xygraph{\n!{0;\/r1.3pc\/:}\n!{\\xunderv=<}\n[u(0.5)] [l(1)]\n!{\\xbendl}\n[u(2)]\n!{\\hcap[-2]}\n!{\\xbendr-}\n[r(1)]\n!{\\xbendr}\n[u(2)]\n!{\\hcap[2]}\n[l(1)]\n!{\\xbendl-}} \\\\ [-1.1cm]\n\\multicolumn{2}{c}{+} \\\\ [-0.95cm]\n\\hspace{-95pt} & \\hspace{-95pt} \\\\ [-0.9cm]\na preon & \\hspace{-1pt} c preon \\\\ [-0.4cm]\n\\hspace{2pt}$r=0$ & \\hspace{2pt}$r=0$ \\\\ [-0.4cm]\n$w_a = +1$ & $w_c=-1$ \\\\ [-0.4cm]\n$ Q_a = -\\frac{e}{3}$ & $ Q_c = 0$  \\\\\n\\end{tabular}\n\\end{center}\n\nIn the scenario suggested by Figure 3 the opposing states are quantum mechanically entangled and may undergo gravitational exchange scattering. \n\nThe $\\binom{c}{a}$ doublet of Fig. 3 is similar to the Higgs doublet which is independently required by the mass term of the Lagrangian described in reference 3 to be a SLq(2) singlet $(j=0)$ and a SU(2) charge doublet $(t=\\frac{1}{2})$. The matrix elements connecting the preon a and c states in Fig. 3 is obviously fundamental for the model. Since the Higgs mass contributes to the inertial mass, one expects a fundamental connection with the gravitational field at this point. \n\n\\begin{comment}\nA fraction of the preons $(j= \\frac{1}{2})$ produced by the fusion of two yons might in turn combine to form two preon $(j=1)$ and then three preon $(j = \\frac{3}{2})$ states with two and three crossings respectively. The $j = \\frac{3}{2}$ states would be recognized in the present universe as leptons and quarks. Since, however, the $j = \\frac{1}{2}$ and $j=1$ particles with one and two crossings, respectively, are not topologically stable in three dimensions and can relapse into a $j=0$ state with no crossings, the building up process from yons would not produce topologically stable particles before the $j=\\frac{3}{2}$ leptons and quarks with three crossings are reached. These remarks apply equally to the e and g sectors.\n\\end{comment}\n \nIf at an early cosmological time, only a fraction of the initial gas of quantum loops was converted to preons and these in turn led to a still smaller number of leptons and quarks, then most of the mass and energy of the universe would at the present time still reside in the dark loops while charge and current and visible mass would be confined to structures composed of leptons and quarks. \\emph{In making experimental tests for particles of dark matter one might expect the SLq(2) $j=0$ dark loops to be different in mass than the dark neutrino trefoils where $j= \\frac{3}{2}$, although both $j=0$ and $j=\\frac{3}{2}$ would contribute to the dark matter.}\n\n\\bigskip\n\n\\section*{Acknowledgements}\nI thank E. Abers, C. Cadavid, J. Smit, and S. Mackie for comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}