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{"text":"\\section{#1}}\n\\renewcommand{\\theequation}{\\arabic{section}.\\arabic{equation}}\n\\numberwithin{equation}{section}\n\\def\\N{{\\mathcal N}}\n\\def\\L{{\\mathcal L}}\n\\def\\E{{\\mathcal E}}\n\\def\\S{{\\mathcal S}}\n\\def\\D{{\\mathcal D}}\n\\def\\J{{\\mathcal J}}\n\\def\\O{{\\mathcal O}}\n\\def\\A{{\\mathcal A}}\n\\def\\F{{\\mathcal F}}\n\\def\\V{{\\mathrm V}}\n\\def\\r{\\rho}\n\\def\\a{\\alpha}\n\\def\\be{\\beta}\n\\def\\d{\\delta}\n\\def\\g{\\gamma}\n\\def\\G{\\Gamma}\n\\def\\s{\\sigma}\n\\def\\t{\\tau}\n\\def\\ti{\\tilde}\n\\def\\x{\\xi}\n\\def\\z{\\zeta}\n\\def\\m{\\mu}\n\\def\\n{\\nu}\n\\def\\k{\\kappa}\n\\def\\f{\\phi}\n\\def\\vf{\\varphi}\n\\def\\e{\\eta}\n\\def\\ep{\\varepsilon}\n\\def\\eps{\\epsilon}\n\\def\\l{\\lambda}\n\\def\\th{\\theta}\n\\def\\h{\\hat}\n\\def\\o{\\omega}\n\\newcommand{\\tr}{\\mbox{Tr}}\n\\newcommand{\\ud}{\\mbox{d}}\n\\newcommand{\\am}{\\mbox{\\bf am}}\n\\newcommand{\\sn}{\\mbox{\\bf sn}}\n\\newcommand{\\sd}{\\mbox{\\bf sd}}\n\\def\\p{\\partial}\n\\def\\rb{\\right}\n\\def\\lb{\\left}\n\\def\\axs{{\\rm AdS}_5\\times S^5}\n\\newcommand{\\eq}[1]{\\begin{equation} #1 \\end{equation}}\n\\newcommand{\\al}[1]{\\begin{align} #1 \\end{align}}\n\\newcommand{\\ml}[1]{\\begin{multline} #1 \\end{multline}}\n\\newcommand{\\arr}[1]{\\begin{array} #1 \\end{array}}\n\\begin{document}\n\\begin{titlepage}\n\\title{Three-point correlation functions from pulsating strings in AdS$_5\\times S^5$}\n\\author{D.~Arnaudov${}^{a}$\\thanks{\\texttt{[email protected]}}\\ \\ and R.~C.~Rashkov${}^{b,a}$\\thanks{\\texttt{[email protected]}}\n\\ \\\\ \\ \\\\\n${}^a$ Department of Physics, Sofia University,\\\\\n5 J. Bourchier Blvd, 1164 Sofia, Bulgaria\n\\ \\\\ \\ \\\\\n${}^b$ Institute for Theoretical Physics,\\\\\nVienna University of Technology,\\\\\nWiedner Hauptstr. 8-10, 1040 Vienna, Austria\n}\n\\date{}\n\\end{titlepage}\n\n\\maketitle\n\n\\begin{abstract}\nOne of the most important problems in any conformal field theory is the calculation of three-point functions of primary operators. In this paper we provide explicit examples of correlators with two scalar operators in ${\\cal N}=4$ super-Yang--Mills theory at large $N$, corresponding to pulsating semiclassical strings in AdS$_5\\times S^5$, and an operator with small quantum numbers at strong coupling.\n\\end{abstract}\n\n\\sect{Introduction}\n\nAn extremely active area of research in theoretical high-energy physics in recent years has been the correspondence between gauge and string theories. Following the impressive conjecture made by Maldacena \\cite{Maldacena} that type IIB string theory on $\\axs$ is dual to ${\\cal N}=4$ super-Yang--Mills theory with a large number of colors, an explicit realization of the AdS\/CFT correspondence was provided in \\cite{GKP}. Many convincing results have been achieved thereafter, paving the way for the subject to become an indispensable tool in probing such diverse areas as the dynamics of quark-gluon plasma and high-temperature superconductivity.\n\nA key feature of the duality is the connection between planar correlation functions of conformal primary operators in the gauge theory and correlators of corresponding vertex operators of closed strings with $S^2$ worldsheet topology. Recently, some progress was accomplished in the study of three- and four-point functions with two and three ``heavy'' vertex operators with large quantum numbers at strong coupling. The remaining operators were chosen to be various ``light'' states (with quantum numbers and dimensions of order one). It was shown that the large $\\sqrt{\\lambda}$ behavior of such correlators is fixed by a semiclassical string trajectory governed by the heavy operator insertions, and with sources provided by the vertex operators of light states.\n\nInitially this approach was utilized in the calculation of two-point functions of heavy operators in \\cite{Polyakov:2002}--\\cite{Buchbinder:2010vw}. More recently the above procedure was extended to certain three-point correlators in \\cite{Janik:2010gc,Zarembo:2010,Costa:2010}. A method based on heavy vertex operators was proposed in \\cite{Roiban:2010}. Further developments in the computation of correlators with two string states are presented in~\\cite{Hernandez:2010}. The main goal of these investigations is elucidation of the structure of three-point functions of three semiclassical operators \\cite{Klose:2011}.\n\nRecently the authors of \\cite{Bajnok:2014} noticed that the precise formulation of such correlators should involve string energy eigenstates, which necessitates a slight modification of previous methods. Namely, one should average over all string solutions with a given energy. Although this alteration does not invalidate the results for the correlation functions obtained so far, it turns out that in the case of pulsating strings \\cite{Minahan:2002rc,Engquist:2003rn} we need to apply strictly the procedure, described in \\cite{Bajnok:2014}, in order to get the correct answer. In the present paper we consider the three-point correlation function of two heavy operators, corresponding to a pulsating string solution in $S^3\\subset S^5$ \\cite{Kruczenski:2004}, and one BPS (dilaton or chiral primary) operator. We provide some limiting cases and recover known results.\n\nThe paper is organized in the following way. In Section \\ref{sec2} we present a brief review of the procedure for calculating semiclassically two- and three-point correlation functions via vertex operators. In Section \\ref{sec3} we proceed with the derivation of three-point correlators for a particular pulsating string solution, taking either the dilaton or the chiral primary operator~(CPO) as the light operator. We study a number of limiting cases of the correlation functions. We conclude with a short discussion on the results.\n\n\\sect{Correlation functions with two heavy operators}\\label{sec2}\n\nWe commence with a review of the method for obtaining two-point correlation functions. Their computation in the leading semiclassical approximation is closely related to utilizing an adequate classical string solution \\cite{Tseytlin:2003}--\\cite{Janik:2010gc}. If $V_{H1}(\\xi_1)$ and $V_{H2}(\\xi_2)$ are the two heavy vertex operators, which are inserted at the $\\xi_1$ and $\\xi_2$ points on the string worldsheet, the corresponding two-point correlator in the limit of large 't Hooft coupling is obtained from the stationary point of the action\n\\eq{\n\\langle V_{H1}(\\xi_1)V_{H2}(\\xi_2)\\rangle\\sim e^{-I},\n}\nwhere $I$ is the action of the $\\axs$ string sigma model in the usual embedding coordinates\n\\al{\n&I=\\frac{\\sqrt{\\lambda}}{4\\pi}\\int d^2\\xi\\ \\Big(\\partial Y_M\\bar{\\partial}Y^M+\\partial X_k\\bar{\\partial}X_k+{\\rm fermions}\\Big)\\,,\\\\\n&Y_MY^M=Y_0^2+Y_1^2+Y_2^2+Y_3^2+Y_4^2-Y_5^2=-R^2,\\quad\\ X_kX_k=1\\,.\\nonumber\n}\nThroughout the paper we apply conformal gauge and use a worldsheet with Euclidean signature. Correspondingly, the two-dimensional derivatives are $\\partial=\\partial_1+i\\partial_2,\\,\\bar{\\partial}=\\partial_1-i\\partial_2$. We also work with the Euclidean continuation of AdS$_5$. The embedding, global and Poincar\\'e coordinates of AdS$_5$ assume the following form\n\\al{\n&Y_5+iY_0=R\\,\\cosh\\rho\\ e^{it},\\quad\nY_1+iY_2=R\\,\\sinh\\rho\\,\\cos\\Theta\\,e^{i\\phi_1},\\quad\nY_3+iY_4=R\\,\\sinh\\rho\\,\\sin\\Theta\\,e^{i\\phi_2},\\nonumber\\\\\n&Y_m=\\frac{R\\,x_m}{z}\\,,\\qquad\nY_4=\\frac{1}{2z}(-R^2+z^2+x^mx_m)\\,,\\qquad\nY_5=\\frac{1}{2z}(R^2+z^2+x^m x_m)\\,,\n\\label{poincare}\n}\nwhere $x^mx_m=x_0^2+x_ix_i\\ (m=0,1,2,3;\\ i=1,2,3)$.\n\nThe stationary solution satisfies the string equations of motion with singular sources given by $V_{H1}(\\xi_1)$ and $V_{H2}(\\xi_2)$. Utilizing the conformal symmetry of the theory, we are able to map the $\\xi$-plane worldsheet to a Euclidean cylinder with $(\\tau,\\sigma)$ coordinates\n\\eq{\ne^{\\tau+i\\sigma}=\\frac{\\xi-\\xi_2}{\\xi-\\xi_1}\\,.\n\\label{confmap}\n}\nUnder this Schwarz--Christoffel mapping the singular solution on the $\\xi$-plane goes to a smooth solution on the cylinder \\cite{Tseytlin:2003,Buchbinder:2010,Buchbinder:2010vw} with $t=\\kappa\\tau$, where $\\kappa$ is a constant parameter proportional to the string energy. The quantum numbers of the latter solution coincide with the quantum numbers of the heavy vertex operators, guaranteeing that there is no loss of information.\n\nThe considerations above can also be applied to a physical integrated vertex operator dependent on a point ${\\rm x}$ on the boundary of the Poincar\\'e patch of AdS$_5$ \\cite{Polyakov:2002,Tseytlin:2003}\n\\eq{\n{\\rm V}_H({\\rm x})=\\int d^2\\xi\\ V_H(\\xi;{\\rm x})\\,,\\qquad V_H(\\xi;{\\rm x})\\equiv V_H(z(\\xi),x(\\xi)-{\\rm x},X_k(\\xi))\\,.\n}\nAgain the semiclassical two-point correlation function $\\langle\\V_{H1}({\\rm x}_1)\\V_{H2}({\\rm x}_2)\\rangle$ is determined by the classical action evaluated on the stationary point solution. Applying the conformal mapping \\eqref{confmap}, we obtain the corresponding smooth spinning string solution in terms of Poincar\\'e coordinates, with the boundary conditions\\footnote{We refer to \\cite{Buchbinder:2010vw} for details.}\n\\eq{\n\\tau\\rightarrow-\\infty\\ \\ \\Longrightarrow\\ \\ z\\rightarrow0\\,,\\ \\ x\\rightarrow{\\rm x}_1\\,,\\qquad\\tau\\rightarrow+\\infty\\ \\ \\Longrightarrow\\ \\ z\\rightarrow0\\,,\\ \\ x\\rightarrow{\\rm x}_2\\,.\n\\label{boundcond}\n}\n\nIn a similar fashion we can calculate three-point correlation functions with two heavy and one light operators \\cite{Zarembo:2010,Roiban:2010}\n\\al{\nG_3({\\rm x}_1,{\\rm x}_2,{\\rm x}_3)&=\\langle\\V_{H1}({\\rm x}_1)\\V_{H2}({\\rm x}_2)\\V_L({\\rm x}_3)\\rangle\\\\\n&=\\int{\\cal D}\\mathbb{X}^\\mathbb{M}\\ e^{-I}\\int d^2\\xi_1d^2\\xi_2d^2\\xi_3\\ V_{H1}(\\xi_1;{\\rm x}_1)V_{H2}(\\xi_2;{\\rm x}_2)V_L(\\xi_3;{\\rm x}_3)\\,,\\nonumber\n}\nwhere $\\int{\\cal D}\\mathbb{X}^\\mathbb{M}$ is the integral over $(Y_M,X_k)$. We note that the contribution of the light operator in the stationary point equations can be neglected, so that one can use the same classical string solution as in the case of the two-point function of two heavy operators. In this way we obtain \\cite{Roiban:2010}\n\\eq{\n\\frac{G_3({\\rm x}_1,{\\rm x}_2,{\\rm x}_3)}{G_2({\\rm x}_1,{\\rm x}_2)}=\\int d^2\\xi\\ V_L(z(\\xi),x(\\xi)-{\\rm x}_3,X_k(\\xi))\\,,\n\\label{strconstxi}\n}\nwhere $(z(\\xi),x(\\xi),X_k(\\xi))$ denote the respective string solution with the same quantum numbers as the heavy vertex operators, and with the boundary conditions in \\eqref{boundcond} mapped to the $\\xi$-plane by the Schwarz--Christoffel mapping \\eqref{confmap}. Using the two-dimensional conformal invariance, we can also provide \\eqref{strconstxi} in terms of the cylinder ($\\int d^2\\sigma=\\int^\\infty_{-\\infty}d\\tau\\int^{2\\pi}_0d\\sigma$)\n\\al{\n\\label{strconstst}\n&\\frac{G_3({\\rm x}_1,{\\rm x}_2,{\\rm x}_3)}{G_2({\\rm x}_1,{\\rm x}_2)}=\\\\\n&=\\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\int_{-T\/2}^{T\/2}d\\tau_0\\int d^2\\sigma\\ V_L(z(\\tau-\\tau_0,\\sigma),x(\\tau-\\tau_0,\\sigma)-{\\rm x}_3,X_k(\\tau,\\sigma))\\,e^{-(\\Delta_2-\\Delta_1)\\kappa\\tau_0},\n\\nonumber\n}\nwhere, as was detailed in \\cite{Bajnok:2014}, we have averaged over all solutions in AdS (parameterized by different values of $\\tau_0$) with the same energy in order to obtain the needed energy eigenstate. We have denoted the conformal dimension of $V_{H1}$ with $\\Delta_1$ and that of $V_{H2}$ with $\\Delta_2$.\n\nThe global conformal $\\textrm{SO}(2,4)$ symmetry fixes the spacetime dependence of two- and three-point functions\\footnote{We assume that $\\V_{H2}=\\V^*_{H1}$, which is valid for the correlation functions we are interested in.}\n\\al{\\label{2point}\nG_2({\\rm x}_1,{\\rm x}_2)&=\\frac{C_{12}\\ \\delta_{\\Delta_1\\!,\\Delta_2}}{{\\rm x}_{12}^{\\Delta_1+\\Delta_2}}\\,,\\qquad{\\rm x}_{ij}\\equiv|{\\rm x}_i-{\\rm x}_j|\\,,\\\\\nG_3({\\rm x}_1,{\\rm x}_2,{\\rm x}_3)&=\\frac{C_{123}}{{\\rm x}_{12}^{\\Delta_1+\\Delta_2-\\Delta_3}{\\rm x}_{13}^{\\Delta_1+\\Delta_3-\\Delta_2}\n{\\rm x}_{23}^{\\Delta_2+\\Delta_3-\\Delta_1}}\\,,\n\\label{3point}\n}\nwhere $\\Delta_i$ are the dimensions of corresponding operators. Choosing properly ${\\rm x}_i$, we can suppress the dependence on ${\\rm x}_{ij}$ in \\eqref{strconstst}, and apply the prescription given in \\eqref{strconstst} to compute the structure constant $C_{123}$~\\cite{Zarembo:2010,Roiban:2010}. Having in mind that $\\Delta_1\\approx\\Delta_2$ and setting $C_{12}=1$ in \\eqref{2point}, we determine that\n\\eq{\n\\frac{G_3({\\rm x}_1,{\\rm x}_2,{\\rm x}_3=0)}{G_2({\\rm x}_1,{\\rm x}_2)}=C_{123}\\left(\\frac{{\\rm x}_{12}}{|{\\rm x}_1||{\\rm x}_2|}\\right)^{\\Delta_3}\\!.\n\\label{norm3point}\n}\nFor further details we refer the interested reader to \\cite{Buchbinder:2010,Buchbinder:2010vw,Roiban:2010,Bajnok:2014}.\n\n\\sect{Three-point correlators from pulsating strings in $\\mathbb{R}\\times S^3$}\\label{sec3}\n\nIn the present Section we use the approach outlined above for the calculation of specific three-point correlators. Without loss of generality we can fix ${\\rm x}_1=(-1,0,0,0)$ and ${\\rm x}_2=(1,0,0,0)$, from which follows that $R=1$. We consider a particular pulsating string in $\\mathbb{R}\\times S^3\\subset\\axs$ \\cite{Kruczenski:2004} as the string solution that describes the semiclassical trajectory. Using that the string energy is $E=\\sqrt{\\lambda}\\kappa$ and the spin is $J=\\sqrt{\\lambda}{\\cal J}$, the solution is defined as\n\\al{\\label{sol}\nt&=\\kappa\\tau\\,,\\ \\ \\rho=0\\,,\\ \\ \\cos\\theta(\\tau)=a_-\\sn\\!\\left(ima_+\\tau,\\frac{a_-}{a_+}\\right),\\\\ \\nonumber\n\\varphi_1(\\tau)&=-\\frac{\\cal J}{ma_+}\\Pi\\!\\left[\\am\\!\\left(ima_+\\tau,\\frac{a_-}{a_+}\\right),a_-^2,\\frac{a_-}{a_+}\\right],\\ \\ \\varphi_2=m\\sigma\\,,\\\\\n\\kappa^2&=-\\dot{\\theta}^2+\\frac{{\\cal J}^2}{\\sin^2\\theta}+m^2\\cos^2\\theta\\,,\\quad a_{\\pm}^2=\\frac{\\kappa^2+m^2\\pm\\sqrt{(\\kappa^2-m^2)^2+4m^2{\\cal J}^2}}{2m^2}\\,,\n\\nonumber\n}\nwhere we have assumed the notation of \\cite{Gradshteyn} for Jacobi elliptic functions, and $(\\theta,\\varphi_1,\\varphi_2)$ parameterize $S^3\\subset S^5$ with metric\n\\eq{\nds_{S^3}^2=d\\theta^2+\\sin^2\\theta\\,d\\varphi_1^2+\\cos^2\\theta\\,d\\varphi_2^2\\,.\n}\nIn Poincar\\'e coordinates \\eqref{poincare} the AdS part of the solution is\n\\eq{\nz=\\frac{1}{\\cosh[\\kappa(\\tau-\\tau_0)]}\\,,\\quad x_0=\\tanh[\\kappa(\\tau-\\tau_0)]\\,,\\quad x_i=0\\,,\n}\nwhere we have left the integration constant $\\tau_0$ unfixed, because we will need to average our expressions over it. It can be shown that the above solution possesses the right asymptotic behavior, namely, $\\lim_{\\tau\\rightarrow\\pm\\infty}z=0$ and $\\lim_{\\tau\\rightarrow\\pm\\infty}x_0=\\pm1$. Note that by taking ${\\cal J}=0$ we would get the original solution for pulsating strings in $\\mathbb{R}\\times S^2$ \\cite{Minahan:2002rc}.\n\nWe will proceed with the study of the corresponding three-point correlation functions with two heavy and one light operators. We will examine two choices for the light operator~-- dilaton or superconformal primary scalar (chiral primary operator).\n\n\\subsection{Dilaton as light operator}\nIt is known that the ten-dimensional dilaton field is decoupled from the metric in the Einstein frame \\cite{Kim:1985}. Consequently, it is described by a free massless ten-dimensional Laplace equation in $\\axs$. The respective string vertex operator is proportional to the worldsheet Lagrangian ($j\\geq0$ is the $S^5$ momentum of the dilaton)\n\\al{\\label{dilvertex}\nV_L({\\rm x}=0)&=V^{(\\rm dil)}_j(0)=\\hat{c}_{\\Delta}K_{\\Delta}\\,{\\rm X}^j\\big[(\\partial x_m\\bar{\\partial}x^m+\\partial z\\bar{\\partial}z)\/z^2+\n\\partial X_k\\bar{\\partial}X_k+{\\rm fermions}\\big]\\,,\\\\\nK_{\\Delta}&\\equiv\\left(\\frac{z}{z^2+x^mx_m}\\right)^{\\Delta}\\!,\\qquad{\\rm X}\\equiv X_1+iX_2=e^{i\\varphi_1},\\nonumber\n}\nwhere $\\hat{c}_\\Delta$ is a constant determined by the normalization of the dilaton. The conformal dimension of the dilaton is $\\Delta=4+j$ to the leading order in the large 't Hooft coupling expansion. The corresponding operator in the dual gauge theory is proportional to ${\\rm tr}(F^2_{mn}Z^j+\\ldots)$. For $j=0$ it is given by the SYM Lagrangian.\n\nFrom \\eqref{strconstst}, \\eqref{norm3point} and \\eqref{dilvertex} we obtain that\n\\al{\n\\label{dilstrconst}\nC_{123}&=4c_{\\Delta}\\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\int_{-T\/2}^{T\/2}d\\tau_0\\int^{\\infty}_{-\\infty}d\\tau\n\\int^{\\pi\/2}_0d\\sigma\\,K_{\\Delta}\\,U\\,e^{-(\\Delta_2-\\Delta_1)\\kappa\\tau_0}\\\\\nU&={\\rm X}^j\\big[(\\partial x_m\\bar{\\partial}x^m+\\partial z\\bar{\\partial}z)\/z^2+\\partial X_k\\bar{\\partial}X_k\\big],\\qquad c_{\\Delta}=2^{-\\Delta}\\hat{c}_{\\Delta}\\,.\n}\nThe authors of \\cite{Roiban:2010} calculated the normalization constant of the dilaton $\\hat{c}_\\Delta$ as\n\\eq{\n\\hat{c}_\\Delta=\\hat{c}_{4+j}=\\frac{\\sqrt{\\lambda}}{8\\pi N}\\sqrt{(j+1)(j+2)(j+3)}\\,.\n}\nEvaluating $U$ on the pulsating string solution \\eqref{sol}, we get\n\\eq{\nU=(\\kappa^2+\\dot{\\theta}^2-\\frac{{\\cal J}^2}{\\sin^2\\theta}+m^2\\cos^2\\theta)\\,e^{ij\\varphi_1}=2m^2\\cos^2\\theta\\,e^{ij\\varphi_1},\n}\nso that the expression in \\eqref{dilstrconst} takes the following form\n\\al{\\nonumber\nC_{123}&=8m^2c_{\\Delta}\\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\int_{-T\/2}^{T\/2}d\\tau_0\\int^{\\infty}_{-\\infty}d\\tau\n\\int^{\\pi\/2}_0d\\sigma\\,\\frac{\\cos^2\\theta\\,e^{ij\\varphi_1-j{\\cal J}\\tau_0}}{\\cosh^{4+j}[\\kappa(\\tau-\\tau_0)]}\\\\\n&=4\\pi m^2c_{\\Delta}\\int^{\\infty}_{-\\infty}d\\tau'\\,\\frac{e^{-j{\\cal J}\\tau'}}{\\cosh^{4+j}(\\kappa\\tau')} \\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\int_{-T\/2}^{T\/2}d\\tau\\,\\cos^2\\theta\\,e^{ij\\varphi_1-j{\\cal J}\\tau},\n\\label{dilstrconst'}\n}\nwhere in the integral over $\\tau_0$ we have changed the integration variable to $\\tau'=\\tau_0-\\tau$. The first integral in the second line could be computed in terms of hypergeometric functions. The second integral, however, is difficult to calculate analytically due to the presence of an elliptic integral of the third kind in the exponent. Therefore, we will study the structure constant for particular values of the parameters. First, we note that when $m=0$ we get the three-point function with light operator corresponding to a point-like string. In this case, as has been explained in \\cite{Kruczenski:2004}, the equation of motion for $\\theta$ leads to $\\theta=\\pi\/2$. Thus, it follows that $\\kappa={\\cal J}$, which means that $E=J$ as expected for a BPS solution. It can be easily seen that if we set $m=0$ in \\eqref{dilstrconst'}, we will indeed get a vanishing structure constant. Next, let us concentrate on the most significant case of $j=0$. It can be obtained that\n\\al{\\nonumber\nC_{123}&=\\frac{16\\pi}{3}\\frac{m^2}{\\kappa}c_{\\Delta}\\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\int_{-T\/2}^{T\/2}d\\tau\\,\\cos^2\\theta\\\\\n&=\\frac{16\\pi}{3}\\frac{m^2a_-^2}{\\kappa}c_{\\Delta}\\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\int_{-T\/2}^{T\/2}d\\tau\\,\\sn^2\\!\\left(ima_+\\tau,\\frac{a_-}{a_+}\\right).\n}\nThe resulting integral is divergent. In order to obtain a finite result, we analytically continue $m\\rightarrow-im$. We will reverse this operation in the end. We get for the integral\n\\al{\\nonumber\nC_{123}&=-\\frac{16\\pi}{3}\\frac{m^2a_-^2}{\\kappa}c_{\\Delta}\\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\int_{-T\/2}^{T\/2}d\\tau\\,\\sn^2\\!\\left(ma_+\\tau,\\frac{a_-}{a_+}\\right)\\\\\n&=-\\frac{16\\pi}{3}\\frac{m^2a_-^2}{\\kappa}c_{\\Delta}\\lim_{T\\rightarrow\\infty}\\frac{1}{ma_+T}\\int_{-ma_+T\/2}^{ma_+T\/2}dx\\,\\sn^2\\!\\left(x,\\frac{a_-}{a_+}\\right).\n}\nThe integrand is a periodic function over the real numbers, so we need to integrate over only one period in order to obtain the average\n\\al{\\nonumber\nC_{123}&=-\\frac{16\\pi}{3}\\frac{m^2a_-^2}{\\kappa}c_{\\Delta}\\frac{1}{2{\\bf K}\\big(\\frac{a_-}{a_+}\\big)}\\int_{-{\\bf K}\\big(\\frac{a_-}{a_+}\\big)}^{{\\bf K}\\big(\\frac{a_-}{a_+}\\big)}dx\\,\\sn^2\\!\\left(x,\\frac{a_-}{a_+}\\right)\\\\\n&=-\\frac{16\\pi}{3}\\frac{m^2a_-^2}{\\kappa}c_{\\Delta}\\frac{a_+^2}{a_-^2}\\!\\left(1-\\frac{{\\bf E}\\big(\\frac{a_-}{a_+}\\big)}{{\\bf K}\\big(\\frac{a_-}{a_+}\\big)}\\right).\n}\nWe go back to real $m$, and finally get\n\\eq{\nC_{123}=\\frac{16\\pi}{3}\\frac{m^2a_+^2}{\\kappa}c_{\\Delta}\\left(1-\\frac{{\\bf E}\\big(\\frac{a_-}{a_+}\\big)}{{\\bf K}\\big(\\frac{a_-}{a_+}\\big)}\\right).\n\\label{dilstrconstj}\n}\nAs pointed out in \\cite{Costa:2010,Bajnok:2014}, the structure constant should be proportional to the derivative of the string energy with respect to the square root of the 't Hooft coupling\n\\eq{\nC_{123}=\\frac{16\\pi}{3}c_{\\Delta}\\frac{\\partial E(J,I_{\\theta},m,\\sqrt{\\lambda})}{\\partial\\sqrt{\\lambda}}\\,,\n\\label{derenergy}\n}\nwhere $I_{\\theta}$ is the action variable corresponding to $\\theta$. Differentiating the expression for $I_{\\theta}$, obtained in \\cite{Kruczenski:2004}, we are able to confirm the validity of \\eqref{derenergy}.\n\nLet us describe two particular cases of \\eqref{dilstrconstj}. If we consider the case of large energy, namely large $\\kappa=E\/\\sqrt{\\lambda}$, we will get for the structure constant\n\\eq{\nC_{123}=\\frac{8\\pi}{3}\\kappa c_{\\Delta}\\left(1-\\frac{8{\\cal J}^2-m^2}{8\\kappa^2}-\\frac{4m^2{\\cal J}^2-m^4}{16\\kappa^4}+\\ldots\\right).\n}\nAnother interesting case is when ${\\cal J}\\ll\\kappa$. Then we get\n\\eq{\nC_{123}\\approx\\frac{16\\pi}{3}\\kappa c_{\\Delta}\\!\\left(1-\\frac{{\\bf E}\\!\\left(\\frac{m}{\\kappa}\\right)}{{\\bf K}\\!\\left(\\frac{m}{\\kappa}\\right)}\\right).\n}\n\n\\subsection{Superconformal primary scalar as light operator}\nThe string state that corresponds to the chiral primary operator results from the trace of the graviton in the $S^5$ section of the geometry \\cite{Kim:1985,Lee}. As detailed in \\cite{Zarembo:2010,Berenstein:1998}, the bosonic part of the respective operator takes the form\\footnote{We neglect derivative terms that will not influence our calculations since we have made the restriction ${\\rm x}_1=-{\\rm x}_2$.}\n\\al{\\label{cpovertex}\nV_L({\\rm x}=0)&=V^{(\\rm CPO)}_j(0)=\\hat{c}_{\\Delta}K_{\\Delta}\\,{\\rm X}^j\\big[(\\partial x_m\\bar{\\partial}x^m-\\partial z\\bar{\\partial}z)\/z^2-\n\\partial X_k\\bar{\\partial}X_k\\big]\\,,\\\\\nK_{\\Delta}&\\equiv\\left(\\frac{z}{z^2+x^mx_m}\\right)^{\\Delta}\\!,\\qquad{\\rm X}\\equiv X_1+iX_2=e^{i\\varphi},\\nonumber\n}\nwhere $\\hat{c}_\\Delta$ is again given by the normalization. The corresponding operator in the dual gauge theory is the BMN operator ${\\rm tr}Z^j$ with dimension $\\Delta=j$.\n\nWe can infer from \\eqref{strconstst}, \\eqref{norm3point} and \\eqref{cpovertex} that\n\\al{\n\\label{cpostrconst}\nC_{123}&=4c_{\\Delta}\\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\int_{-T\/2}^{T\/2}d\\tau_0\\int^{\\infty}_{-\\infty}d\\tau\n\\int^{\\pi\/2}_0d\\sigma\\,K_{\\Delta}\\,U\\,e^{-(\\Delta_2-\\Delta_1)\\kappa\\tau_0}\\\\\nU&={\\rm X}^j\\big[(\\partial x_m\\bar{\\partial}x^m-\\partial z\\bar{\\partial}z)\/z^2-\\partial X_k\\bar{\\partial}X_k\\big],\\qquad c_{\\Delta}=2^{-\\Delta}\\hat{c}_{\\Delta}\\,,\n}\nwhere the constant $\\hat{c}_\\Delta$ of the superconformal scalar is \\cite{Zarembo:2010,Berenstein:1998}\n\\eq{\n\\hat{c}_\\Delta=\\hat{c}_j=\\frac{\\sqrt{\\lambda}}{8\\pi N}(j+1)\\sqrt{j}\\,.\n}\nThe expression for $U$, evaluated on the solution \\eqref{sol}, leads to\n\\eq{\nU=2\\!\\left(\\frac{\\kappa^2}{\\cosh^2[\\kappa(\\tau-\\tau_0)]}-\\frac{{\\cal J}^2}{\\sin^2\\theta}-m^2\\cos^2\\theta\\right)\\!\ne^{ij\\varphi_1},\n}\nso that \\eqref{cpostrconst} gives\n\\eq{\nC_{123}=4\\pi c_{\\Delta}\\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\!\\int_{-T\/2}^{T\/2}\\!\\!\\!\\!d\\tau_0\\!\n\\int^{\\infty}_{-\\infty}\\!\\!\\!\\!d\\tau\\,\\frac{e^{ij\\varphi_1-j{\\cal J}\\tau_0}}{\\cosh^j[\\kappa(\\tau-\\tau_0)]}\\!\n\\left(\\frac{\\kappa^2}{\\cosh^2[\\kappa(\\tau-\\tau_0)]}-\\frac{{\\cal J}^2}{\\sin^2\\theta}-m^2\\cos^2\\theta\\!\\right).\n}\nAnalogously to the dilaton case the integrals cannot be calculated analytically, so we take ${\\cal J}$ to be small and consider only the first term in the resulting series. We also change the variable $\\tau_0$ to $\\tau'=\\tau_0-\\tau$ and get\n\\al{\\nonumber\nC_{123}&=4\\pi c_{\\Delta}\\int^{\\infty}_{-\\infty}\\frac{d\\tau'}{\\cosh^j(\\kappa\\tau')} \\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\int_{-T\/2}^{T\/2}d\\tau\\left(\\frac{\\kappa^2}{\\cosh^2(\\kappa\\tau')}-m^2\\cos^2\\theta\\right)\\\\\n&=4\\pi\\kappa^2c_{\\Delta}\\int^{\\infty}_{-\\infty}\\frac{d\\tau'}{\\cosh^{j+2}(\\kappa\\tau')}-\n4\\pi m^2c_{\\Delta}\\int^{\\infty}_{-\\infty}\\frac{d\\tau'}{\\cosh^j(\\kappa\\tau')} \\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\int_{-T\/2}^{T\/2}d\\tau\\,\\cos^2\\theta\\nonumber\\\\\n&=4\\pi^{3\/2}\\kappa c_{\\Delta}\\frac{\\Gamma[\\frac{j}{2}]}{\\Gamma[\\frac{1+j}{2}]}\n\\left(\\frac{{\\bf E}\\!\\left(\\frac{m}{\\kappa}\\right)}{{\\bf K}\\!\\left(\\frac{m}{\\kappa}\\right)}-\\frac{1}{1+j}\\right).\n}\n\n\\sect{Conclusion}\n\nThe AdS\/CFT correspondence has been through significant development in recent years. One of the active areas of research has been the holographic calculation of three-point functions at strong coupling. The correlation functions of three massive string states escape full comprehension so far~\\cite{Klose:2011}, but we have uncovered almost all features of correlators containing two heavy and one light states in the semiclassical approximation~\\cite{Zarembo:2010}--\\cite{Hernandez:2010}.\n\nIn the present paper we calculated three-point correlation functions of two string and one supergravity states from string theory in $\\axs$ at strong coupling, applying the approach of \\cite{Roiban:2010} for computing correlators using the respective vertex operators. We examined the method, which had been correctly modified by the authors of \\cite{Bajnok:2014}, for the occasion of a particular pulsating string solution, providing some limiting cases.\n\nOne of the possible future directions for exploration is the connection of our work to recent developments in the calculation of correlation functions with heavy states based on integrability methods in ${\\cal N}=4$ SYM \\cite{Escobedo:2010}.\n\n\\section*{Acknowledgments}\nThe authors would like to thank N. Bobev and H. Dimov for valuable discussions. This work was supported in part by the BNSF grant DFNI T02\/6.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\n\nYoung nearby moving groups (MGs hereafter) are sparse stellar associations whose members were formed together in loose environment and share common proper motions.\nTherefore, members of MGs spread out over time in space, and, after hundreds of millions of years, they are not easily distinguishable against old field stars.\nAfter the first identification of the nearby young MG, TW Hydrae Association \\citep{kas97}, about ten additional MGs were identified \\citep{web99, zuc00, tor00, zuc01a, zuc04, tor08, zuc11}.\n \nNearby, young MGs are unique objects in astronomy because of their proximity and youth.\nMG members are prime targets for exoplanet imaging \\citep{lag10, cha04, mar08}, because orbiting young planets are brighter and more widely separated compared to those around old, distant stars.\nThey are also useful in calibrating stellar age-dating methods and in studying the evolution of low-mass stars (Binks \\& Jeffries 2016; Malo et al. 2014; Murphy, Lawson \\& Bento 2015).\nIn addition, they are essential to understanding the recent star formation history in the solar neighbourhood \\citep{schn12, tor03}.\nThese studies are critically dependant on assignments of MG membership because they rely on MG group properties such as age, space motion, and location from Earth that are derived from known MG members.\n\nBecause of their importance, identification of MG members has been intensively investigated (Song, Zuckerman \\& Bessell 2003; Zuckerman \\& Song 2004; Torres et al. 2006; Schlieder, L\\'{e}pine \\& Simon 2010; Kiss et al. 2011; Rodriguez et al. 2011; Shkolnik et al. 2012; Malo et al. 2013; Gagn\\'{e} et al. 2014; Murphy et al. 2015). \nAmong methods applied to searching for MG members, a statistical approach based on a Bayesian framework has become popular recently \\citep{mal13, gag14}.\nWhile there are many advantages in using the Bayesian method (e.g., an intuitively easier interpretation of the result due to the quantitative membership probability and the availability of marginalising over nuisance parameters), the resultant membership probability needs to be carefully adopted because of its sensitive dependence on input models. \n\nIn this paper, we examine the impact of models and prior knowledge in the Bayesian MG membership probability, focusing on the accepted member list and the distribution functions for Beta-Pictoris moving group (BPMG hereafter) and field stars.\nWe then suggest improvements to the membership calculations, and finally provide a list of confirmed and probable members of the BPMG.\n\n\n\n\\section{Bayesian moving group membership calculations}\nOur method is based on the same Bayesian principle as used in BANYAN II \\citep{gag14}, and one of our main purposes of this paper is to demonstrate the careful treatments of model parameters.\nTherefore, to minimize any possible confusion, we decide to use the BANYAN II notation in describing various terms related to the Bayesian MG membership probability calculation.\nThroughout this paper, in developing, validating, and comparing our method and result, we used BPMG as the main test case because this MG is one of the youngest, nearest MGs with many discovered members spread over a large area of the sky.\nFor various purposes, different lists of stars for BPMG were used (see Table~\\ref{tab1}).\n\n\n\\subsection{Validation of our calculation: comparison against the BANYAN II result}\nWe developed a \\textsc{python} script to calculate Bayesian MG membership probability.\nBayesian probability (the posterior probability) is proportional to the product of the prior probability and the likelihood, which \nare both derived from models for MGs and field stars.\nTo validate our code, we compare our membership calculation results (the Bayesian probability) against those of BANYAN II using identical parameters.\nPrior probability distribution functions (PDFs) for observables such as distance, radial velocity (RV), magnitude of proper motion, and galactic latitude are reproduced, and two of these are compared in Fig.~\\ref{fig1}.\nPrior PDFs from BANYAN II and those from this study are similar but not exactly the same because these prior PDFs are generated from random simulated stellar distributions.\nWe also used the BANYAN II group size ratios between MGs and the field stars.\n\nThe membership probability calculations from BANYAN II and those from our code are compared in Fig.~\\ref{fig2} using a list of BPMG members taken from the BANYAN II webpage ({\\it BANYAN II list}}%{{\\it list 1}, see Table~\\ref{tab1}).\nThis figure shows that our code replicates almost the same result of BANYAN II except for a handful of abnormal cases where the difference between the two calculations is likely caused by the small difference in prior PDFs shown in Fig.~\\ref{fig1}. \n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{fig1.eps}\n\\caption{Prior probability distribution functions (PDFs) of RV and distance for BPMG from BANYAN II (red) and those from this study (blue) assuming identical model parameters.\nThe differences of two prior PDFs are presented with green lines.}\n\\label{fig1}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{fig2.eps}\n\\caption{A comparison of membership probabilities from BANYAN II and those from this study ({\\it Case I}) utilizing BPMG members from the {\\it BANYAN II list}}%{{\\it list 1}\\ (Table~\\ref{tab1}).\nBoth calculations are based on the same models (Table~\\ref{tab2}).}\n\\label{fig2}\n\\end{figure}\n\n\n\\begin{landscape}\n\\begin{table}\n\\caption{Star lists used in this study.}\n\\label{tab1}\n\\begin{threeparttable}\n\\begin{tabular}{p{.1\\linewidth}p{.035\\linewidth}p{.35\\linewidth}p{.445\\linewidth}\n\\hline\nName & N & Description & Purpose of usage \\\\ \\hline\n{\\it BANYAN II list}}%{{\\it list 1} & 50 & A list of previously known BPMG members taken from the BANYAN II webpage \\newline (www.astro.umontreal.ca\/$\\sim$gagne\/banyanII.php; \\citet{gag14}). & To validate our calculation by comparing it to BANYAN II's calculation (Section 2.1). \\newline To create lists of initially accepted BPMG members ({\\it exclusive list}}%{{\\it list 2 (exclusive)}\\ and {\\it inclusive list}}%{{\\it list 2 (inclusive)}) (Section 2.2.1). \\\\\n\\hline\n{\\it exclusive list}}%{{\\it list 2 (exclusive)} \\newline {\\it inclusive list}}%{{\\it list 2 (inclusive)} & 39 \\newline 47 & Lists of initially accepted BPMG members based on the membership assessment criteria (exclusive and inclusive). \\newline Subsets of the {\\it BANYAN II list}}%{{\\it list 1}. & To construct improved BPMG models (Section 2.2.1 and 2.2.2). \\\\\n\\hline\n{\\it SIMBAD list}}%{{\\it list 3} & 275 & A list of BPMG members in SIMBAD database as of 2017 April \\newline (http:\/\/simbad.u-strasbg.fr\/simbad\/), which constitutes the whole {\\it BANYAN II list}}%{{\\it list 1}. & To test effects of models (Section 3.1). \\newline To provide the updated list of bona fide and probable members of BPMG (Section 3.2).\\\\\n\\hline\nbona fide member list & 51(57\\tnote{a} ) & A confirmed BPMG member list in this study (Table~\\ref{tab5}). & To derive revised model parameters for BPMG (Section 3.2). \\\\\n\\hline\n\\end{tabular}\n\\begin{tablenotes}\n\\item[a] Including 6 classical BPMG members showing moderate signs of youth, discussed in section 3.2.1.\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\n\\begin{table}\n\\caption{Combinations of different factors in building models.}\n\\label{tab2}\n\\begin{threeparttable}\n\\begin{tabular}{llcccccccc}\n\\hline\n\\multirow{2}{*}{Model factors} & \\multirow{2}{*}{Type} & & \\multicolumn{7}{c}{cases in this study} \\\\\n\\cline{4-10} \n& & BANYAN II & {\\it Case I}\\ & {\\it Case II (Excl.)}\\ & {\\it Case II (Incl.)}\\ & {\\it Case III (Excl.)}\\ & {\\it Case III (Incl.)}\\ & {\\it Case IV}\\ & {\\it Case V}\\ \\\\\n\\hline\n\\multirow{3}{*}{Selection of BPMG members} & BANYAN II & O & O & & & & &O& \\\\ \\cdashline{2-10}\n & Exclusive & & & O& &O & & &O\\\\ \\cdashline{2-10}\n & Inclusive & & & &O& &O& & \\\\\n\\hline\n\\multirow{2}{*}{$XYZ$ distribution function of BPMG members} & Gaussian & O &O &O&O& & &O& \\\\ \\cdashline{2-10}\n & Uniform & & & & &O&O& &O\\\\\n\\hline\n\\multirow{2}{*}{Distribution models of field stars} & BANYAN II & O & O&O&O&O&O& &\\\\ \\cdashline{2-10}\n & A new model & & & & & & &O&O\\\\\n\\hline\n\\end{tabular}\n\\end{threeparttable}\n\\end{table}\n\\end{landscape}\n\n\n\n\n\n\\subsection{Improvement over BANYAN II: updating models}\n\n\nDifferent models modify likelihoods and prior PDFs, making changes in membership probabilities. \nIn this study, we consider three important factors in building models: \n(1) a list of adopted initial MG members, \n(2) a distribution function of BPMG members in $XYZ$, and \n(3) new distribution functions of field stars in $XYZ$ and $UVW$ \\footnote{$U$, $V$, and $W$ are positive toward the directions of the Galactic centre, Galactic rotation, and the North Galactic Pole, respectively..\nTo investigate the effects of these three factors, we carried out various combinations of them and compared the result against that of BANYAN II (see Table~\\ref{tab2}).}\n\n\\subsubsection{Re-examination of MG membership}\nTo define the characteristics of a MG, one has to start with a certain list of MG members.\nFor example, to measure the extent of the distribution of MG members in $XYZ$ and $UVW$, one has to model the distribution of accepted members in such 3D spaces, which means that a different set of stars will produce different distribution model parameters for the MG.\nThis seemingly straightforward task of establishing a list of accepted MG members is more complicated because the assignment of membership status is an iterative process.\nStarting with a stringent initial list of accepted members, a MG will have tighter distributions in $XYZ$ and $UVW$ which in turn forces any candidate members need to fit the tighter parameters.\nOn the other hand, if the membership assessment starts with a more lenient list of members, the distribution model of the MG will become more inclusive, accepting more marginal members as true members.\n\n\nIn this section, we reassess membership of previously known BPMG members from the {\\it BANYAN II list}}%{{\\it list 1}.\nA true member should possess not only similar kinematic characteristics ($XYZ$ and $UVW$) but also a similar age with other members.\n\n\nFirstly, we tried to flag outliers in $XYZ$ and $UVW$ by calculating standard deviations ($\\sigma$) in each $X, Y, Z, U, V$, and $W$.\nTo calculate standard deviations that properly represent the MG as a whole, we excluded obvious outliers that are above the upper limit \\footnote{(3rd quartile [75\\%-ile] + 1.5$\\times$ interquartile range (IQR))} or below the lower limit \\footnote{(1st quartile [25\\%-ile] - 1.5$\\times$ IQR)}. \nHIP 11360 and $\\eta$ Tel A deviate $\\gtrsim$5$\\sigma$ in $UVW$ from the mean.\nIncluding $\\eta$ Tel B, these 3 stars were determined to be kinematic outliers.\nAdditionally, HIP 50156 was determined as a marginal outlier because this star has Z value about 4.5$\\sigma$ away from the mean.\n\nAge outliers were determined based on age indicators.\n Genuine BPMG members should show clear signs of youth appropriate to the age of BPMG (12$-$25 Myr: \\cite{zuc04, ort02, bin16, mam14}).\nYoung ($<$100 Myr), Sun-like or lower mass (mid-F to M types) stars can be distinguished from older counterparts because they have brighter photometric magnitude, NUV excess, X-ray excess, and\/or strong Li absorption \\citep{sod10, rod11, zuc04, bin16}.\nFourty-two out of 50 stars in the {\\it BANYAN II list}}%{{\\it list 1}\\ were estimated to be $\\lesssim$25 Myr.\nAmong the remaining 8 stars, no age-related data exists for $\\eta$ Tel B, \nand the age estimation for the other 7 stars involves significant uncertainties (HIP 10679, HIP 10680, 2M J06085283-2753583, HIP 21547, HIP 88726A, HIP 88726B, and HIP 92024).\nBoth HIP 10679 and HIP 10680 show strong Li absorption features but can be $\\sim$100 Myr old based on a $V-K$ versus $M_K$ colour-magnitude diagram (CMD).\nThe location of 2M J06085283-2753583 on the CMD is ambiguous, making it difficult to unambiguously assess if this star is $\\lesssim$ 25 Myr.\nBecause HIP 21547, HIP 88726 A, HIP 88726 B, and HIP 92024 have early spectral types (F0 to A5), reliable age estimates are difficult.\nEven though the age estimates of these 7 stars are ambiguous, none of them are marked as evident old star interlopers. \n\n\nBased on age and kinematic data, one can try to build two different extreme cases for the initial accepted BPMG member list: (1) an {\\it inclusive list}}%{{\\it list 2 (inclusive)}\\ containing all marginal members with large age uncertainties and outlying kinematics and (2) an {\\it exclusive list}}%{{\\it list 2 (exclusive)}\\ containing only unambiguous members in age and kinematics. \nUsing these two lists, parameters for two BPMG distribution models (BPMG$_{\\rm Excl}$\\ and BPMG$_{\\rm Incl}$\\ in Table~\\ref{tab3}) are estimated \n by fitting a single Gaussian model utilizing the Gaussian mixture model algorithm from \\textsc{python scikit-learn} package \\citep{ped11}. \nThe distance prior PDFs for both models and 2D projections of BPMG$_{\\rm Excl}$\\ are presented in Fig.~\\ref{fig3}, ~\\ref{fig5}, ~\\ref{fig6}, and ~\\ref{fig7}.\n\n\\begin{figure}\n\\includegraphics[width=0.99\\columnwidth]{fig3.eps}\n\\caption{Prior PDFs of distance for BPMG models. \nInset is an enlarged view of probabilities at around 100 pc. \nModels using minimum volume enclosing ellipsoid (uniform distribution in $XYZ$; BPMG$_{\\rm Unif,Excl}$\\ and BPMG$_{\\rm Unif,Incl}$$-${\\it Case III}) reach zero probability at around 100 pc, while other models, assuming Gaussian distribution, do not.}\n\\label{fig3}\n\\end{figure}\n\n\n\\begin{figure*}\n \\includegraphics[width=0.99\\textwidth]{fig4.eps}\n \\caption{Normalized density of stars in the {\\it exclusive list}}%{{\\it list 2 (exclusive)}\\ in each $X$, $Y$, and $Z$.\nThe density function using a kernel density estimation (KDE), which shows binning-free distribution, is presented with a red dashed line overlaid on the data histogram.\nIn the boundary region, KDE underestimates the density because there is no data (boundary effects) and this boundary effects are corrected by truncating the kernel at the outermost boundaries (minimum and maximum values for each $X$, $Y$, $Z$).\nGaussian and uniform distribution of pseudo data are presented as shaded area with blue and green colors, respectively.\n}\n\\label{fig4}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\\begin{subfigure}{0.66\\textwidth}\n \\includegraphics[width=\\textwidth]{fig5a.eps}\n\\end{subfigure}\n\\begin{subfigure}{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{fig5b.eps}\n\\end{subfigure}\n\\caption{2D projections of BPMG models in $XYZ$.\nBPMG$\\rm_{BANYANII}$, BPMG$_{\\rm Excl}$, and BPMG$_{\\rm Unif,Excl}$\\ models are presented with blue, red, and grey, respectively.\nStars in the {\\it exclusive list}}%{{\\it list 2 (exclusive)}\\ are presented as dots.\nWe plotted 1.2 times the MVEE semi-major axes for BPMG$_{\\rm Unif,Excl}$\\ and 2-$\\sigma$ boundaries for the other two Gaussian models.\nThe former should have 100 per cent of integrated probability within the plotted boundaries while the latter should have 87 per cent of integrated probabilities within the plotted boundaries.\nRight panel presents the number density (i.e., probability density function) along principal major axes ($X'$, $Y'$, and $Z'$) for each model.}\n\\label{fig5}\n\\end{figure*}\n\n\\begin{figure*}\n \\includegraphics[width=\\textwidth]{fig6.eps}\n\\caption{2D projections of BPMG models in $UVW$.\nBPMG$\\rm_{BANYANII}$ and BPMG$_{\\rm Excl}$\\ models are presented with blue and red, respectively. \nStars in the {\\it exclusive list}}%{{\\it list 2 (exclusive)}\\ are presented as dots.\nThe model boundaries are 2-$\\sigma$ values in each semi-major axis.}\n\\label{fig6}\n\\end{figure*}\n\n\\subsubsection{$XYZ$ Distribution models of BPMG members}\nIn this section, we discuss a proper probability distribution of MG members as a function of distance from the centre of the MG.\nShould a star closer to the MG centre be assigned with a higher membership probability, or should any star within a given distance limit be treated with the same probability?\nA proper way to handle the model probability distribution near the boundary of a MG should be closely related to our understanding of the origin of MGs.\nIf a MG was formed in a loosely bound environment at birth without any noticeable over-density of their members against field stars, a uniform probability distribution model, i.e., assigning the same membership probability for all candidate members within the MG boundary, makes more sense.\nHowever, if a MG was formed in a more gravitationally bound environment with a central concentration of members, a Gaussian probability distribution model may be more appropriate.\nBetween uniform or Gaussian distribution models in $XYZ$, to investigate which model is more appropriate to represent the actual distribution of MG members, one has to check the existence of an over-dense region in the spatial distribution of the models.\nIt is difficult to scrutinize any over-density in an already dispersed low-density group of stars.\nTherefore, checking such an over-density in the back-traced positions in time is easier than a case of using the current positions.\nHowever, the propagation of errors backward in time makes such analyses very difficult.\nFor given current knowledge and data of young MGs in the solar neighborhood, it is very difficult to distinguish between these two models.\n\nTherefore, the virtue of using a uniform or a Gaussian $XYZ$ distribution model needs to be evaluated by the current distribution of members.\nFig.~\\ref{fig4} shows the normalized number density of BPMG members (stars in the {\\it exclusive list}}%{{\\it list 2 (exclusive)}) in each $X$, $Y$, and $Z$.\nFor eliminating the binning effect in the investigation of the distributions, a kernel density estimation (KDE) was applied to these data.\nIt is well known that KDE underestimates the density near the boundary.\nAfter correcting the boundary effect by truncating the kernel at the outermost boundaries (minimum and maximum values for each $X$, $Y$, and $Z$), $X$ and $Y$ values do not appear to be centrally concentrated.\nHowever, $Z$ values appear to be more centrally concentrated.\nIt is interesting to note that BPMG members are concentrated in the $Z$-direction.\nBecause of the vertical structure of Milky Way with the scale height of about 300 pc, a certain degree of concentration of stars is expected but not at this level.\nThe peak of the $Z$-concentration is around $-$15 pc which may be the manifestation of the Sun being located 10$-$30 pc above the Galactic plane \\citep{hum95, jos07}.\n\nAs shown in Fig.~\\ref{fig4}, the difference in $XYZ$ distribution models is not that significant and the best model seems to be a combined one (uniform in $X$ \\& $Y$ and Gaussian in $Z$).\nIf a future survey for BPMG members especially close to the known BPMG boundary is carried out, such a combined $XYZ$ distribution model is recommended.\nIn this paper, we select a version of the uniform distribution model in all three ($XYZ$) directions for simplicity.\n\n\nWe compared results from two different distribution models: Gaussian and uniform models (in Table~\\ref{tab2}, {\\it Case II}\\ and {\\it Case III}, respectively).\nThroughout the paper, the distribution function of BPMG members in $UVW$ is assumed to be a Gaussian similar to the treatment in BANYAN II.\n\n{\\it Case III (Excl.)}\\ and {\\it Case III (Incl.)}\\ adopt members in the {\\it exclusive \\rm and \\it inclusive lists}, to construct uniform $XYZ$ models for BPMG utilizing a minimum volume enclosing ellipsoid (MVEE) algorithm \\citep{kum05}.\nThe model parameters for these two cases are presented in Table~\\ref{tab3} (BPMG$_{\\rm Unif,Excl}$\\ and BPMG$_{\\rm Unif,Incl}$), \n and the distribution of one model (BPMG$_{\\rm Unif,Excl}$) is shown in Fig.~\\ref{fig5}.\nThe uniform membership probability functions fitted as MVEEs would be step functions; a constant within each ellipsoid, but zero outside of the ellipsoid.\nTo consider candidate members sitting barely outside of the distance limit of known members, we increase the lengths of principal axes of the MVEEs by 1.2 times.\nFig.~\\ref{fig3} shows that probabilities from these two uniform models drop to zero above a certain distance, while those from Gaussian models never reach zero.\n\n\n\n\n\n\n\\begin{landscape}\n\\begin{table}\n \\caption{Model parameters for BPMG. Parameters for other MGs (i.e., TWA, Tuc-Hor, AB Doradus, Columba, Carina, and Argus) are taken from BANYAN II \\citep{gag14}.}\n \\label{tab3}\n \\begin{threeparttable}\n \\begin{tabular}{crrrrrrrrrrrrrrrrrr}\n \\hline\n Name & $X$ & $Y$ & $Z$ & $\\sigma_{X}$ & $\\sigma_{Y}$ & $\\sigma_{Z}$ & $\\phi_{XYZ}$ & $\\theta_{XYZ}$ & $\\psi_{XYZ}$ & $U$ & $V$ & $W$ & $\\sigma_{U}$ & $\\sigma_{V}$ & $\\sigma_{W}$ & $\\phi_{UVW}$ & $\\theta_{UVW}$ & $\\psi_{UVW}$ \\\\ \n & (pc) & (pc) & (pc) & (pc) & (pc) & (pc) & ($^\\circ$) & ($^\\circ$) & ($^\\circ$) & (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & ($^\\circ$) & ($^\\circ$) & ($^\\circ$) \\\\\n \\hline\n \\multicolumn{19}{c}{Model parameters from BANYAN II} \\\\ \\hline\n BPMG$_{\\rm BANYANII}$ & 7.6 & -3.5 & -14.5 & 8.2 & 13.5 & 30.7 & -90.2 & 65.1 & -77.9& -11.0 & -15.6 & -9.2 & 1.4 & 1.7 & 2.5 & -113.0 & -70.3 & 76.6\\\\ \\hline\n\\multicolumn{19}{c}{Model parameters (Section 2)} \\\\ \\hline\nBPMG$_{\\rm Excl}$\\tnote{a}& 9.4 & -5.6 & -13.5 & 7.8 & 13.5 & 29.1 & 88.7 & -62.1 & 78.6 & -10.7 & -16.0 & -9.3 & 1.4 & 1.5 & 2.4 & -105.9 & -51.2 & 84.5 \\\\\nBPMG$_{\\rm Incl}$\\tnote{b}& 7.7 & -5.2 & -12.8 & 8.3 & 13.6 & 29.3 & 86.9 & -60.5 & 80.0 & -10.7 & -16.0 & -9.2 & 1.3 & 1.6 & 2.4 & -107.9 & -49.3 & 81.0 \\\\\nBPMG$_{\\rm Unif,Excl}$\\tnote{a}& 19.7 & -2.8 & -14.4 & 22.0 & 32.5 & 67.0 & -24.9 & 77.2 &-18.4 & \\multicolumn{9}{c}{same to BPMG$_{\\rm Excl}$} \\\\\nBPMG$_{\\rm Unif,Incl}$\\tnote{b}& 13.1 & -3.9 & -10.0 & 29.1 & 33.2 & 68.5 & -93.6 & 62.1 & 86.8 & \\multicolumn{9}{c}{same to BPMG$_{\\rm Incl}$} \\\\ \\hline\n\\multicolumn{19}{c}{Revised model parameters using the confirmed member list of BPMG in Table 5 (Section 3)} \\\\ \\hline\nBPMG$\\rm_{revised}$& 13.4 & -3.4 & -18.1 & 19.1 & 32.0 & 71.2 & -71.6 & 74.8 & 111.2 & -10.4 & -15.9 & -9.1 & 1.2 & 1.4 & 2.2 & -93.1 & -45.5 & -81.5 \\\\\n\\hline\n\\end{tabular}\n\\begin{tablenotes}\n \\item[a] The properties are obtained from the {\\it exclusive list}}%{{\\it list 2 (exclusive)}.\n \\item[b] The properties are obtained from the {\\it inclusive list}}%{{\\it list 2 (inclusive)}.\n \\item Notes. \n \\item 1. $X, Y, Z$ and $U, V, W$ are central positions of the ellipsoidal models.\n \\item 2. $\\sigma_X$, $\\sigma_Y$, $\\sigma_Z$, and $\\sigma_U$, $\\sigma_V$, $\\sigma_W$ are the lengths of semi-major axes in the direction of the principal axes. \n Values from uniform distribution models ($XYZ$ models for BPMG$_{\\rm Unif,Excl}$, BPMG$_{\\rm Unif,Incl}$, and BPMG$\\rm_{revised}$) are the lengths of semi-principal axes of the minimum volume enclosing ellipsoid (MVEE), while other values are 1-$\\sigma$ lengths from the Gaussian models.\nThe $\\sigma$ values from MVEE and Gaussian model are not comparable because all members should be enclosed within the $\\sigma$ in MVEE, while a large portion of members ($\\sim$80 per cent) are located outside of 1-$\\sigma$ in the Gaussian model.\nIn the 3D Gaussian distribution, 20, 74, and 97 per cent of data are within 1, 2, and 3-$\\sigma$ boundaries, respectively.\n \\item 3. $\\phi$, $\\theta$, $\\psi$ are Euler rotational angles of the ellipsoids in degrees.\n \\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\n\\begin{table}\n \\caption{Group properties of the field star model in $UVW$ in this study.}\n \\label{tab4}\n \\centering\n \\begin{threeparttable}\n \\begin{tabular}{crrrrrrrrrr}\n \\hline \nName & $U$ & $V$ & $W$ & $\\sigma_{U}$ & $\\sigma_{V}$ & $\\sigma_{W}$ & $\\phi$ & $\\theta$ & $\\psi$ & Weight\\tnote{a} \\\\\n& (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & ($^\\circ$) & ($^\\circ$) & ($^\\circ$) \\\\\n\\hline\n FLD1 & -17.1 & -18.0 & -6.3 & 6.8 & 8.1 & 14.2 & 48.5 & -86.2 & -56.2 & 0.29\\\\\nFLD2 & -32.0 & -16.2 & -8.1 & 15.8 & 16.5 & 19.4 & -54.7 & -80.7 & 112.6 & 0.26\\\\ \nFLD3 & 2.4 & -0.8 & -6.7 & 8.8 & 9.8 & 13.9 & 88.8 & 1.2 & -97.4 & 0.24\\\\\nFLD4 & 22.1 & -16.8 & -9.5 & 16.1 & 16.8 & 21.3 & 110.9 & -67.8 & 104.1 &0.21\\\\\n\\hline \n\\end{tabular}\n\\begin{tablenotes}\n \\item[a] The relative number density.\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\\end{landscape}\n\n\\clearpage\n\n\n\n\n\\subsubsection{Distribution models of field stars}\nSimilar to the MG properties, field star properties can be acquired by finding the best-fit distribution model to the actual distribution of field stars in $XYZ$ and $UVW$.\nIn $XYZ$, we assumed that stars are uniformly distributed although this uniform spherical distribution may not be perfect in the Z direction at large distance because the scale height of the Galactic disk is about 300 pc.\nThis uniform field star model in $XYZ$ explains the actual distribution of nearby stars better than the case of BANYAN II.\nUtilizing the Besan\\c{c}on galaxy model \\citep{rob03, rob12}, \\citet{gag14} created a field star model by fitting the $XYZ$ distribution of all field stars inside of 200 pc with a single Gaussian model for the young ($<$1 Gyr) and old ($\\ge$1 Gyr) field stars.\nThe field star model used in BANYAN II predicts a high concentration of field stars at $\\sim$120 pc, \nwhile the largest stellar catalogue with measured parallactic distances at present ($Tycho-Gaia$ Astrometric Solution (TGAS); Michalik, Lindegren \\& Hobbs 2015) shows no such concentration (Fig.~\\ref{fig7}), which supports the uniform field star distribution in $XYZ$.\n\n\n\\begin{figure*}\n\\includegraphics[width=0.99\\columnwidth]{fig7.eps}\n\\caption{Prior PDFs of distance for field star models used in BANYAN II (dashed line) and in this study (solid line).\nBANYAN II includes two field star models (young and old), but their distance PDFs are similar and we present the old one.\nData from $Hipparcos$ (dot-dashed line) and TGAS (dotted line) catalogues are presented for comparisons.\nProbabilities are normalized to $Hipparcos$ at 30 pc that would complete down to early M type stars.\nWe note that $Hipparcos$ data especially suffers from the limited survey depth, causing an apparent peak around $\\sim$60 pc.}\n\\label{fig7}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.9\\linewidth]{fig8.eps}\n\\caption{Stellar distribution in $UVW$. First column represents the distribution of $Hipparcos$ stars with measured RV ($\\sim$50,000 stars, \\citet{and12}; references therein).\nSecond column shows density maps of TGAS stars of possible $UVW$ values over the RV range of $-$100 to $+$100 km s$^{-1}$.\nResidual contour plots at 3rd and 4th columns represent the differences between $Hipparcos$ stars and simulated stars generated by field star model from this study (3rd column), and those from BANYAN II (4th column).\nColors correspond to the colorbar, which scales from the minimum to the maximum values of the differences, that appear in the BANYAN II$-$HIP map on the $VW$ plane.}\n\\label{fig8}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n \\includegraphics[width=\\textwidth]{fig9.eps}\n \\caption{Field star distribution models from this study (grey) and those from BANYAN II (red and blue represent old and young field stars, respectively.).\n Plotted are 1-$\\sigma$ ellipses.}\n \\label{fig9}\n\\end{figure*}\n\nThe field star distribution in $UVW$ was examined using $Hipparcos$ and TGAS catalogues.\nTo examine the distribution of stars in $UVW$, one has to know six parameters: RA, Dec., distance, proper motions in RA and Dec., and RV.\nAproximately 50,000 stars in the $Hipparcos$ catalogue have all six parameters, and they clearly show 3-4 subgroups in $UVW$ (Fig.~\\ref{fig8}).\nBy applying a wavelet analysis technique on the 2D $UVW$ plots, Skuljan, Hearnshaw \\& Cottrell (1999) and Chereul, Cr\\`{e}z\\`{e} \\& Bienaym\\`{e} (1997) identified $\\sim$4 kinematic clusters of nearby field stars in the $Hipparcos$ catalogue.\nBecause the $Hipparcos$ catalogue is limited to stars mostly within $<$100 pc from the Sun, one has to use a larger kinematic catalogue to check if the apparent 4 kinematic clusters of nearby stars exist beyond 100 pc.\nThe TGAS catalogue provides 5 parameters for $\\sim$2 million stars, and the only missing parameter to calculate $UVW$ is RV.\nSince almost all nearby stars do not travel faster than $\\sim$100 km s$^{-1}$\\ with respect to the Sun, we can assume that most TGAS stars have RVs in the range of $-$100 and $+$100 km s$^{-1}$.\nCalculating possible $UVW$s over the range of these RVs, we can examine the kinematic clustering of field stars using a much larger sample of stars ($\\sim$2 million) than the $Hipparcos$ data ($\\sim$50,000).\nEven though the stellar distribution becomes diluted because of the $\\pm$100 km s$^{-1}$\\ RV range instead of a single value of RV for a star, the kinematic clustering of TGAS stars looks similar to that of $Hipparcos$ stars (first and second columns in Fig.~\\ref{fig8}).\n\n\nAdopting these two possible improvements described above, we created a new field star model.\nThe new field star model in $UVW$ is obtained from the best-fit Gaussian ellipsoidal models of $Hipparcos$ stars (Table~\\ref{tab4} and 2D projections in Fig.~\\ref{fig9}), and the model in $XYZ$ is a uniform distribution within a sphere of 200 pc in radius.\nAs can be seen in Fig.~\\ref{fig7}, the distance distribution of the field star model in BANYAN II peaks at $\\sim$120 pc, while the PDF of the new field star model expects more stars at larger distance.\nA field star distribution model is involved in the overall normalisation of the calculation, therefore, this uniform field star distribution model would increase the MG membership probability for nearby MG members;\nhowever, it would decrease the membership probability of more distant MG candidate members.\nFig.~\\ref{fig8} (3rd and 4th columns) compares the field star $UVW$ model used in BANYAN II to the new model, showing that the latter fits the $Hipparcos$ more closely.\n\n\n\n\n\\section{Results}\nUsing the updated models mentioned in the previous section, we now calculate membership probabilities of all BPMG candidate members available from SIMBAD \\citep{zuc01a, son03, zuc04, moo06, tor06, tor08, lep09, tei09, sch10, sch12a, sch12b, shk12, mal13, moo13, mal14, bes15, gag15a, gag15b}.\nThe result shows a sensitive dependence on models.\nBased on the calculated membership probabilities, we updated the list of BPMG members adding 12 new bona fide members.\nWith this updated membership list, we revised the BPMG model as described below.\n\n\\subsection{Effects of improved models}\nWe examine the effects of membership by comparing cases that adopt the {\\it exclusive \\rm and \\it inclusive lists}\\ (Table~\\ref{tab2}). \nSince BANYAN II used a Gaussian $XYZ$ distribution ({\\it Case I}), we compare {\\it Case I}, {\\it Case II (Excl.)}\\ and {\\it Case II (Incl.)}, using a Gaussian distribution for members in $XYZ$.\nSince the BPMG model based on {\\it inclusive list}}%{{\\it list 2 (inclusive)}--({\\it Case II (Incl.)})--allows more marginal members to start with, this flexibility would increase the membership probability of a marginal member, while the model from {\\it exclusive list}}%{{\\it list 2 (exclusive)}--({\\it Case II (Excl.)})--would decrease the probability.\nFigs.~\\ref{fig10}$-$~\\ref{fig12} compare membership probabilities from {\\it Case I}\\ and those from {\\it Case II}.\nOverall, the effects of the member list appear to be small (less than 5 per cent of test stars having a probability difference of greater than 20 per cent), and it is likely due to the similarities of the model extents (see $\\sigma$ values for BPMG$_{\\rm Excl}$, BPMG$_{\\rm Incl}$, BPMG$\\rm_{BANYANII}$ in Table~\\ref{tab3}).\nHowever, for a few stars, the effect of the member list is significant causing the membership probability changes up to $\\sim$40 per cent.\n\n\\begin{figure*}\n \\includegraphics[width=0.9\\linewidth]{fig10.eps}\n \\caption{Effects of the BPMG member selection (BANYAN II ({\\it Case I}) versus {\\it exclusive list}}%{{\\it list 2 (exclusive)}\\ ({\\it Case II})). \nLeft panel shows a histogram of the membership probability differences between these two cases.\nStars showing large differences in the membership probabilities ($\\Delta p >$ 18 per cent, grey area in the left panel) are presented in the right panel.\n Test stars are from the {\\it SIMBAD list}}%{{\\it list 3}.}\n \\label{fig10}\n\\end{figure*}\n\n\\begin{figure*}\n \\includegraphics[width=0.9\\linewidth]{fig11.eps}\n \\caption{Effects of the BPMG member selection (BANYAN II ({\\it Case I}) versus {\\it inclusive list}}%{{\\it list 2 (inclusive)}\\ ({\\it Case II})). \n Left panel shows a histogram of the membership probability differences between these two cases.\nStars showing large differences in the membership probabilities ($\\Delta p >$ 18 per cent, grey area in the left panel) are presented in the right panel.\n Test stars are from the {\\it SIMBAD list}}%{{\\it list 3}.}\n \\label{fig11}\n\\end{figure*}\n\n\\begin{figure*}\n \\includegraphics[width=0.9\\linewidth]{fig12.eps}\n \\caption{Effects of the BPMG member selection ({\\it inclusive list}}%{{\\it list 2 (inclusive)}\\ versus {\\it inclusive list}}%{{\\it list 2 (inclusive)}).\n Left panel shows a histogram of the membership probability differences between these two cases.\nStars showing large differences in the membership probabilities ($\\Delta p >$ 18 per cent, grey area in the left panel) are presented in the right panel.\n Test stars are from the {\\it SIMBAD list}}%{{\\it list 3}.}\n \\label{fig12}\n\\end{figure*}\n\n\\begin{figure*}\n \\includegraphics[width=0.9\\linewidth]{fig13.eps}\n \\caption{Effects of the distribution model of BPMG members (a Gaussian ({\\it Case II (Excl.)}) versus a uniform ({\\it Case III (Excl.)}) distribution in $XYZ$). \n Left panel shows a histogram of the membership probability differences between these two cases.\nStars showing large differences in the membership probabilities ($\\Delta p >$ 60 per cent, grey area in the left panel) are presented in the right panel.\n Test stars are from the {\\it SIMBAD list}}%{{\\it list 3}.}\n \\label{fig13}\n\\end{figure*}\n\n\\begin{figure*}\n \\includegraphics[width=0.9\\linewidth]{fig14.eps}\n \\caption{Effects of the distribution model of BPMG members (a Gaussian ({\\it Case II (Incl.)}) versus uniform ({\\it Case III (Incl.)}) distribution in $XYZ$). \n Left panel shows a histogram of the membership probability differences between these two cases.\nStars showing large differences in the membership probabilities ($\\Delta p >$ 60 per cent, grey area in the left panel) are presented in the right panel.\nTest stars are from the {\\it SIMBAD list}}%{{\\it list 3}.}\n \\label{fig14}\n\\end{figure*}\n\n\n\nFigs.~\\ref{fig13} and ~\\ref{fig14} compare the membership probabilities from cases assuming Gaussian distribution ({\\it Case II}) or uniform distribution ({\\it Case III}) of members in $XYZ$.\nAs expected, the Gaussian distribution models decrease membership probabilities of candidate members located near the boundary.\nThe membership probabilities of HD 168210 and $\\delta$ Sco were increased by $\\sim$60 per cent under the uniform spatial distribution model.\nThese two stars are located far from the centre of the BPMG model ($\\sim$70 and $\\sim$40 pc, respectively).\nStars relatively close to the centre, such as G76-8 ($\\sim$20 pc) on the other hand, have larger membership probabilities under the Gaussian distribution model ($>$80 per cent in {\\it Case II}\\ versus $<$10 per cent in {\\it Case III}).\nThis difference can be important in investigating the membership status of candidate members around or beyond the assumed initial MG boundary.\n\n\\begin{figure*}\n \\includegraphics[width=0.9\\linewidth]{fig15.eps}\n \\caption{Effects of the field star model (BANYAN II ({\\it Case I}) versus a new field star model ({\\it Case IV})).\n Left panel shows a histogram of the membership probability differences between these two cases.\nStars showing large differences in the membership probabilities ($\\Delta p >$ 60 per cent, grey area in the left panel) are presented in the right panel.\n Test stars are from the the {\\it SIMBAD list}}%{{\\it list 3}.}\n \\label{fig15}\n\\end{figure*}\n\n\\begin{figure*}\n \\includegraphics[width=0.9\\linewidth]{fig16.eps}\n\\caption{A comparison of membership probabilities from {\\it Case V}\\ (using all updated models simultaneously) and those from {\\it Case I}\\ (BANYAN II).\n Left panel shows a histogram of the membership probability differences between these two cases.\nStars showing large differences in the membership probabilities ($\\Delta p >$ 70 per cent, grey area in the left panel) are presented in the right panel.\n Test stars are from the {\\it SIMBAD list}}%{{\\it list 3}.}\n\\label{fig16}\n\\end{figure*}\n\n\n\\begin{figure}\n \\includegraphics[width=0.9\\columnwidth]{fig17.eps} \n\\caption{Membership probabilities for stars from the {\\it SIMBAD list}}%{{\\it list 3}\\ using all updated models ({\\it Case V}).}\n\\label{fig17}\n\\end{figure}\n\nBecause there are many more old field stars than MG members in a given range of $XYZUVW$, a small change in the $XYZUVW$ distribution model of field stars can significantly affect membership probabilities of candidate MG members.\nFig.~\\ref{fig15} shows the effect of the field star model by comparing membership probabilities from {\\it Case IV}\\ and those from {\\it Case I}.\nMembership probabilities tend to increase under the new model of field star distribution ({\\it Case IV}), by up to 80 per cent.\n{\\it Case IV}\\ assumes a uniform field star distribution in $XYZ$, expecting a smaller field star number density, within $\\sim$140 pc, compared to BANYAN II (Fig.~\\ref{fig7}).\nSince almost all known BPMG candidate members are located within $\\sim$100 pc, membership probabilities from {\\it Case IV}\\ generally increase compared to those from {\\it Case I}.\n\n\n\nWe have shown that each case listed in Table~\\ref{tab2} significantly affects the membership probability.\nUsing all of these updated models simultaneously ({\\it Case V}), we calculated membership probabilities of stars in the {\\it SIMBAD list}}%{{\\it list 3}.\nThese membership probabilities are compared to those from {\\it Case I}\\ (Fig.~\\ref{fig16}), showing a significant difference in membership probability.\nAbout 40 stars show changed membership probabilities larger than $\\sim$50 per cent compared to values from {\\it Case I}.\nThe majority of stars in {\\it SIMBAD list}}%{{\\it list 3}\\ ($\\sim$60 per cent) have membership probabilities of less than 50 per cent, implying the high contamination rate of false members in the {\\it SIMBAD list}}%{{\\it list 3}\\ (Fig.~\\ref{fig17}). \n\n\\clearpage\n\n\\subsection{Membership assessment and a revised BPMG model based on the improvement}\n\nWe can reconstruct a new list of bona fide members of BPMG based on our updated scheme of using several updated models simultaneously (the updated field star model, uniform $XYZ$ distribution of BPMG members, and either the {\\it exclusive} or {\\it inclusive} list of initial adopted members).\nWhen this new scheme was applied to 275 candidate members from the {\\it SIMBAD list}}%{{\\it list 3}, only about 40 per cent stars are believed to be kinematically associated with BPMG (p $>$50 per cent).\nA more stringent selection of candidate members (p$>$80 per cent) indicates only one third of the suggested members can be retained.\nMoreover, a kinematic similarity is not a sufficient condition for being a true member because of the large contamination of field stars with similar kinematics.\nTo be deemed as a bona fide member, any survived kinematic candidates should also show a clear signs of youth ($\\lesssim$25 Myr).\nIn this section, we discuss details on how we reject or include a particular star in a list of updated bona fide members.\n\n\n\\subsubsection{Method of rejection}\nStars with small kinematic membership probability ($<$80 per cent) or lacking clear signs of youth (age $\\lesssim$25 Myr) were rejected.\nWe present five cases of rejected stars from the {\\it BANYAN II list}}%{{\\it list 1}.\n\n\\paragraph{HIP 11360}\nHIP 11360 was initially suggested as a BPMG member by \\citet{moo06}, while \\citet{mal13} suggested it as a Columba member.\nAlthough the star shows strong Li absorption, it should not be considered as a BPMG member due to the low membership probability (0 per cent) in agreement with \\citet{mal13}.\nInstead, we suggest that HIP 11360 is a Tuc-Hor member because of the large membership probability (p(Tuc-Hor)$\\sim$100 per cent).\n\n\n\\paragraph{HIP 50156}\nHIP 50156 was initially proposed as a member of BPMG by \\citet{sch12a}, while \\citet{mal13} suggested that it is likely to be a member of Columba.\nIn spite of showing unambiguous youth ($\\lesssim$25 Myr; based on X-ray luminosity, photometric magnitude, and NUV-excess), this star should not be considered as a BPMG member due to the low membership probability (0 per cent).\n\n\n\\paragraph{2M J06085283-2753583}\n2M 06085283-2753583 was initially suggested as a BPMG member by Rice, Faherty \\& Cruz (2010). \n However, this star has a low kinematic membership probability (1 per cent).\nIn addition, unambiguous young age cannot be demonstrated based on a CMD position (e.g., $V-K$ versus $M_K$) because of the large model uncertainty of PMS evolution models at this young age and low mass range.\nFurthermore, an empirical comparison against other known young stars is implausible because of lacking comparison stars with demonstrated youth.\n\n\n\\paragraph{$\\eta$Tel A\\&B}\n$\\eta$ Tel A\\&B were originally proposed as members of Tuc-Hor by \\citet{zuc00}, but their memberships were later revised to BPMG by \\citet{zuc01a}.\n$\\eta$ Tel A shows unambiguous youth on the CMD, but it should not be considered a BPMG member due to the low kinematic membership probability (4 per cent).\nIts companion, $\\eta$ Tel B, has a large membership probability ($\\sim$90 per cent), which is obtained by marginalising over RV.\nIn order for $\\eta$ Tel B to be a BPMG member, its predicted RV must be $\\sim$1 km s$^{-1}$, which is significantly different from the consistently measured value of 12$-$14 km s$^{-1}$\\ for $\\eta$ Tel A \\citep{cam28, wil53, eva64, eva67, gon06}.\nThus, these stars should not be considered as BPMG members.\n\n\\subsubsection{Method of inclusion}\nIn order to be considered as bona fide members, stars should show large kinematic membership probability ($>$80 per cent) and clear signs of age younger than or similar to 25 Myr.\n39 stars in the {\\it BANYAN II list}}%{{\\it list 1}\\ are confirmed as bona fide BPMG members.\nThey are listed in the first part of Table~\\ref{tab5}.\nAmong these stars, HIP 86598 and HIP 89829 are located far from Earth compared to other members ($\\sim$70$-$80 pc; other members are located within 50 pc).\nHIP 86598 was suggested as a Upper Scorpius member by Song, Zuckerman \\& Bessell (2012), while \\citet{mal13} suggested this star as a BPMG member.\nThe other star, HIP 89829, was identified as a BPMG member by \\citet{tor08} and \\citet{mal13}.\nBoth HIP 86598 and HIP 89829 can be BPMG members because their Z position (about $-$10 pc) is in the range for BPMG members ($-$40 to $+$10 pc) rather than for stars of Upper Scorpius ($+$20 to $+$100 pc).\nTheir positions in $XYZ$ and\/or $UVW$ appear to be close to the edge of the BPMG model ($X, Y, U$, and $V$ for HIP 86598, $X$ and $Z$ for HIP 89829).\nTherefore, these stars can be treated marginal BPMG members.\n\nIn addition to these confirmed members in the {\\it BANYAN II list}}%{{\\it list 1}, we added 12 new members from the {\\it SIMBAD list}}%{{\\it list 3}\\ spanning spectral types K1 to M4.5.\nThese members have kinematic membership probabilities larger than 95 per cent and clear signs of youth.\nThey are shown in Table~\\ref{tab5} designated as newly confirmed members.\nThese 12 stars were suggested as BPMG candidate members in previous researches \\citep{tor06, lep09, mes10, kis11, mal13, mal14, rie14}.\n2MASS J21212873-6655063 was initially suggested as a member of Tuc-Hor by Zuckerman, Song \\& Webb (2001b);\n however, subsequent works \\citep{mal13, mal14} and the kinematic membership probability (100 per cent) based on the updated models in this study support that this star should be a bona fide member of BPMG.\nIn addition, CD-54 7336 and CD-31 16041 were initially proposed as members of Upper Scorpius by \\citet{son12} based on the direction in the sky and large assumed distances.\nNow, we also conclude that these two stars are members of BPMG instead of Upper Scorpius based on their updated membership probabilities (97 to 100 per cent), Z positions ($-$12 to $-$13 pc), and trigonometric distances ($\\sim$50 to 70 pc).\n\n\n\\subsubsection{Ambiguous cases}\nAmong well-known members of BPMG (the {\\it BANYAN II list}}%{{\\it list 1}), 6 stars (HIP 10679, HIP 10680, HIP 21547, HIP 88726A and B, HIP 92024) are ambiguous in their membership determination because they show moderate signs of youth ($\\lesssim$100 Myr but not $\\lesssim$25 Myr).\nWe retained them in our bona fide member list for the lacking clear evidence of their non-memberships.\n\n\\subsubsection{Suggestion of probable members}\nFinally, there are 17 highly probable BPMG candidate members.\nWe separately listed these stars in 3 groups in Table 5 according to age and data constraints.\nGroup 1 consists of stars showing unambiguous youth with large membership probabilities ($>$85 per cent), but lacking partial kinematic parameters such as distance or RV.\nThe 4 stars in this group were suggested as candidate members of BPMG in several studies \\citep{zuc04, tor06, mes10, mal13, mal14}.\nHD 161460 was suggested as a member of Upper Scorpius by \\citet{son12}.\nGroup 2 contains stars having large membership probabilities ($>$80 per cent), but showing only moderate signs of youth and missing distance or RV.\nStars in this group were suggested as members of BPMG in multiple studies \\citep{lep09, sch10, kis11, mal13, mal14, gag15a, gag15b}.\nHowever, PYC J00390+1330 and UPM J1354-7121 were also suggested as AB Dorarus members by \\citet{sch12a} and \\citet{mal13}, respectively.\n\n\nA single star in group 3, TYC 6872-1011-1, has the full 6 kinematic parameters showing unambiguous youth, but its kinematic probability is slightly low ($\\sim$60 per cent).\nThis star was also suggested to be a BPMG member in several studies \\citep{tor06, mes10, mal13, mal14}.\n\n\n\\subsubsection{Summary}\nAmong all 275 BPMG candidate members in the {\\it SIMBAD list}}%{{\\it list 3}, 57 stars can be confirmed as bona fide BPMG members.\n39 stars are from the {\\it BANYAN II list}}%{{\\it list 1}, and 12 stars are newly confirmed.\nWe additionally include traditional 6 BPMG members showing ambiguity in youth.\nThese stars should be removed in the future if they show clear evidence of non-memberships.\nFive stars from the {\\it BANYAN II list}}%{{\\it list 1}\\ are rejected, mainly due to updated low kinematic membership probabilities. \nWe note that some of the false members were used in several age-related studies (e.g., absolute isochronal age scale; Bell, Mamajek \\& Naylor (2015), lithium depletion boundary age; \\citet{bin16}), which could have biased the results.\n\n\nThe list of updated bona fide members was utilized to revise a BPMG model (BPMG$\\rm_{revised}$ in Table~\\ref{tab3}), which, in turn, can be used in future searches for members based on new data from the $Gaia$ mission.\n\n\n\\begin{table*} \n\\caption{The updated list of BPMG members.}\n\\label{tab5}\n \\centering \n \\tiny \n \\setlength\\tabcolsep{2pt}\n \\begin{threeparttable} \n \\begin{tabular}{cccccccccccccccccccc} \n \\hline \nName & SpT. & R.A. & Dec. & $\\mu\\rm_{\\alpha}$ & $\\mu\\rm_{\\delta}$ & $\\mu$ & $\\pi$ & $\\pi$ & RV & RV & B-V & V-K & K & NUV & log$L_{\\rm X}\/L_{\\rm bol}$\\ & Li\\tnote{a} & Li & p(BPMG)\\tnote{b} \\\\ \n & & (hh:mm:ss) & (dd:mm:ss) & (mas yr$^{-1}$) & (mas yr$^{-1}$) & Ref. & (mas) & Ref. & (km s$^{-1}$) & Ref. & (mag) & (mag) & (mag) & (mag) & & (m\\AA) & Ref. & (\\%) \\\\ \\hline \n \\multicolumn{20}{c}{Confirmed Members from a Previously Known BPMG Member List (the {\\it BANYAN II list}}%{{\\it list 1})} \\\\ \\hline\n HIP 560 & F2IV & 00:06:50 & -23:06:27 & 97.1 $\\pm$ 0.03 & -47.3 $\\pm$ 0.02 & 5 & 25.2 $\\pm$ 0.4 & 5 & 6.5 $\\pm$ 3.5 & 7 & 0.38 & 0.93 & 5.24 & $-$ & -5.34 & 87 & 25 & 100 \\\\ \n 2MASS J01112542+1526214 & M5 & 01:11:25 & +15:26:21 & 192.0 $\\pm$ 8.0 & -130.0 $\\pm$ 8.0 & 30 & 45.8 $\\pm$ 1.8 & 17 & 4.0 $\\pm$ 0.1 & 17 & 1.76 & 6.22 & 8.21 & 19.88 & -3.00 & $-$ & $-$ &100 \\\\ \n 2MASS J01351393-0712517 &M4 & 01:35:13 & -07:12:51 & 93.0 $\\pm$ 1.7 & -48.0 $\\pm$ 2.2 & 30 & 25.9 & 20 & 6.3 $\\pm$ 0.5 & 12 & 1.50 & 5.35 & 8.08 & 19.05 & -2.56 & $-$ & $-$ & 99\\\\ \n HIP 10679 & G2V & 02:17:24 & +28:44:31 & 85.1 $\\pm$ 0.4 & -70.8 $\\pm$ 0.3 & 5 & 25.5 $\\pm$ 0.2 & 5 & 5.7 $\\pm$ 0.3 & 33 & 0.62 & 1.50 & 6.26 & 12.94 & -3.91 & 163 & 13,14,4,25,22 & 100\\\\ \n HIP 10680 & F5V & 02:17:25 & +28:44:43 & 87.1 $\\pm$ 0.2 & -74.1 $\\pm$ 0.2 & 5 & 25.2 $\\pm$ 0.2 & 5 & 5.4 $\\pm$ 0.5 & 37 & 0.51 & 1.24 & 5.79 & 12.63 & -4.18 & 132 & 13,14,4,25,22 &100 \\\\ \n HIP 11152 & M3V & 02:23:26 & +22:44:06 & 98.5 $\\pm$ 0.2 & -112.5 $\\pm$ 0.1 & 5 & 36.9 $\\pm$ 0.3 & 5 & 10.4 $\\pm$ 2.0 & 18 & 1.56 & 3.93 & 7.35 & 17.77 & -3.16 & 0 & 32 & 100\\\\ \n HIP 11437 B & M0 & 02:27:28 & +30:58:41 & 81.5 $\\pm$ 4.6 & -69.1 $\\pm$ 3.2 & 30 & 24.3 $\\pm$ 0.2 & 5 & 4.7 $\\pm$ 1.3 & 22 & 1.50 & 4.63 & 7.92 & 18.98 & -2.37 & 115 & 14,4,25,22 &100 \\\\ \n HIP 11437 A & K8 & 02:27:29 & +30:58:25 & 79.5 $\\pm$ 0.2 & -72.1 $\\pm$ 0.1 & 5 & 24.3 $\\pm$ 0.2 & 5 & 6.5 $\\pm$ 0.4 & 27 & 1.33 & 2.99 & 7.08 & 17.64 & -3.10 & 227 & 14,4,25,22 & 100\\\\ \n HIP 12545 AB & K6V & 02:41:25 & +05:59:19 & 79.5 $\\pm$ 3.1 & -53.9 $\\pm$ 1.7 &27 & 23.8 $\\pm$ 1.5 & 27& 10.0 & 25 & 1.09 & 3.15 & 7.07 & 17.47 & -2.94 & 433 &28,14,4,25,22 & 97 \\\\ \n 2MASS J03350208+2342356 & M8.5 & 03:35:02 & +23:42:36 & 54$\\pm$10 & -56 $\\pm$ 10 & 20 & 23.1 & 20 & 15.5 $\\pm$1.7 & 20 & $-$ & $-$ & 11.26 & 22.12& $-$ & 615 & 20 & 85 \\\\\n HIP 21547 & F0V & 04:37:36 & -02:28:25 & 44.2 $\\pm$ 0.4 & -64.4 $\\pm$ 0.3 & 27 & 34.0 $\\pm$ 0.3 & 27 & 12.6 $\\pm$ 0.3 & 27 & 0.28 & 0.67 & 4.54 & $-$ & $-$ & 0 & 25& 94 \\\\ \n GJ 3305 AB & M0V & 04:37:37 & -02:29:28 & 45.9 $\\pm$ 1.3 & -63.6 $\\pm$ 1.2 & 30 & 34.0 $\\pm$ 0.3 & 27 & 21.7$\\pm$ 0.3 & 20& 1.45 & 4.18 & 6.41 & $-$ & -2.52 & 99 & 14,25 &100 \\\\ \n HIP 23200 & M0V & 04:59:34 & +01:47:00 & 39.3 $\\pm$ 0.2 & -95.0 $\\pm$ 0.1 &5 & 40.7 $\\pm$ 0.3 &5 & 19.3 $\\pm$ 0.2 & 34& 1.38 & 3.84 & 6.26 & $-$ & -3.05 & 270 & 2,25& 100\\\\ \n HIP 23309 & M0 & 05:00:47 & -57:15:26 & 35.3 $\\pm$ 0.1 & 74.1 $\\pm$ 0.1 & 5 & 36.9 $\\pm$ 0.3 &5 & 19.4 $\\pm$ 0.3 & 25& 1.38 & 3.73 & 6.24 & 17.51 & -3.33 & 325 & 28,14,4,25,31 &100\\\\ \n HIP 23418 & M3V & 05:01:58 & +09:58:60 & 12.1 $\\pm$ 9.9 & -74.4 $\\pm$ 5.7 & 27& 30.1 $\\pm$ 9.6 & 27& 14.9 $\\pm$ 3.5 & 27& 1.52 & 5.14 & 6.37 & 16.50 & -2.82 & 0 &25,22 &93\\\\ \n GJ 3331 BC & M3.5V+M4V & 05:06:49 & -21:35:04 & 33.1 $\\pm$ 2.7 & -33.2 $\\pm$ 2.0 & 30 & 50.7 $\\pm$ 0.3 & 5 & 23.7 $\\pm$ 1.7 & 6 & 1.66 & 5.39 & 6.11 & 16.77 & -2.97 & 20 &2 &96 \\\\ \n GJ 3331 A & M1V & 05:06:49 & -21:35:09 & 46.6 $\\pm$ 0.6 & -16.3 $\\pm$ 1.0 &5 & 50.7 $\\pm$ 0.3 & 5& 21.2 $\\pm$ 0.9 &6 & 1.42 & 4.32 & 6.12 & 16.73 & -2.80 & 20 &2 &99 \\\\ \n HIP 25486 & F7 & 05:27:04 & -11:54:04 & 17.1 $\\pm$ 0.03 & -49.2 $\\pm$ 0.02 &5 & 37.4 $\\pm$ 0.3 &5 & 20.2 $\\pm$ 0.5 & 29& 0.53 & 1.37 & 4.93 & 12.86 & -3.53 & 162 &28,14,25 &100 \\\\ \n HIP 27321 & A5V & 05:47:17 & -51:03:59 & 4.7 $\\pm$ 0.1 & 83.1 $\\pm$ 0.2 & 27 & 51.4 $\\pm$ 0.1 & 27 & 20.0 $\\pm$ 0.7 & 7& 0.16 & 0.32 & 3.53 & $-$ & $-$& 0 & 2,25 &100 \\\\ \n HIP 29964 & K4V & 06:18:28 & -72:02:43 & -7.7 $\\pm$ 0.1 & 74.4 $\\pm$ 0.1 & 5 & 25.6 $\\pm$ 0.2 &5 & 16.2 $\\pm$ 1.0 &16 & 1.07 & 3.18 & 6.81 & 16.70 & -2.72 & 400 &28,14,4,25,31& 100 \\\\ \n TWA 22 B & M6V & 10:17:26 & -53:54:26 & -175.8 $\\pm$ 0.8 & -21.3 $\\pm$ 0.8 &24 & 57.0 $\\pm$ 0.7 & 24& 14.8 $\\pm$ 2.1 & 24& 1.73 & 6.27 & 7.69 & 19.55 & -2.89 & 580 & 19,22& 100 \\\\ \n TWA 22 A & M6V & 10:17:26 & -53:54:26 & -175.8 $\\pm$ 0.8 & -21.3 $\\pm$ 0.8 &24 & 57.0 $\\pm$ 0.7 & 24& 14.8 $\\pm$ 2.1 & 24& 1.73 & 6.27 & 7.69 & 19.55 & -2.89 & 580 & 19,22&100\\\\ \n HIP 76629 BC & M4.5 & 15:38:56 & -57:42:18 & -52.9 & -106.0 & 25 & 27.1 $\\pm$ 0.3 & 5 & 0.1 $\\pm$ 2.0 & 25 & 1.71 & 5.61 & 9.19 & $-$ & -1.6 & 425 &25,2,22 & 100 \\\\ \n HIP 76629 A & K0V & 15:38:57 & -57:42:26 & -49.9 $\\pm$ 0.06 & -97.9 $\\pm$ 0.1 &5 & 27.1 $\\pm$ 0.3 & 5& 3.1 $\\pm$ 0.8 &27 & 0.85 & 2.30 & 5.85 & 14.18 & -3.24 & 280 &28,14,25,31 & 100 \\\\ \n HIP 79881 & A0 & 16:18:17 & -28:36:51 & -31.2 $\\pm$ 0.3 & -100.9 $\\pm$ 0.2 & 27& 24.2 $\\pm$ 0.2 & 27& -13.0 $\\pm$ 0.8 & 7& 0.02 & 0.04 & 4.74 & $-$ & $-$ & 0 & 32& 99 \\\\ \n HIP 84586 A & G5IV & 17:17:25 & -66:57:03 & -21.5 $\\pm$ 0.02 & -137.3 $\\pm$ 0.03 &5 & 32.8 $\\pm$ 0.4 &5 & 5.9 $\\pm$ 0.2 &27 & 0.81 & 2.17 & 4.70 & 12.78 & -3.20 & 250 &25 & 100\\\\ \n HIP 84586 B & K0IV & 17:17:25 & -66:57:03 & -21.5 $\\pm$ 0.02 & -137.3 $\\pm$ 0.03 & 5& 32.8 $\\pm$ 0.4 & 5& 5.9 $\\pm$ 0.2 & 27& 0.81 & 2.17 & 4.70 & 12.78 & -3.20 & 250 &25 & 100 \\\\ \n HIP 84586 C & M3V & 17:17:31 & -66:57:05 & -11.0 $\\pm$ 2.0 & -143.0 $\\pm$ 2.0 & 30& 32.8 $\\pm$ 0.4 & 5& 2.7 $\\pm$ 1.8 &25 & 1.54 & 5.19 & 7.63 & $-$ & -1.45 & 20 &2,25 &100 \\\\ \n HIP 86598 & F9V & 17:41:48 & -50:43:28 & -3.7 $\\pm$ 1.1 & -65.7 $\\pm$ 0.9 & 27& 13.8 $\\pm$ 0.9 &27 & 1.7 $\\pm$ 1.7 & 23& 0.55 & 1.37 & 6.99 & $-$ & -3.64 & 130 &9 & 85\\\\ \n HIP 88399 A & F5V & 18:03:03 & -51:38:54 & 2.3 $\\pm$ 0.04 & -86.1 $\\pm$ 0.03 & 5& 19.8 $\\pm$ 0.3 &5 & -0.4 $\\pm$ 0.5 & 27& 0.43 & 1.10 & 5.91 & $-$ & -4.53 & 107&25 &100 \\\\ \n HIP 88399 B & M2V & 18:03:04 & -51:38:56 & 2.3 $\\pm$ 0.04 & -86.1 $\\pm$ 0.03 & 5& 19.8 $\\pm$ 0.3 & 5& -2.4 $\\pm$ 1.3 & 25& 1.52 & 4.23 & 8.27 & $-$ & -2.93 & 70&25 & 100 \\\\ \n HIP 88726 A & A5V & 18:06:49 & -43:25:30 & 10.7 $\\pm$ 1.1 & -106.6 $\\pm$ 0.5 & 30& 23.9 $\\pm$ 0.7 & 27& -7.8 $\\pm$ 0.4 & 7& 0.22 & 0.55 & 4.39 & $-$& $-$ & 0 &11 & 100\\\\ \n HIP 88726 B & A5V & 18:06:49 & -43:25:29 & 10.7 $\\pm$ 1.1 & -106.6 $\\pm$ 0.5 & 30& 23.9 $\\pm$ 0.7 & 27& -7.8 $\\pm$ 0.4 & 7& 0.24 & 0.55 & 4.39 & $-$ & $-$ & 0 &11& 100 \\\\ \n HIP 89829 & G5V & 18:19:52 & -29:16:32 & 4.6 $\\pm$ 0.1 & -46.4 $\\pm$ 0.1 & 5& 12.6 $\\pm$ 0.3 & 5& -7.0 $\\pm$ 2.6 & 25& 0.64 & 1.75 & 7.05 & $-$ & -3.23 & 290 & 2,25 & 88 \\\\ \n HIP 92024 A & A7 & 18:45:26 & -64:52:16 & 32.4 $\\pm$ 0.2 & -149.5 $\\pm$ 0.2 & 27& 35.0 $\\pm$ 0.2 & 27& 2.0 $\\pm$ 4.2 &27 & 0.20 & 0.47 & 4.30 & $-$ & -5.70 & 0&25 & 99 \\\\ \n HIP 92024 BC & K7V & 18:45:36 & -64:51:45 & 25.9 $\\pm$ 8.0 & -184.2 $\\pm$ 8.0 & 30& 35.0 $\\pm$ 0.2 & 27& 1.0 $\\pm$ 3.0 & 25& 1.12 & 3.30 & 6.10 & $-$ & -2.98 & 477 &14,25 & 98\\\\ \n HIP 92680 & G9IV & 18:53:05 & -50:10:47 & 17.6 $\\pm$ 1.1 & -83.6 $\\pm$ 0.8 & 27& 19.4 $\\pm$ 1.0 &27 & -4.2 $\\pm$ 0.2 &7 & 0.81 & 2.04 & 6.37 & 14.48 & -3.23 & 279 &14,25&100 \\\\ \n HIP 95270 & F5.5 & 19:22:58 & -54:32:15 & 24.5 $\\pm$ 0.04 & -82.2 $\\pm$ 0.03 &5 & 20.6 $\\pm$ 0.5 & 5& 0.1 $\\pm$ 0.4 &7 & 0.46 & 1.13 & 5.91 & $-$ & $-$ & 117 &28,14,25 & 100 \\\\ \n HIP 99273 & F5V & 20:09:05 & -26:13:26 & 40.4 $\\pm$ 0.04 & -67.5 $\\pm$ 0.03 &5 & 19.6 $\\pm$ 0.3 &5 & -6.4 $\\pm$ 1.7 &3 & 0.44 & 1.10 & 6.08 & 12.07 & -4.90 & 95 & 2& 100 \\\\ \n HIP 102141 B & M4V & 20:41:50 & -32:26:10 & 286.2 $\\pm$ 8.0 & -377.2 $\\pm$ 8.0 & 30& 93.5 $\\pm$ 3.7 & 27& -5.2 & 25& 1.6 & 5.39 & 4.94 & 15.96 & -2.63 & 0 &25& 100 \\\\ \n HIP 102141 A & M4V & 20:41:51 & -32:26:07 & 270.5 $\\pm$ 4.6 & -365.6 $\\pm$ 3.5 & 27& 93.5 $\\pm$ 3.7 & 27& -3.7 $\\pm$ 3.0 & 15& 1.55 & 5.39 & 4.94 & 15.96 & -2.63 & 0&25 & 100 \\\\ \n HIP 102409 & M1V & 20:45:09 & -31:20:27 & 281.4 $\\pm$ 0.1 & -360.1 $\\pm$ 0.04 & 5& 102.1 $\\pm$ 0.4 & 5& -4.5 $\\pm$ 0.3 & 1 & 1.45 & 4.23 & 4.53 & 15.61 & -2.77 & 68 & 28,14,25& 100 \\\\ \n HIP 103311 AB & F8V & 20:55:47 & -17:06:51 & 58.8 $\\pm$ 0.8 & -62.8 $\\pm$ 0.7 & 27& 21.9 $\\pm$ 0.8 & 27& -4.5 $\\pm$ 2.1 &25 & 0.52 & 1.51 & 5.81 & $-$ & -3.41 & 110 &28,25,8& 100 \\\\ \n HIP 112312 A & M4IV & 22:44:57 & -33:15:02 & 179.9 $\\pm$ 0.2 & -123.3 $\\pm$ 0.1 & 5& 48.2 $\\pm$ 0.6 & 5& 3.2 $\\pm$ 0.5 &20 & 1.52 & 5.14 & 6.93 & 18.26 & -2.36 & 0 &21,25 & 100 \\\\ \n HIP 112312 B & M5IV & 22:45:00 & -33:15:26 & 171.1 $\\pm$ 1.3 & -125.2 $\\pm$ 4.3 & 30 & 48.2 $\\pm$ 0.6 & 5& 2.0 $\\pm$ 5.2 &10 & 1.60 & 5.56 & 7.79 & 19.77 & -2.22 & 336 &14,4,25,26 & 100\\\\ \n\\hline\n \\multicolumn{19}{c}{Newly Confirmed Members} \\\\ \\hline\n 2MASS J00172353-6645124 & M2.5 & 00:17:24 & -66:45:13 & 102.9 $\\pm$ 1.0 & -15.0 $\\pm$ 1.0 & 30 & 25.6 $\\pm$ 1.7 & 17 & 10.8 $\\pm$ 0.2 & 12 & 1.54 & 4.65 & 7.70& 19.12 & -3.00 & $-$& $-$ & 100 \\\\ \n GJ 2006 A & M4 & 00:27:50 & -32:33:06 & 99.2 $\\pm$ 1.3 & -61.3 $\\pm$ 2.6 & 30 & 30.1 $\\pm$ 2.5 & 17 & 9.5 $\\pm$ 0.3 & 11 & 1.38 & 4.94 & 8.01 & 19.48 & -2.18 & $-$ & $-$ &100 \\\\ \n GJ 2006 B & M3.5 & 00:27:50 & -32:33:24 & 117.2 $\\pm$ 4.1 & -31.5 $\\pm$ 5.8 & 30 & 31.8 $\\pm$ 2.5 & 17 & 8.5 $\\pm$ 0.2 & 12 & 1.41 & 5.13 & 8.12 & 18.87 & -2.20 & $-$ & $-$ &100\\\\ \n 2MASS J16572029-5343316 & M3 & 16:57:20 & -53:43:32 & -13.0 $\\pm$ 6.3 & -85.1 $\\pm$ 2.2 & 30 & 19.4 $\\pm$ 0.7 & 5 & 1.4 $\\pm$ 0.2 & 12 & 1.46 & 4.62 & 7.79 & $-$ & -3.23 & $-$ & $-$ & 100 \\\\ \n CD-54 7336 & K1V & 17:29:55 & -54:15:49 & -9.8 $\\pm$ 3.2 & -60.0 $\\pm$ 1.7 & 30 & 14.4 $\\pm$ 0.2 & 5 & -0.2 $\\pm$ 0.9 & 3 & 0.77 & 2.25 & 7.36 & $-$ & -3.13 & 360 & 2,25 & 97 \\\\ \n CD-31 16041 & K8V & 18:50:44 & -31:47:47 & 16.4 $\\pm$ 1.6 & -72.8 $\\pm$ 1.1 & 30 & 20.1 $\\pm$ 0.3 & 5 & -6.0 $\\pm$ 1.0 & 25 & 1.06 & 3.73 & 7.46 & 18.04 & -2.79 & 492 & 2,25 & 100 \\\\ \n TYC 7443-1102-1 & K9IV & 19:56:04 & -32:07:38 & 31.9 $\\pm$ 1.4 & -65.1 $\\pm$ 1.2 & 30 & 19.9 $\\pm$ 0.3 & 5 & -7.1 $\\pm$ 2.2 & 10 & 1.36 & 3.74 & 7.85 & 19.09 & -2.89 & 110 & 13,9 & 99 \\\\ \n UCAC3 124-580676 & M3 & 20:10:00 & -28:01:41 & 40.7 $\\pm$ 3.0 & -62.0 $\\pm$ 1.7 & 30 & 20.9 $\\pm$ 1.3 & 17 & -5.8 $\\pm$ 0.6 & 12 & 1.50 & 5.26 & 7.73 & 18.43 & -2.66 & $-$ & $-$ & 99\\\\ \n 2MASS J20333759-2556521 & M4.5 & 20:33:38 & -25:56:52 & 52.8 $\\pm$ 1.7 & -75.9 $\\pm$ 1.3 & 30 & 20.7 $\\pm$ 1.4 & 17 & -6.0 $\\pm$ 0.5 & 12 & 1.71 & 5.99 & 8.88 & 20.25 & -3.04 & $-$ & $-$ & 100\\\\ \n 2MASS J21212873-6655063 & K7V & 21:21:29 & -66:55:06 & 97.2 $\\pm$ 1.1 & -104.1 $\\pm$ 1.6 & 30 & 31.1 $\\pm$ 0.8 & 5 & 3.3 & 25 & 1.34 & 3.59 & 7.01 & $-$ & -2.92 & 15 & 25 &100\\\\ \n CPD-72 2713 & K7V & 22:42:49 & -71:42:21 & 92.7 $\\pm$ 0.8 & -51.1 $\\pm$ 0.8 & 30 & 27.4 $\\pm$ 0.3 & 5 & 8.6 $\\pm$ 0.5 & 25 & 1.32 & 3.67 & 6.89 & 17.55 & -2.80 & 440 & 2,25 &100 \\\\ \n BD-13 6424 & M0V & 23:32:31 & -12:15:51 & 137.4 $\\pm$ 1.0 & -81.0 $\\pm$ 1.0 & 30 & 36.0 $\\pm$ 0.5 & 5 & 1.8 $\\pm$ 0.7 & 25 & 1.74 & 4.07 & 6.57 & 17.82 & -3.68 & 185 & 2,25 & 100\\\\ \n \\hline \n \\multicolumn{19}{c}{Probable Members} \\\\ \\hline\n \\multicolumn{19}{c}{Group1: young, but missing $\\pi$ or RV} \\\\\n GSC 08350-01924 & M3V & 17:29:21 & -50:14:53 & -5.8 $\\pm$ 1.5 & -62.7 $\\pm$ 5.1 & 30 & $-$ & $-$ & 0.3 $\\pm$ 1.1 & 12 & 1.46 & 4.87 & 7.99 & $-$ & -2.94 & 50 & 2 &88 \\\\ \n HD 161460 & K0IV & 17:48:34 & -53:06:43 & -3.6 $\\pm$ 1.0 & -58.4 $\\pm$ 1.3 & 30 & $-$ & $-$ & -0.2 $\\pm$ 1.5 & 23 & 0.97 & 2.31 & 6.78 & $-$ & -3.14 & 320 & 2,25 & 89 \\\\ \n Smethells 20 & M1V & 18:46:53 & -62:10:37 & 13.6 $\\pm$ 1.4 & -79.4 $\\pm$ 1.4 & 30 & $-$ & $-$ & 0.3 $\\pm$ 3.2 & 10 & 1.24 & 3.98 & 7.85 & $-$ & -3.00 & 332 & 2,25 &97 \\\\ \n AZ Cap & K7 & 20:56:03 & -17:10:54 & 57.6 $\\pm$ 1.1 & -59.9 $\\pm$ 1.2 & 30 & $-$ & $-$ & -6.9 & 25 & 1.12 & 3.44 & 7.04 & $-$ & -3.25 & 235.5 & 25,14 & 99 \\\\ \n\n\\multicolumn{19}{c}{Group2: probably young, but missing $\\pi$ or RV} \\\\ \n 2MASS J00281434-3227556 & M5 & 00:28:14 & -32:27:56 & 110.1 $\\pm$ 1.8 & -43.0 $\\pm$ 3.3 & 30 & $-$ & $-$ & 5.9 $\\pm$ 3.4 & 12 & 1.58 & 5.95 & 9.28 & 20.85 & -2.55 & $-$ & $-$ & 89 \\\\ \n PYC J00390+1330 & M4 & 00:39:03 & +13:30:17 & 85.5 $\\pm$ 3.2 & -68.0 $\\pm$ 3.9 & 30 & $-$ & $-$ & $-$& $-$ & 1.60 & 5.64 & 10 06 & 21.29 & -2.74 & $-$ & $-$ & 97\\\\ \n BD+17 232A & $-$ & 01:37:39 & +18:35:33 & 68.6 $\\pm$ 0.8 & -47.3 $\\pm$ 0.6 & 30 & $-$ & $-$& 3.2 $\\pm$ 1.0 & 18 & 1.03 & 3.86 & 6.72 & 14.37 & -2.62 & $-$ & $-$ & 97 \\\\ \n UCAC3 176-23654 & M3 & 05:34:00 & -02:21:32 & 12.3 $\\pm$ 1.2 & -61.3 $\\pm$ 2.4 & 30 & $-$ & $-$ & 21.0 $\\pm$ 0.2 & 12 & 1.49 & 4.72 & 7.70 & $-$ & -2.57 & 0 & 32 &96 \\\\ \n 2MASS J08173943-8243298 & M3.5 & 08:17:39 & -82:43:30 & -80.3 $\\pm$ 1.1 & 102.5 $\\pm$ 0.8 & 30 & $-$ & $-$ & 17.5 $\\pm$ 1.6 & 12 & 1.58 & 5.03 & 6.59 & 17.59 & -2.94 & 0 & 32 & 94 \\\\ \n UPM J1354-7121 & M2.5 & 13:54:54 & -71:21:48 & -165.0 $\\pm$ 8.0 & -132.7 $\\pm$ 8.0 & 30 & $-$ & $-$ & 5.7 $\\pm$ 0.2 & 12 & 1.48 & 4.57 & 7.67 & 18.53 & -3.10 & $-$ & $-$ & 100 \\\\ \n 2MASS J17150219-3333398 & M0 & 17:15:02 & -33:33:40 & 7.8 $\\pm$ 1.0 & -175.9 $\\pm$ 1.2 & 30 & $-$ & $-$ & -14.6 $\\pm$ 3.5 & 12 & 1.41 & 3.87 & 7.07 & $-$ & -2.99 & $-$ & $-$ & 87 \\\\\n 2MASS J18420694-5554254 & M3.5 & 18:42:07 & -55:54:26 & 9.7 $\\pm$ 12.1 & -81.2 $\\pm$ 2.8 & 30 & $-$ & $-$ & 0.3 $\\pm$ 0.5 & 12 & 1.58 & 4.95 & 8.58 & 19.70 & -2.72 & 0 & 32 & 98 \\\\ \n 2MASS J19102820-2319486 & M4 & 19:10:28 & -23:19:49 & 16.6 $\\pm$ 1.4 & -51.8 $\\pm$ 1.4 & 30 & $-$ & $-$ & -8.0 $\\pm$ 0.8 & 12 & 1.53 & 5.01 & 8.21 & 19.08 & -2.69 & $-$ & $-$ & 86 \\\\ \n 2MASS J19243494-3442392 & M4 & 19:24:35 & -34:42:39 & 22.1 $\\pm$ 1.8 & -71.7 $\\pm$ 1.8 & 30 & $-$ & $-$ & -3.2 $\\pm$ 0.3 & 12 & 1.58 & 5.52 & 8.79 & 20.00 & -3.11 & $-$ & $-$ & 94 \\\\ \n UCAC3 116-474938 & M4 & 19:56:03 & -32:07:19 & 35.2 $\\pm$ 1.8 & -59.9 $\\pm$ 1.5 & 30 & $-$ & $-$ & -2.8 $\\pm$ 1.8 & 12 & 1.56 & 5.12 & 8.1 & 19.60 & -2.73 & 0 & 9 & 81 \\\\\n GSC 06354-00357 & M2 & 21:10:05 & -19:19:57 & 89.0 $\\pm$ 0.9 & -89.9 $\\pm$ 1.8 & 30 & $-$& $-$ & -5.5 $\\pm$ 0.5 & 12 & 1.52 & 4.46 & 7.20 & 18.58 & -2.83 & $-$ & $-$ & 100 \\\\ \n\\multicolumn{19}{c}{Group3: young with full 6 kinematic parameters, but low membership probability} \\\\ \n TYC 6872-1011-1 & M0V & 18:58:04 & -29:53:05 & 12.2 $\\pm$ 1.3 & -45.7 $\\pm$ 2.5 & 30 & 12.8 $\\pm$ 0.4 & 5 & -4.9 $\\pm$ 1.0 & 25 & 2.16 & 3.78 & 8.02 & $-$ & $-$ & 483 & 2,25 & 60\\\\ \n \\hline \n \\end{tabular} \n \\begin{tablenotes}\n \\item[a] If multiple data are available, weighted mean is used.\n \\item[b] The membership probability for BPMG is calculated using the improved models (the new field star model and uniformly distributed BPMG model in XYZ using {\\it exclusive list}}%{{\\it list 2 (exclusive)}--{\\it Case V}).\n \n \\item Notes.\n \\item References to the table:\n(1) \\citet{chu11}; (2) \\citet{sil09}; (3) \\citet{des15}; (4) Fern\\`{a}ndez, Figueras \\& Torra (2008); (5) \\citet{gai16}; (6) Gizis, Reid \\& Hawley (2002); (7) \\citet{gon06};\n (8) \\citet{kai04}; (9) \\citet{kis11}; (10) \\citet{kor13}; (11) \\citet{kra14}; (12) \\citet{mal14}; (13) \\citet{mcc12};\n (14) \\citet{men08}; (15) \\citet{mon01a}; (16) \\citet{mon01b}; (17) \\citet{rie14}; (18) \\citet{sch10}; (19) \\citet{shk11};\n (20) \\citet{shk12}; (21) Song, Bessell \\& Zuckerman (2002); (22) \\citet{son03}; (23) \\citet{son12}; (24) \\citet{tei09}; (25) \\citet{tor06};\n (26) van Altena, Lee \\& Hoffleit (1995); (27) \\citet{lee07}; (28) \\citet{wei10}; (29) White, Gabor \\& Hillenbrand (2007); (30) \\citet{zac12}; (31) \\citet{zuc01a}; (32) Internel data\n\n \\end{tablenotes}\n \\end{threeparttable} \n \\end{table*} \n\n\n\n\n\n\\section{Summary and conclusions}\n\nDeploying the same formulation that BANYAN II used, we examine the impact of three assumptions on models in the MG membership probability calculation: accepted initial member lists, distribution models of MG members, and distribution models of field stars.\nReassessment of membership of BPMG members in the {\\it BANYAN II list}}%{{\\it list 1}\\ results in a refined kinematic model for BPMG.\nDepending on the membership assessment criteria (exclusive and inclusive), membership probabilities of stars in a test set (the {\\it SIMBAD list}}%{{\\it list 3}) change up to $\\sim$40 per cent.\nLacking evidence of a central concentration of BPMG members in $XYZ$, we suggest to use the uniform distribution model in $XYZ$.\nThis uniform spatial distribution model changes the membership probabilities of the test stars up to $\\sim$80 per cent compared to the Gaussian distribution.\nFor field star models, assuming the uniform distribution in $XYZ$ is more realistic compared to the Gaussian distribution model;\nthe uniform distribution model expects more field stars at larger distance, while the Gaussian model expects the maximum stellar number density at $\\sim$120 pc, which seems to be artificial.\nIn $UVW$, field stars show distinct subgroups, and the model properties of these subgroups are obtained using a Gaussian mixture model.\nCombined effect of these model modifications show changes in membership probabilities of the test stars up to $\\sim$80 per cent.\nThese comparisons show significant membership probability changes especially for some marginal members, indicating the sensitivity to prior knowledge on the MG membership calculation and the importance of using reliable models.\n\n\nWe confirm 57 (51, if we exclude 6 classical members showing ambiguity in youth) BPMG members from the {\\it SIMBAD list}}%{{\\it list 3}.\nOnly about 90 stars from the {\\it SIMBAD list}}%{{\\it list 3}\\ seem to be kinematically associated with BPMG (p $>$80 per cent), and 51 (12 stars are new compared to the {\\it BANYAN II list}}%{{\\it list 1}) out of these $\\sim$90 stars show unambiguous signs of youth with 6 full kinematic parameters, which allow us to confirm them as bona fide BPMG members.\nAdditionally, we suggest 17 probable BPMG members.\n\n\nIn this study, we considered only kinematic properties in the MG membership probability calculation.\nBecause the number density of field stars is much larger than those of MG members and there are many field stars with similar $UVW$ to that of MGs, the contamination (old interlopers) rate has to be significantly large without considering age-related information.\nIn the future, we will formally incorporate the age-related information into the Bayesian scheme developed in this study to provide a more reliable MG membership calculation (Lee \\& Song in preparation).\n\n\n\\clearpage\n\n\n\\section*{Acknowledgements}\n\nWe thank the anonymous referee for valuable comments and suggestions that helped to significantly increase the quality of this work.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nGalactic globular clusters (GCs) show distinctive chemical and \nphotometric signatures that can vary over a large range from cluster to cluster.\nThe complex chemical composition patterns of GCs are nearly non-existent \namong halo field stars of our Galaxy \\citep{gra00} and in its satellites \n\\citep{tol09}. \nIn most cases the basic nucleosynthesis is attributable to light-element \nproton-capture reactions in H-burning at high temperature, which leads\neventually to large stellar surface variations in the abundances of\nC, N, O, Na, Al, and in carbon isotopic ratios. \nThese variations can be most easily understood in terms \nof multiple populations within each cluster; see recent reviews by \n\\cite{gra12} and \\cite{pio10} for more detailed discussion.\n\nThe most notable chemical signature of multiple populations, the Na-O \nanti-correlation (discovered by the Lick-Texas group, see \\citealt{kra94}), \nwas recently proposed as a definition of a genuine GC \\citep{car10a}. \nExtensive surveys of light element abundances in GCs (see e.g. \n\\citealt{car06,car10a}) can be understood with a common series of evolutionary\nevents: formation of a first stellar generation, pollution of \nintra-cluster medium by a fraction of the most massive stars of this \ngeneration, formation of a more spatially compact second generation, with the\nloss of most of the first stellar population (see e.g. \\citealt{der08,decr08}).\nThese actions occurred in each GC in slightly different ways.\n\nA notable example of GC abundance anomalies is NGC~2419. \nThis is the third most massive cluster in our Galaxy ($M_V = -9.42$, \\citealt{har96}). \nUnfortunately it is located at a very large distance \n\\citep[about 88 kpc,][]{dic11}, and thus even its brightest members\nare faint ($V$~$>$ 17) for high-resolution spectroscopy. \nDespite this daunting observational difficulty, the chemical compositions of \nmany NGC~2419 red giants have been recently investigated by \n\\cite{coh10,coh11,coh12} and \\cite{muc12}.\nStudies based on both medium-resolution DEIMOS \\citep{muc12} and high-resolution\nHIRES \\citep{coh12} spectra have revealed two populations of stars in this\ncluster that are characterized by large differences in magnesium and potassium\nabundances. \nOne population is made of giants with normal low-metallicity overabundances\nof [Mg\/Fe] and nearly solar [K\/Fe] abundance ratios, and the other \nincludes stars with large enhancements of K and huge depletions of Mg. \nStars of the latter composition have not been observed to date in other GCs. \nThe reality of this anti-correlation has been discussed thoroughly by\n\\cite{muc12} and \\cite{coh12}. \nPossible spurious effects due to the analysis (non-LTE corrections, \ncontamination from telluric lines, strong velocity fields in the upper \nstellar atmosphere, uncertainties in the adopted atmospheric parameters) \nwere thoroughly scrutinized in those papers, and disregarded as possible \nsources of this unusual Mg\/K abundance pattern.\n\nThe findings by Cohen et al. and Mucciarelli et al. have stimulated \ntheoretical studies on this peculiar cluster.\nIn the framework of multiple populations in GCs, \\cite{ven12} propose\nthat a Mg-K anti-correlation might be produced by the simultaneous activation \nof the Mg-Al-Si and Ar-K cycles in massive asymptotic giant branch (AGB) stars\nat the low metallicity of NGC~2419, provided that the relevant cross-sections \nand\/or the efficiency of hot bottom burning are larger than \ncommonly believed.\n\\cite{coh12} found a rather clear correlation between Na and Al abundances \nin NGC~2419, which likely results from the coupling of the NeNa and MgAl \ncycles that enhance Na and Al in typical GCs (e.g. \\citealt{den89,lan93}, \nsee also \\citealt{car09} for a large survey of GCs). \nHowever, they conclude that the K-Mg anti-correlation should have a different \norigin, because the Na-rich\/Al-rich stars (the second-generation stars formed \nin normal GCs, see \\citealt{car10a}) do not correspond to the stars with \nstrongly depleted Mg and enhanced K abundances.\nPossibly this argument is not as conclusive as proposed by \\cite{coh12}, since\n\\cite{ven12} show that stars that are extremely depleted in Mg \n(and K-rich) may be not very Al-rich, because at very high temperature \nMg is almost entirely transformed into Si rather than Al.\n\nMore K abundances in GCs would be clearly welcome.\nWe wish to confirm, using new data available to us, the growing evidence \nthat NGC~2419 is really unique among the GCs in our Galaxy and discuss if \nthe observed K-Mg anti-correlation should be included in the set of \nsignatures typical of proton-capture burning in normal GCs. \nThis second issue can be treated following various paths. \nFirst, to better understand the nucleosynthesis role of this\npoorly-studied element, it is important to derive its abundance in the\nsame stars as those with the full set of light element (at least \nO, Na, Mg, Al) abundances.\n\\cite{tak09} measured the K abundance in five bright red giant\nbranch (RGB) stars in each of the three clusters M~4, M~13 and M~15.\nUnfortunately, they did not provide abundances of Mg or other proton-capture\nelements. \nTheir analysis only showed a remarkable small scatter in [K\/Fe] ratios\nwithin each cluster, apart from one star in M~4 and one in M~13, whose peculiar\nabundances were explained as an effect of the increased velocity field in the\nupper atmospheric layers (the net result is to increase the strength of the\nalready saturated K resonance lines). \nNote, however, that a reanalysis by \\cite{coh12} of the discrepant giant \nin M~13 found normal [Mg\/Fe] and [K\/Fe] values for that star. \nRecently, \\cite{roe11} measured the K abundance in six giants\nof M~22 (three for each sub-population in this peculiar cluster). \nThe [K\/Fe] ratio is higher in the Ca-rich, $s-$process rich group, but all \nsix stars lie in the region populated by Mg-normal stars in NGC~2419.\nSecond, even if found in other GCs, it would be important to verify that the\nK-Mg anti-correlation is a phenomenon not restricted to the photospheric\nabundances of giant stars only. \nFinding such a pattern also in relatively unevolved cluster \nstars would be a guarantee that no spurious (such as chromospheric activity) \nor evolutionary (like mixing) effects are at work.\n\nTo shed light on these issues we have looked for available observational\nmaterial apt to derive accurate K abundances in (i) RGB stars, (ii) \nmain sequence turn-off and subgiant stars of normal GCs, and \n(iii) field stars, both dwarfs and giants. \nWe present K abundances in 25 red giants of four GCs (NGC~6752, NGC~6121, \nNGC~1904, and the most massive GC in the Galaxy, $\\omega$ Cen), and in \n17 turn-off and 35 subgiants in four GCs (NGC~6752, 47~Tuc, NGC~7099, \nand NGC~6397). \nWith this wealth of data we increase the evidence that NGC~2419\nis presently an unique object among GCs, and likely represents a case where\nproton-capture reactions occurred under very peculiar conditions, as advocated\nby \\cite{ven12}. \nMoreover, we discuss the evidence that NGC~2419 is also\ndifferent from the typical metal-poor population of dwarf galaxies.\n\n\n\n\\section{Available datasets, observations and analysis}\n\nAll K abundances discussed in the present paper are based on \nequivalent widths (EWs) of the resonance K {\\sc i} line at\n7698.98~\\AA; the other observable resonance doublet transition at 7664.91~\\AA\\\nis heavily affected by strong absorption telluric lines.\nThe relevant data are summarized in Table~1.\n\nSpectra for RGB stars are mainly from our program (ESO 085.D-0.205) \ndevoted to study Al abundances for a large sample of stars in selected GCs (see\n\\citealt{car12}). \nWe acquired fiber-fed UVES Red Arm high-resolution ($R\\sim 43,000$) spectra \ncentered at 860 nm and covering the spectral range approximatively from 6730 \nto 10150~\\AA. \nFrom these spectra we only derive the K abundances for stars in NGC~6752, \nM~4, and NGC~1904; the abundances of the other elements discussed \nin the present paper for these giants were taken from \\cite{car09} \nand \\cite{car10c}.\n\nSpectra for six giants in $\\omega$ Cen and for all the less-evolved \nstars are from the ESO Large Program 165.L-0263 (PI Gratton). \nDescription of the observations can be found in \\cite{gra01,pan02,car04,car05}. \nThe spectra for M~30 were acquired with the same setup \nof the other clusters. \n\nSpectra were reduced by the ESO personnel with the dedicated\npipelines, by extracting one-dimensional, wavelength-calibrated\nspectra that were sky subtracted and shifted to zero radial velocity. \n\nAbundances of K were also obtained for 21 field stars with \nmetallicities $-2<$[Fe\/H]$<-1$~dex selected from the sample of \nstars with good parallaxes used by \\cite{gra00} to study the mixing episodes \nin low mass Pop II stars. \nFor that project, high S\/N ($>$ 100), high resolution ($R>50,000$) \nspectra with very broad spectral coverage were obtained with the \nMcDonald 2.7m telescope and Tull coud\\'e echelle spectrograph \n(see \\citealt{gra00} for further details).\n\nSignificant corrections for departures from the LTE assumption should be \nconsidered when deriving K abundances \\citep[e.g.,][]{tak02,tak09}. \nThe corrections are a function of line strength, because strong lines form \nat shallower optical depths, where the non-LTE effects are larger. \nThey increase with decreasing surface gravity (less frequent collisions) and \nincreasing temperature (larger ionizing flux). \nSince our stars span a large range of parameters, appropriate\ncorrections should be considered for each case. \nWe adopted non-LTE abundance corrections which are function of \ntemperature, gravity, metal abundance, and EW of the K {\\sc i} line from \na multivariate interpolation through the about 900 models provided by\n\\cite{tak02}\\footnote{$http:\/\/optik2.mtk.nao.ac.jp\/\\sim takeda\/potassium\\_nonlte$}.\nThe same corrections were applied to the tabulated abundances of \n\\cite{coh12} and \\cite{muc12}. \nFor the latter, we first corrected upward by 0.3 dex their \ntabulated values of [K\/Fe] to recover the LTE values.\n\n\n\\section{Results and discussion}\n\n\\subsection{Setting the stage: the normalization with field stars}\n\nThe derived abundances for K in our program GCs are listed in Table~1, \nwhile in Table~2 we list the derived abundances for field stars.\nReferences for the adopted atmospheric parameters and abundances of the other\nelements are also listed. For each cluster, star-to-star errors were derived \nusing the sensitivities of abundances to changes in the atmospheric parameters\nand the internal uncertainties in each parameter as listed in the original\npapers. \nFor the giants in $\\omega$~Cen we determined the sensitivities using the\nline list adopted in all our recent analyses (from \\citealt{gra03}), since they\nwere not provided in \\cite{pan02}. \nThe [K\/Fe] ratios in Tables~1 and 2 include the non-LTE corrections.\n\nIn Fig.~\\ref{f:fig1} we show the run of the [K\/Fe] ratio as a function of\nthe metallicity for our program stars and the K-poor group in NGC~2419 from\n\\cite{coh12}. \nThe average value of [K\/Fe] seems to be different from cluster to cluster, \nwith a trend to have lower values in more metal-poor clusters. \nThis trend is similar to what we found in field stars of our\nsample, regardless from the evolutionary state. \nThe clear trend of our [K\/Fe] with [Fe\/H] persists even when \nthe value for HD~2665 (one of the most metal-poor stars of our sample, \nwith very low [K\/Fe]) is excluded.\nThe linear correlation coefficient is $r=0.70$\\ over 21 stars, which is \nhighly significant. \nOnce HD~2665 is dropped, the best fit linear regression line is ${\\rm\n[K\/Fe]}=(0.29\\pm 0.07){\\rm [Fe\/H]} + (0.51\\pm 0.09)$. \nNote that while our mean value of [K\/Fe]=0.10 (r.m.s.=0.12 dex) \nagrees fairly well with literature values from \\cite{tak02}, \\cite{zha06}, \nand \\cite{and10}, none of these studies seems to support a trend of \n[K\/Fe] with [Fe\/H].\nTo check if our trend is an artifact of our analysis, we examined stars \nin common with those previous works. \nWe have no stars in common with the samples of \\cite{zha06} and\n\\cite{and10}, and only three stars in common with the sample by \n\\cite{tak02}, listed in Table~3. \nOur measured $EW$s agree well with the values of \\cite{tak02}. \nMoreover, using their Table~1, we verified that all the differences \nin the [K\/Fe] ratio (even those as large as in the case of stars\nHD~103095 and HD~122956) can be explained by the differences in the atmospheric\nparameters, the NLTE corrections and, more importantly for dwarf stars, the\ntreatment of damping. \nWe defer a full discussion of K abundances in metal-poor field stars\nto a future study, where additional effects like the impact of 3-d model atmospheres will be treated in more detail.\nHowever, since they were obtained in homogeneous way, we may use present results as a reference against which K abundances in GCs are compared. \nUsing the average [Fe\/H] value of each cluster we derived an offset in \n[K\/Fe] with respect to the linear fit of field stars. \nThis offset was then applied to each individual value in the GC.\nNote that this is only a second order effect, aimed to obtain more uniform\nabundances of K, and does not affect in any significant way our main\nconclusions.\n\n\n\\subsection{Results for globular clusters}\n\nWe focused on the star-to-star K abundance variations. \nThe [K\/Fe] ratios are plotted against [Mg\/Fe] ratios \nin Fig.\\ref{f:fig2} and Fig.~\\ref{f:fig3}.\nIn Fig.~\\ref{f:fig2} we show the abundances of giants in four clusters\n(NGC~6752, M~4, NGC~1904, and $\\omega$~Cen) superimposed on these\nabundances for NGC~2419 from \\cite{coh12} and \\cite{muc12}.\nThe K and Mg abundances derived for less evolved stars in NGC~6752, \n47~Tuc, M~30 and NGC~6397 are in Fig.~\\ref{f:fig3}.\nOur basic result is that {\\it in all other GCs, stars occupy only the region\nwhere a population with canonical Mg overabundance and moderate K abundances \nlie in NGC~2419}.\nNone out of 77 stars in seven different GCs share the [Mg\/Fe] and [K\/Fe] ratios\nof the super Mg-poor\/K-rich group observed in NGC~2419, not even those in\n$\\omega$~Cen. \nThis was expected for Mg (see Fig. 10 in \\citealt{muc12}), but our study\nextends this knowledge also to K abundances\\footnote{A scrutiny of\nunpublished K abundances was done in \\cite{coh11} (see their summary), who found\nno star similar to star S1131 in NGC~2419 in any other cluster. However, no\nfurther details are given.}. \nAdmittedly, our data show a hint of K-Mg anti-correlation in stars of NGC~6752; \nhowever the range of K abundances is very small and might be largely due to \nthe possible offsets of the result we obtained for SGB stars with \nrespect to TO and RGB stars. \nNo trend is discernible for the remaining clusters (see e.g. the case of M~4, \nwhere our data confirm earlier analysis by \\citealt{tak09}), whatever\nevolutionary phase is considered.\n\nIn the scenario proposed by \\cite{ven12}, the Mg-K anti-correlation could be \ndue to the simultaneous activation of the Mg-Al-Si and Ar-K cycles, so we might \nexpect a corresponding large production of Al, although further $p-$capture on\n$^{27}$Al might transform it into $^{28}$Si (see \\citealt{ven12}). \nA comparison between the abundances of the elements involved in the two cycles \nis fundamental.\nThe classical Al-Mg anti-correlation in our GCs is compared to the pattern in\nNGC~2419 in Fig.~\\ref{f:fig4} and Fig.~\\ref{f:fig5}. \nAl-Mg anti-correlations of moderate extent are seen in normal clusters, \nmost clearly in the more metal-poor GCs like NGC~6752 and NGC~1904. \nM~4 shows the well known small star-to-star variations in Al \nabundances \\citep{iva99,mar08,car09}. \nThe present sample in $\\omega$~Cen does not present such anti-correlation, \nbut it is not representative of the whole cluster, because half of it was \nchosen by \\cite{pan02} to be on the metal-rich ``RGB-a'' branch. \nMore extensive data shows that a clear Mg-Al anti-correlation is present in \n$\\omega$ Cen (see e.g., \\citealt{ndc95a}). \nFor NGC~2419, \\cite{muc12} did not measure Al abundances while \n\\cite{coh12} derived Al abundances for a subset of eight giants\nout of 13 in their sample. \nAl seems to be roughly constant, or even correlated\nto Mg, over the large range of Mg abundances in NGC~2419.\n\n\n\n\\section{Is the chemical inventory in NGC~2419 really unique?}\n\nOur present results, as well as previous extensive studies \nof NGC~2419, obviously point to the fundamental question of why this peculiar \nchemical inventory is only found in this particular cluster.\nMore massive clusters, like $\\omega$~Cen, or GCs as metal-poor as \nNGC~2419 (like NGC~6397 or M~30), do not show the same extreme\nchemical pattern.\n\nA possibility to be explored is that the pollution for the extreme and normal\npopulations originated from two different sources. \nLet us assume that the Mg-rich population in NGC~2419 was polluted by \nclassical core-collapse supernovae (SNe), and suppose as a working hypothesis \nthat the part of the stellar population with low Mg abundances was instead \ncontaminated by the ejecta of a single very peculiar SN. \nA possible example of this class could be the so called pair instability \nSN (PISN). \nThis is a very rare event \\citep{ren12} theorized to end the life \nof a very massive population III star (see e.g. \\citealt{heg05}).\nThe extreme rarity of these events (\\citealt{ren12} identified 18 candidate\nstars possibly contaminated by PISNe over a sample of 12,300 stars with\nspectroscopy from SDSS) could explain why only a GC out of about 150 objects in\nthe Galaxy shows this peculiar pattern. \nThese stars are so massive ($140 \\leq M \\leq 260 M_\\odot$) that one single \nexplosion could well have provided all the $\\sim 80 M_\\odot$ of metals \nrequired for the proto-cluster from which NGC~2419 formed.\n\nSince the majority of stars incorporating a dominant contribution from PISNs are\npredicted to show a strong overabundance of Ca with respect to iron, we can\ncheck this hypothesis by looking at the run of Ca and K as a function of Mg. \nIf the scenario is correct, Ca should be enhanced and Mg depleted in the putative \npopulation contaminated by the PISN in NGC~2419. \nThis is just what we observe (see Fig.~\\ref{f:fig6}) from the data of \nboth \\cite{coh12} and \\cite{muc12}. \n\nHowever, other predictions from the nucleosynthesis associated to PISN\nexplosions clash with the observations for NGC~2419. \nA characteristic chemical signature associated to PISNe is a strong odd-even \neffect, which is clearly absent \\citep{coh12}. \nMoreover, neutron-capture elements are\npresent, while they are predicted to be absent in the ejecta of PISNs.\n\nOn the other hand, the relations plotted in Fig.~\\ref{f:fig6} may be explained\nalso by the hypothesis of proton-capture reactions occurring in a temperature\nrange much higher than usually observed in more normal cluster stars\n(\\citealt{ven12}). \nIn these particular circumstances, normal intermediate steps such as\nthe production of Al from destruction of Mg are bypassed favoring the\nsynthesis of heavier elements, and the production of K, Ca, and even Sc\nmay be activated by p-captures on Ar nuclei. \nIn this case, we should observe an anti-correlation between Mg and the \nelements enhanced in this burning, and this is just what observed in NGC~2419 \nfrom the large dataset of species analyzed by \\cite{coh12}. \nAs seen in Fig.~\\ref{f:fig7}, Si, K, Ca, and Sc are nicely \nanti-correlated with Mg. \nThe magnitude of the effect is larger for K and Sc, but this might simply \nbe due to Ca and Si being much more abundant species than\nK and Sc: transformation of even a small fraction of Ar is enough to produce\nlarge enhancements of these last elements. \nThe scenario put forward by \\cite{ven12} agrees with the\nobservations\\footnote{On the other hand, in the data of M~22 \n\\citep{roe11} K and Ca seem to be correlated. \nThis simply corresponds to the evidence that the two populations in this \ncluster also differ in Ca, higher in the more metal-rich population (see\n\\citealt{mar11,roe11}). \nFrom the data of \\cite{roe11} we also found that K is higher in the $r+s$ \ngroup of stars with respect to the $r$ only stars, with the\n[K\/Fe] ratio increasing when Sr, Y, Zr, Ba, La, Ce, Nd, and Pb (but not Eu) are\nmore abundant.}.\n\nOf course, we still need to understand why the observed extreme processing is\nonly observed in NGC~2419. \nThis is likely not due to the small size of our sample. \nIn fact, while there is still a paucity of determinations of K abundances, \ndata for Ca and Mg are available for many more stars, and can then be used \nto check whether the products of the Ar-K-Ca cycle can be seen elsewhere. \nIn Fig.~\\ref{f:fig8} we compare the pattern observed in NGC~2419\n(from the combined studies of \\citealt{coh12} and \\citealt{muc12}) with those\nof several old stellar populations. \nNGC~2419 stands out from both the other GCs and the dwarf spheroidals. \nWhile \\cite{coh12} suggested that being the nucleus of a disrupted dwarf \ngalaxy could represent the explanation for the chemical signature of this \ncluster, no (present-day) dwarf presents Mg depletions as large as those \nobserved in NGC~2419. \nOn the other hand, such extremely Mg-poor stars are not observed even in \nthe most massive GC in the Galaxy, $\\omega$~Cen, whose Mg depletions are \nconsistent with the pattern shown by dwarf spheroidals. \nAmong the other GCs, only NGC~2808 shows three stars with very large Mg\ndepletions (at [Ca\/H]$\\sim -0.8$ dex, see Fig.~\\ref{f:fig8}). \nThese giants are likely the progeny of the most He-rich \nmain-sequence population. \nWe note, however, that not even these stars have comparably extreme [Ca\/Mg] \nratios as those observed in NGC~2419. \nThe only star that resembles the Mg-poor population in NGC~2419 \nis one giant in M~54 \\citep{car10b}, the second most massive cluster \nin the Milky Way, sitting in the nucleus of the disrupting dwarf \ngalaxy Sagittarius. \nUnfortunately, no K abundance is available for stars in M~54; \na dedicated survey in this cluster could reveal precious information.\n\nIn conclusion, more massive objects, as well as systems with similar\nmetallicities, and stellar aggregates with different Galactocentric distances\ndo not seem able to match the extreme chemistry shown by a sizable part of the\nstellar population of NGC~2419. \nIt is possible that the peculiar combination of low metallicity, large mass, \nand large distance from the main parent Galaxy could explain the \nobserved signatures of this object.\nMore observations of K for large samples of stars, like those assembled as\ncalibrators in \\cite{kir11}, in a large number of GCs are required\nto properly explore the mass-metallicity-distance parameter space.\nUntil then, NGC~2419 continues to be unique among GCs and old stellar systems.\n\n\n\\acknowledgements\nFunding is acknowledged from PRIN INAF 2011 \"Multiple populations in\nglobular clusters: their role in the Galaxy assembly\" (PI E. Carretta), and\nPRIN MIUR 2010-2011 ``The Chemical and Dynamical Evolution of the Milky Way and\nLocal Group Galaxies'' (PI F. Matteucci), prot. 2010LY5N2T. \nWe are grateful also for funding from the U.S. National Science \nFoundation (grant AST-1211585).\nWe thank A. Mucciarelli for sending us the unpublished EWs for K in NGC~2419, \nand I. Roederer for the EW of giants in M~22.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Main results, motivations and generalizations}\n\\label{sec0}\n\nAll algebraic objects in this note will be defined over a field $\\bfk$\nof characteristic zero and $V$ will always denote a \n$\\bfk$-vector space. We will sometimes use the formalism of operads\nexplained, for example,\nin~\\cite{markl-shnider-stasheff:book}. \nSections~2,~3\nand~4 containing the\nmain results, however, do not rely on this language.\n\nLet $\\pL(V)$ denote the free pre-Lie algebra generated by $V$ and\n$\\pL(V)_L$ the associated Lie algebra. We will focus on the Lie\nalgebra $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V) \\subset \\pL(V)$ generated in $\\pL(V)_L$ by $V$, called\nthe {\\em subalgebra of Lie elements\\\/} in $\\pl(V)$. It is\nknown~\\cite{dzhu-lof:HHA02} that $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)$ is (isomorphic to) the free\nLie algebra generated by $V$; we will give a new short proof of this\nstatement in Section~3. Our main result, Theorem~\\ref{.},\ndescribes $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)$ as the kernel of a map \n\\begin{equation}\n\\label{psano_v_Srni}\nd : \\pL(V) \\to \\pL^1(V),\n\\end{equation}\nwhere $\\pL^1(V)$ is the subspace of degree $+1$ elements in the free\ngraded pre-Lie algebra $\\pL^*(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$ generated by $V$ and a degree\n$+1$ `dummy' variable ${\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to$.\nThe map~(\\ref{psano_v_Srni}) is later in the paper identified \nwith a very simple map between\nspaces of trees, see Proposition~\\ref{abych_neonemocnel} and\nCorollary~\\ref{,}.\n\nTheorem~\\ref{.} and Corollary~\\ref{,} have \nimmediate applications to the analysis of natural operations on the\nChevalley-Eilenberg complex of a Lie algebra. In a future work we also\nplan to prove that Theorem~\\ref{.} implies that the only natural\nmultilinear operations on vector fields on smooth manifolds are, in stable\ndimensions, iterations of the standard Jacobi bracket. There is\nalso a possible relation of the results of this paper with Loday's\ntheory of triplettes. In the rest of this introductory\nsection, we discuss some of these applications and \nmotivations in more detail. \n\n\n\\begin{odstavec}\n\\label{mot}\n{\\rm\n{\\it Motivations.\\\/}\nIn~\\cite{markl:de} we studied, among other things, the\ndifferential graded (dg-) operad ${\\cal B}_{\\it Lie}}\\def\\Bshl{{\\cal B}_{L_\\infty}^*$ of natural operations on\nthe Chevalley-Eilenberg complex of a Lie algebra with coefficients in\nitself, along with its homotopy version \n$\\Bshl^*$, the operad of natural operations on the\nChevalley-Eilenberg complex of an $L_\\infty$-algebra (=~strongly\nhomotopy Lie algebra, see~\\cite{lada-markl:CommAlg95}).\nWe proposed:\n\n\\begin{problem}\n\\label{-\/}\nDescribe the homotopy types, in the non-abelian derived category,\nof the dg-operads ${\\cal B}_{\\it Lie}}\\def\\Bshl{{\\cal B}_{L_\\infty}^*$ and $\\Bshl^*$ of natural operations on the\nChevalley-Eilenberg complex.\n\\end{problem}\n\nThe following conjecture was proposed by D.~Tamarkin.\n\n\\begin{conjecture}\n\\label{Tamarkin}\nThe operad ${\\cal B}_{\\it Lie}}\\def\\Bshl{{\\cal B}_{L_\\infty}^*$ has the homotopy type of the operad\n${\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A}$ for Lie algebras.\n\\end{conjecture}\n\n\nIt turns out that $\\Bshl^*$, which is tied to\n${\\cal B}_{\\it Lie}}\\def\\Bshl{{\\cal B}_{L_\\infty}^*$ by the `forgetful' map $c : \\Bshl^* \\to {\\cal B}_{\\it Lie}}\\def\\Bshl{{\\cal B}_{L_\\infty}^*$, contains a\ndg-sub-operad $\\rpL^* = (\\rpL^*,d)$ generated by symmetric\nbraces~\\cite{lm:sb} such that $\\rpL^0$ (the sub-operad of degree $0$\nelements) is the operad $p{\\cal L{\\it ie\\\/}}}\\def\\Brace{{\\cal B{\\it race\\\/}}$ governing pre-Lie algebras. Moreover,\nboth $\\rpL^0$, $\\rpL^1$ and the differential $d : \\rpL^0 \\to \\rpL^1$\nhave very explicit descriptions in terms of planar trees.\nOur conviction in Conjecture~\\ref{Tamarkin} made us believe that the sub-operad\n$$\nH^0(\\rpL^*) = \\Ker \\left(\\rule{0em}{1em} \nd : \\rpL^0 \\to \\rpL^1\\right)\n$$\nof $\\rpL^0 \\cong p{\\cal L{\\it ie\\\/}}}\\def\\Brace{{\\cal B{\\it race\\\/}}$ equals the operad ${\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A}$,\n\\begin{equation}\n\\label{.,}\nH^0(\\rpL^*) \\cong {\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A}.\n\\end{equation}\nThe main result of this paper, equivalent to isomorphism~(\\ref{.,}), is\ntherefore a step towards a solution of Problem~\\ref{-\/}.\n}\n\\end{odstavec}\n\n\n\\begin{odstavec}\n\\label{gen}\n{\\rm\n{\\it Generalizations.\\\/}\nLet us slightly reformulate the above reflections and indicate\npossible generalizations. Let ${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}$ be a quadratic \nKoszul operad~\\cite[Section~II.3.3]{markl-shnider-stasheff:book} and\n$\\calB_{{\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}_\\infty} = (\\calB_{{\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}_\\infty},d)$ the dg-operad of\nnatural operations on the complex defining the operadic cohomology of\n${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}_\\infty$ (= strongly homotopy\n${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}$-algebras~\\cite[Definition~II.3.128]{markl-shnider-stasheff:book}) \nwith coefficients in itself. In~\\cite{markl:de} we conjectured that \n\\begin{equation}\n\\label{oooo}\nH^0(\\calB^*_{{\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}_\\infty})\n\\cong {\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A}\n\\end{equation} \nfor each quadratic Koszul operad ${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}$.\n\nThe operad $\\calB^*_{{\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}_\\infty}$ has a suboperad\n$\\calS^*_{{\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}_\\infty}$ generated by a restricted class of operations\nwhich generalize the braces on the Hochschild cohomology complex of an\nassociative algebra~\\cite{gerstenhaber-voronov:FAP95}. The operad\n$\\calS^0_{{\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}_\\infty}$ of degree~$0$ elements in\n$\\calS^*_{{\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}_\\infty}$ always contains the operad ${\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A}$ for Lie\nalgebras that represents the intrinsic brackets. \nThe conjectural isomorphism~(\\ref{oooo}) would therefore imply: \n\n\\begin{conjecture}\n\\label{jak_to_dopadne}\nFor each quadratic Koszul operad ${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}$,\n$$\n{\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A} \\cong \\Ker\\left(d : \\calS^0_{{\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}_\\infty} \\to \\calS^1_{{\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}_\\infty}\n\\right) .\n$$\n\\end{conjecture}\n\nMoving from operads to free\nalgebras~\\cite[Section~II.1.4]{markl-shnider-stasheff:book}, an\naffirmative solution of this conjecture for a particular operad ${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}$\nwould immediately give a characterization of Lie elements in free\n$\\calS^0_{{\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}_\\infty}$-algebras.\n\n{}From this point of view, the main result of this paper\n(Theorem~\\ref{.}) is a combination of a solution of\nConjecture~\\ref{jak_to_dopadne} for ${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}={\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A}$ with the\nidentification of $\\calS^0_{{\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A}_\\infty} \\cong p{\\cal L{\\it ie\\\/}}}\\def\\Brace{{\\cal B{\\it race\\\/}}$ which expresses\nthe equivalence between symmetric brace algebras and pre-Lie\nalgebras~\\cite{guin-oudom,lm:sb}. Conjecture~\\ref{jak_to_dopadne}\nholds also for ${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S} = {\\cal A{\\it ss\\\/}}} \\def\\Com{{\\cal C{\\it om\\\/}}$, the operad for associative algebras, as\nwe know from the Deligne conjecture in the form proved\nin~\\cite{kontsevich-soibelman}. Since $\\calS^0_{{\\cal A{\\it ss\\\/}}} \\def\\Com{{\\cal C{\\it om\\\/}}_\\infty}$ is the\noperad for (ordinary, non-symmetric)\nbraces~\\cite{gerstenhaber-voronov:FAP95}, one can obtain a description of\n{\\em Lie elements\\\/} in {\\em free brace algebras\\\/}. }\n\\end{odstavec}\n\n\\begin{odstavec}\n{\\rm\n{\\em Loday's triplettes.\\\/}\nTheorem~\\ref{.} can also be viewed as an analog of the characterization of Lie\nelements in the tensor algebra $T(V)$ as primitives of the\nbialgebra $\\calH = (T(V),\\otimes,\\Delta)$ with $\\Delta$ the shuffle\ndiagonal; we recall this classical result as Theorem~\\ref{psano_v_aute} of\nSection~2. The bialgebra $\\calH$ is associative,\ncoassociative cocommutative and its primitives ${\\it Prim\\\/}}\\def\\calH{{\\cal H}(\\calH)$ form a\nLie algebra. To formalize such situations, J.-L.~Loday introduced\nin~\\cite{loday:slides} the notion of a {\\em triplette\\\/}\n$(\\calC,\\spin,\\mbox {${\\cal A}$-{\\tt alg}} \\stackrel{F}\\to \\mbox {${\\cal P}$-{\\tt alg}})$, \nabbreviated $(\\calC, \\calA,{\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S})$,\nconsisting of operads $\\calC$ and $\\calA$, `spin' relations $\\spin$ \nbetween $\\calC$-coalgebras and $\\calA$-algebras defining\n$(\\calC,\\spin,\\calA)$-bialgebras, an operad ${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}$ describing the\nalgebraic structure of the primitives, and a forgetful functor $F : \\mbox {${\\cal A}$-{\\tt alg}}\n\\to \\mbox {${\\cal P}$-{\\tt alg}}$, see\nDefinition~\\ref{jeste_ji_musim_napsat_SMS} in Subsection~\\ref{trip}. \n\n\nThe nature of associative, cocommutative coassociative bialgebras and\ntheir primitives is captured by the triplette $(\\Com,{\\cal A{\\it ss\\\/}}} \\def\\Com{{\\cal C{\\it om\\\/}},{\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A})$. The\nclassical Theorem~\\ref{psano_v_aute} then follows from the fact that\nthe triplette $(\\Com,{\\cal A{\\it ss\\\/}}} \\def\\Com{{\\cal C{\\it om\\\/}},{\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A})$ is {\\em good\\\/}, in the sense which\nwe also recall in Subsection~\\ref{trip}. An interesting question is\nwhether the case of Lie elements in pre-Lie algebras considered in\nthis paper is governed by a good triplette in which $\\calA = p{\\cal L{\\it ie\\\/}}}\\def\\Brace{{\\cal B{\\it race\\\/}}$\nand ${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S} = {\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A}$. See Subsection~\\ref{trip} for more detail. \n}\n\\end{odstavec}\n\n\n\n\\vskip .3em\n\\noindent\n{\\bf Acknowledgments.} I would like to express my thanks to\nF.~Chapoton, M.~Livernet, \\hbox{J.-L.~Loday}, C.~L\\\"ofwall,\nJ.~Stasheff and D.~Tamarkin for many useful comments and\nsuggestions. I am also indebted to M.~Goze and E.~Remm for their\nhospitality during my visit of the University of Mulhouse in the Fall\nof 2004 when this work was initiated.\n\n\n\n\\section{Classical results revisited}\n\\label{sec1}\n\nIn this section we recall some classical results about Lie elements in\nfree associative algebras in a language suitable for the purposes of\nthis paper. Let ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$ be the tensor algebra generated by a vector\nspace $V$,\n$$\n{\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V) = \\bfk \\oplus \\bigoplus_{n=1}^\\infty {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}^n(V), \n$$\nwhere ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}^n(V)$ is the $n$-th tensor power $\\bigotimes^n (V)$ of the\nspace $V$. Let ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)_{L}$ denote the space ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$ considered as a Lie\nalgebra with the commutator bracket\n$$\n[x,y] := x\\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL y - y\\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL x,\\hskip 1em x,y \\in {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V),\n$$ \nand let $\\fL(V) \\subset {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$ be the Lie sub-algebra of\n${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)_{L}$ generated by $V$. It is well-known that $\\fL(V)$ is\n(isomorphic to) the\nfree Lie algebra generated by $V$~\\cite[\\S4, Theorem~2]{serre:65}. \n\nThere are several characterizations of the subspace $\\fL(V)\n\\subset {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$~\\cite{ree:AnM69,serre:65}. \nLet us recall the one which uses the {\\em\nshuffle diagonal\\\/} $\\Delta : {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V) \\to {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$ given, for\n$v_1 \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\cdots \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL v_n \\in {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}^n(V)$,~by\n\\begin{equation}\n\\label{jake_bude_pocasi}\n\\Delta(v_1 \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\cdots \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL v_n) := \n\\sum_{i=0}^n\\sum_{\\sigma \\in\\Sh(i,n-i)} \n[v_{\\sigma(1)} \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\cdots \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL v_{\\sigma(i)} ]\n\\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL [v_{\\sigma(i+1)} \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\cdots \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL v_{\\sigma(n)}],\n\\end{equation}\nwhere $\\Sh(i,n-i)$ denotes the set of all $(i,n-i)$-shuffles,\ni.e.~permutations $\\sigma \\in \\Sigma_n$ such that\n$$\n\\sigma(1) < \\cdots < \\sigma(i)\\ \\mbox { and } \n\\sigma(i+1) < \\cdots < \\sigma(n).\n$$\n\nNotice that, in the right hand side of~(\\ref{jake_bude_pocasi}), the\nsymbol $\\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL$ has two different meanings, the one inside the brackets\ndenotes the tensor product in ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$, the middle one the tensor\nproduct of two copies of ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$. To avoid this ambiguity, we denote\nthe product in ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$ by the dot~$\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em$,~(\\ref{jake_bude_pocasi})\nwill then read as\n$$\n\\Delta(v_1 \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em \\cdots \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em v_n) := \n\\sum_{i=0}^n\\sum_{\\sigma \\in \\Sh(i,n-i)} \n[v_{\\sigma(1)} \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em \\cdots \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em v_{\\sigma(i)}] \n\\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL [v_{\\sigma(i+1)} \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em\\cdots \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em v_{\\sigma(n)}].\n$$\n\nThe triple $({\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V),\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em,\\Delta)$ is a standard example of a unital counital\nassociative coassociative cocommutative Hopf algebra. \nWe will need also the {\\em augmentation ideal\\\/}\n${\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\subset \\fA(V)$ which equals $\\fA(V)$ minus the ground field, \n$$\n{\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) = \\bigoplus_{n=1}^\\infty {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}^n(V),\n$$\nand the {\\em reduced\\\/} diagonal $\\bDelta : {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\to {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL\n{\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)$ defined as \n$$\n\\bDelta(x) := \\Delta(x) - 1 \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL x - x \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL 1,\\ \\mbox { for $x \\in {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)$,}\n$$\nor, more explicitly,\n$$\n\\bDelta(v_1 \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em \\cdots \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em v_n) := \n\\sum_{i=1}^{n-1}\\sum_{\\sigma \\in \\Sh(i,n-i)} \n[v_{\\sigma(1)} \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em \\cdots \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em v_{\\sigma(i)}] \n\\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL [v_{\\sigma(i+1)} \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em\\cdots \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em v_{\\sigma(n)}],\n$$\nfor $v_1,\\ldots,v_n \\in V$ and $n \\geq 1$. Clearly $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V) \\subset\n\\bT(V)$. The following theorem is classical~\\cite{serre:65}.\n\n\\begin{theorem}\n\\label{psano_v_aute}\nThe subspace $\\fL(V) \\subset \\bT(V)$ equals the subspace of\nprimitive elements,\n$$\n\\fL(V) = \\Ker\\left(\\bDelta : {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\to {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)\\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)\\right).\n$$\n\\end{theorem}\n\nThe diagonal $\\Delta : {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V) \\to {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$ is a homomorphism of\nassociative algebras, that~is \n\\begin{equation}\n\\label{to_jsem_zvedav_jesli_budou_pastviny_fungovat}\n\\Delta (x \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em y) = \\Delta(x) \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em \\Delta(y),\\ \\mbox { for $x \\in {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$,}\n\\end{equation}\nwhere the same $\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em$ denotes both the multiplication in ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$ in\nthe left hand side and the induced multiplication of ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$\nin the right hand side. The reduced diagonal $\\bDelta : {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\to\n{\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)$ is, however, of a different nature:\n\n\\begin{proposition}\nFor each $x,y \\in {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)$,\n\\begin{equation}\n\\label{1}\n\\bDelta(x\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em y) = \\Delta(x)\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em\\bDelta(y) + \\bDelta(x)\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em(y \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL 1 +\n1 \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL y) + (x \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL y + y \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL x).\n\\end{equation}\n\\end{proposition}\n\nThe proof is a direct verification which we leave for the reader. We\nare going to reformulate~(\\ref{1}) using an action of $\\bT(V)$ on\n$\\bT(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\bT(V)$ defined as follows. For $\\xi \\in {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL\n{\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)$ and $x \\in {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)$, let\n\\begin{equation}\n\\label{2}\n\\begin{array}{rcl}\nx* \\xi &:=& \\Delta(x)\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em \\xi \\in {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\bT(V),\\ \\mbox { and}\n\\\\\n\\xi* x &:=& \\xi \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em (1 \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL x + x \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL 1) \\in {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\bT(V),\n\\rule{0em}{1.2em}\n\\end{array}\n\\end{equation}\nwhere $\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em$ denotes, as before, the tensor multiplication in\n$\\bT(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\bT(V)$. Observe that, while\n\\begin{equation}\n\\label{3}\n(x\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em y)* \\xi = x *(y* \\xi)\\ \\mbox { and }\\ (x*\\xi)*y = x*(\\xi* y),\n\\end{equation}\n$(\\xi* x)* y \\not=\\xi* (x\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em y)$, therefore the action~(\\ref{2})\n{\\em does not\\\/} make $\\bT(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\bT(V)$ a bimodule over the associative\nalgebra $({\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V),\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em)$. To understand the algebraic properties of\nthe above action better, we need to recall the following important\n\n\\begin{Definition}{\\rm (\\cite{gerstenhaber:AM63})}\nA {\\em pre-Lie\\\/} algebra is a vector space $X$ with a bilinear\nproduct $\\star : X \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL X \\to X$ such that the associator $\\Phi :\nX^{\\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL 3} \\to X$ defined by\n\\begin{equation}\n\\label{dnes_jsem_objel_Polednik}\n\\Phi(x,y,z):=(x \\star y) \\star z-x\\star (y\\star z),\\ \\mbox { for\n$x,y,z \\in X$},\n\\end{equation}\nis symmetric in the last two variables,\n$\\Phi(x,y,z) = \\Phi(x,z,y)$. Explicitly,\n\\begin{equation}\n\\label{aby_to_nezkoncilo_prusvihem}\n(x\\star y) \\star z-x\\star (y\\star z) = \n(x\\star z) \\star y-x\\star (z\\star y)\\ \\mbox { for each $x,y,z \\in X$}.\n\\end{equation}\n\\end{Definition}\n\nThere is an obvious graded version of this definition. Pre-Lie\nalgebras are known also under different names, such as right-symmetric\nalgebras, Vinberg algebras, \\&c. Pre-Lie algebras are particular\nexamples of {\\em Lie-admissible\\\/} algebras~\\cite{markl-remm}, which means\nthat the object $X_L := (X,[-,-])$ with $[-,-]$ the commutator of\n$\\star$, is a Lie algebra. Each associative algebra is clearly\npre-Lie. In the following proposition, ${\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)_{pL}$ denotes the\naugmentation ideal $\\bT(V)$ of the associative algebra ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$\nconsidered as a pre-Lie algebra.\n\n\\begin{proposition}\nFormulas~(\\ref{2}) define on ${\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)$ a structure of a\nbimodule over the pre-Lie algebra ${\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)_{pL}$. This means that\n$$\n(\\xi* x)*y - \\xi * (x \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em y) = (\\xi * y) * x - \\xi* (y \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em x)\n$$\nand\n$$\n(x \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em y) * \\xi - x * (y* \\xi) = (x * \\xi) *y - x*(\\xi* y),\n$$\nfor each $x,y \\in \\bT(V)$ and $\\xi \\in \\bT(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\bT(V)$. In\nparticular, ${\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)$ is a module over the Lie algebra\n${\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)_L$.\n\\end{proposition}\n\n\\begin{Proof}\nTo prove the first equality, notice that\n$$\n(\\xi * x) *y - \\xi *(x \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em y) \n= \\xi\\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em(x \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL y + y \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL x) = (\\xi* y)*x - \\xi*(y \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em x).\n$$\nThe second one immediately follows from~(\\ref{3}).\n\\end{Proof}\n\n\nUsing action~(\\ref{2}), rule~(\\ref{1}) can be rewritten as\n\\begin{equation}\n\\label{4}\n\\bDelta(x \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em y) = \\bDelta(x)*y + x*\\bDelta(y) + R(x,y),\\\nx,y \\in \\bT(V),\n\\end{equation}\nwhere the symmetric bilinear form $R(x,y) := x \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL y + y \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL x$\nmeasures the deviation of $\\bDelta$ from being a pre-Lie algebra\nderivation in \n$$\n{\\it Der}_{\\it pre-Lie}\\left({\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)_{pL},{\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL\n{\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)\\right).\n$$\n\nOn the other hand, since $R: \\bT(V) \\to \\bT(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\bT(V)$ is\nsymmetric, $\\bDelta$ is a derivation of the associated Lie algebra\n${\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)_L$,\n$$\n\\bDelta \\in {\\it Der}_{\\it Lie}\\left({\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)_{L},{\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)\\right),\n$$\nwhich implies that $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V) \\subset \\Ker(\\bDelta)$. The following\nstatement is completely obvious and we formulate it only to motivate\nProposition~\\ref{a} of Section~3.\n\n\\begin{proposition}\n\\label{strasne_pocasi_trva}\nThe map $\\bDelta : \\bT(V) \\to \\bT(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\bT(V)$ is uniquely\ndetermined by the rule~(\\ref{4}) together with the requirement that\n$\\bDelta(v) = 0$ for $v \\in V$.\n\\end{proposition}\n\nObserve that the reduced diagonal $\\bDelta : {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\to {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL\n{\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)$ is the initial differential of the cobar construction\n\\begin{equation}\n\\label{1a}\n{\\it Cob}(\\bT(V),\\bDelta): \\hskip 1em\n{\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\stackrel{d}{\\longrightarrow}\n{\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\stackrel{d}{\\longrightarrow}\n{\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)\\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V)\\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V) \\stackrel{d}{\\longrightarrow}\\cdots\n\\end{equation}\nof the coassociative coalgebra $({\\overline{\\T}}}\\def\\Ker{{\\it Ker}}\\def\\Im{{\\it Im}(V),\\bDelta)$.\nComplex~(\\ref{1a}) calculates the cohomology\n\\begin{equation}\n\\label{ooOoo}\nH^*\\left({\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V),\\Delta\\right) = \n{\\it Cotor}^{*+1}_{({\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V),\\Delta)}(\\bfk,\\bfk)\n\\end{equation} \nof the shuffle coalgebra. \n\nOn the other hand, \nby the Poincar\\'e-Birkhoff-Witt theorem,\nthere is an isomorphism of coalgebras\n$$\n({\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V),\\Delta) \\cong (\\bfk[\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)],\\nabla),\n$$\nwhere the polynomial ring $\\bfk[\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)]$ in the right hand side is equipped\nwith the standard cocommutative comultiplication $\\nabla$. \nDualizing the proof of the\nclassical~\\cite[Theorem~VII.2.2]{maclane:homology},\none obtains the isomorphism\n$$\n{\\it Cotor}^{*}_{(\\bfk[\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)],\\nabla)}(\\bfk,\\bfk) \\cong \\mbox{\\large$\\land$}^*(\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V))\n$$ \nwhere $\\mbox{\\large$\\land$}^*(-)$\ndenotes the exterior algebra functor.\nWe conclude that \n$$\nH^*\\left({\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V),\\Delta\\right) \\cong \\mbox{\\large$\\land$}^{*+1}(\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)).\n$$\n\n\\section{Lie elements in the free pre-Lie algebra}\n\\label{sec2}\n\nIn this section we show that the results reviewed in\nSection~2 translate to pre-Lie algebras.\nLet $\\pL(V) = (\\pL(V),\\star)$ denote the free pre-Lie algebra\ngenerated by a vector space $V$ and let $\\pL(V)_L$ be the associated\nLie algebra. The following proposition is proved\nin~\\cite{dzhu-lof:HHA02}, but we will give a shorter and more\ndirect proof, which was kindly suggested to us by M.~Livernet.\n\n\\begin{proposition}\nThe subspace $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V) \\subset \\pL(V)_L$ generated by $V$ is isomorphic\nto the free Lie algebra on $V$.\n\\end{proposition}\n\n\n\\noindent\n{\\bf Proof}\n(due to M.~Livernet).\\hglue 1.8em \nLet us denote in \nthis proof by $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}'(V)$ the Lie subalgebra of $\\pL(V)_L$ generated by $V\n\\subset \\pL(V)$ and by $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}''(V)$ the Lie subalgebra of ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)_L$ generated by\n$V \\subset {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$. The canonical map $\\pL(V) \\to {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)_{pL}$ clearly\ninduces a map $\\pL(V)_L \\to {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)_L$ which restricts to a Lie algebra\nhomomorphism $\\alpha: \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}'(V) \\to \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}''(V)$.\n\nLet $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)$ be, as before, the free Lie algebra generated by\n$V$. Since $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}'(V)$ is also generated by $V$, the canonical map $\\beta\n: \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V) \\to \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}'(V)$ is an epimorphism. To prove that it is a\nmonomorphism, observe that the composition $\\alpha \\beta : \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V) \\to\n\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}''(V)$ coincides with the canonical map induced by the inclusion $V\n\\hookrightarrow \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}''(V)$. Since $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}''(V)$ is isomorphic to the free\nLie algebra generated by $V$~\\cite[\\S4, Theorem~2]{serre:65}, the\ncomposition $\\alpha \\beta : \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V) \\to \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}''(V)$ is an isomorphism,\ntherefore $\\beta$ must be monic. We conclude that the canonical map\n$\\beta : \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V) \\to \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}'(V)$ is an isomorphism, which finishes the\nproof.\\qed\n\nConsider the free graded pre-Lie algebra $\\pL(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$ generated by $V$\nand one `dummy' variable ${\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to$ placed in degree $+1$. Observe that\n\\begin{equation}\n\\label{5}\n\\pL^*(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) = \\pL(V) \\oplus \\bigoplus_{n \\geq 1} \\pL^n(V), \n\\end{equation}\nwhere $\\pL^n(V)$ is the subset of $\\pL(V)$ spanned by\nmonomials with exactly $n$ occurrences of the dummy variable ${\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to$.\n\nWe need to consider also the graded pre-Lie algebra $\\rpL(V)$ (``r''\nfor ``reduced'') defined as the quotient\n$$\n\\rpL(V) := \\pL(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)\/({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)\n$$\nof the free pre-Lie algebra $\\pL(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$ by the ideal $({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$\ngenerated by ${\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to$. The grading~(\\ref{5}) clearly induces a\ngrading of $\\rpL(V)$ such that $\\rpL^0(V) = \\pL(V)$ and $\\rpL^1(V) =\n\\pL^1(V)$,\n\\begin{equation}\n\\label{koupali-jsme-se-na-Hradistku}\n\\rpL^*(V) = \\pL(V) \\oplus \\pL^1(V) \\oplus \\bigoplus_{n \\geq 2}\n\\rpL^n(V).\n\\end{equation}\nThe following statement, in which $\\Phi$ is the\nassociator~(\\ref{dnes_jsem_objel_Polednik}), is an analog of\nProposition~\\ref{strasne_pocasi_trva}.\n\n\\begin{proposition}\n\\label{a}\nThere exists precisely one degree $+1$ map $d : \\rpL^*(V) \\to\n\\rpL^{*+1}(V)$ such that $d(v) = 0$ for $v \\in V$, $d({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) = 0$ and\n\\begin{equation}\n\\label{0}\nd(a \\star b) = d(a) \\star b + (-1)^{|a|} a \\star d(b) + Q(a,b),\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{6}\nQ(a,b) : = ({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star a ) \\star b - {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star (a \\star b) = \\Phi({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to,a,b),\n\\end{equation}\nfor $a,b \\in \\rpL^*(V)$. Moreover, $d^2 = 0$.\n\\end{proposition}\n\n\\begin{Proof}\nThe uniqueness of the map $d$ with the properties stated\nin the proposition is clear. To prove that such a map exists, we show\nfirst that there exists a degree one map $\\td : \\pL(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) \\to\n\\pL(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$ of graded free pre-Lie algebras such that $\\td(v) = 0$ for $v\n\\in V$, $\\td({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) = 0$ and\n\\begin{equation} \n\\label{0'}\n\\td(x \\star y) = \\td(x) \\star y + (-1)^{|x|} x \\star \\td(y) + Q(x,y),\n\\end{equation}\nwhere $Q(x,y) : = \\Phi({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to,x,y)$ for $x,y \\in \\pL(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$. Let us\nverify that the above rule is compatible with the axiom $\\Phi(x,y,z) =\n(-1)^{|z||y|}\\Phi(x,z,y)$ of graded pre-Lie\nalgebras. Applying~(\\ref{0'}) twice, we obtain\n\\begin{eqnarray}\n\\label{16a}\n\\td\\Phi(x,y,z) \\!\\! &=&\\!\\!\n\\Phi(\\td x,y,z) + (-1)^{|x|} \\Phi(x,\\td y,z) + (-1)^{|x| +\n |y|}\\Phi(x,y,\\td z) \n\\\\ \\nonumber \n&&\n- (-1)^{|x|} x \\star Q(y,z) + Q(x \\star y,z) +Q(x,y) \\star z -\nQ(x,y \\star z),\n\\end{eqnarray}\nfor arbitrary $x,y,z \\in \\pL(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$. \n\nLet us make a small digression and observe that the associator $\\Phi$\nbehaves as a Hochschild cochain, that is\n$$\n{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star\\Phi (x,y,z) - \\Phi({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star x,y,z) + \\Phi({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to, x \\star y,z)\n- \\Phi({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to,x,y\\star z) + \\Phi({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to,x,y) \\star z = 0.\n$$\nIt follows from the definition of the form $Q$ and the above equation\nthat the last three terms of~(\\ref{16a}) equal $\\Phi({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star x,y,z)\n- {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star\\Phi (x,y,z)$, therefore~(\\ref{16a}) can be rewritten as \n\\begin{eqnarray*}\n\\td\\Phi(x,y,z) &=&\n\\Phi(\\td x,y,z) + (-1)^{|x|} \\Phi(x,\\td y,z) + (-1)^{|x| +\n |y|}\\Phi(x,y,\\td z) \n\\\\ \\nonumber \n&&\n- (-1)^{|x|} x \\star Q(y,z) + \\Phi({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star x,y,z)\n- {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star\\Phi (x,y,z).\n\\end{eqnarray*}\nSince the right hand side of the\nabove equality is graded symmetric in $y$ and $z$, we conclude that\n$$\n\\td \\left(\\Phi(x,y,z) - (-1)^{|z||y|}\\Phi(x,z,y)\\right) =0,\n$$\nwhich implies the existence of $\\td : \\pL(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) \\to \\pL(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$ with\nthe properties stated above. It is easy to verify, using~(\\ref{0'})\nand the assumption $\\td({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) = 0$, that\n\\begin{equation}\n\\label{k}\n\\td({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) = \\Phi({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)\n\\end{equation}\nand that\n\\begin{equation}\n\\label{l}\n\\td^2(x \\star y) = \\td^2(x) \\star y + x \\star \\td^2(y) + Q(\\td x,y) +\n(-1)^{|x|} Q(x,\\td y) + \\Phi({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to,x,y)\n\\end{equation}\nfor arbitrary $x,y \\in \\pL(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$. \n\nA simple induction on the number of generators based on~(\\ref{k})\ntogether with the rule~(\\ref{0'}) shows that $\\td$ preserves the ideal\ngenerated by ${\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to$. An equally simple induction based\non~(\\ref{l}) and~(\\ref{0'}) shows that $\\Im(\\td^2)$ is a subspace of\nthe same ideal. We easily conclude from the above facts that $\\td$\ninduces a map $d : \\rpL^*(V) \\to \\rpL^{*+1}(V)$ required by the\nproposition.\n\\end{Proof}\n\n\nLet us remark that each pre-Lie algebra\n$(X,\\star)$ determines a unique {\\em symmetric brace algebra\\\/}\n$(X,-\\langle -,\\ldots,-\\rangle)$ with $x \\langle y \\rangle = x\\star y$\nfor $x,y \\in X$~\\cite{lm:sb,guin-oudom}. \nThe bilinear form $Q$ in~(\\ref{0}) then can be\nwritten as\n$$\nQ(a,b) = {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\langle a,b \\rangle,\\ \\mbox { for } a,b \\in \\rpL(V).\n$$\nThe complex\n\\begin{equation}\n\\label{za_chvili_tam_musim_volat_a_hrozne_se_mi_nechce!}\n\\pL(V) \\stackrel{d}{\\longrightarrow} \\pL^1(V)\n\\stackrel{d}{\\longrightarrow} \\rpL^2(V) \\stackrel{d}{\\longrightarrow} \\cdots \n\\end{equation}\nshould be viewed as an analog of the cobar construction~(\\ref{1a}).\nWe will see in Section~5 that it describes natural\noperations on the Chevalley-Eilenberg cohomology of a Lie algebra. \nThe main result of this paper reads:\n\n\n\\begin{theorem}\n\\label{.}\nThe subspace $\\fL(V) \\subset \\pL(V)$ equals the kernel of the map $d :\n\\pL(V) \\to \\pL^1(V)$,\n$$\n\\fL(V) = \\Ker \\left(\\rule{0em}{1em} d : \\pL(V) \\to \\pL^1(V) \\right).\n$$\n\\end{theorem}\n\nIn Section~4 we\ndescribe the spaces $\\pL(V)$, $\\pL^1(V)$ and the map $d : \\pL(V) \\to\n\\pL^1(V)$ in terms of trees. Theorem~\\ref{.} will be proved in\nSection~6.\n\n\\section{Trees}\n\\label{v_podstate_cela_napsana_v_Moravske_Trebove}\n\nWe begin by recalling a tree description of free pre-Lie algebras due\nto F.~Chapoton and M.~Livernet~\\cite{chapoton-livernet:pre-lie}. By a\n{\\em tree\\\/} we understand a finite connected simply connected graph\nwithout loops and multiple edges. We will always assume that our trees\nare {\\em rooted\\\/} which, by definition, means that one of the\nvertices, called the {\\em root\\\/}, is marked and all edges are\noriented, pointing to the root.\n\nLet us denote by ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n$ the set of all trees with $n$ vertices\nnumbered $1,\\ldots,n$. The symmetric group $\\Sigma_n$ act on\n${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n$ by relabeling the vertices. We define\n$$\n{\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n(V) := {\\rm Span}_{\\bfk}{({\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n)} \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL_{\\Sigma_n} \\otexp Vn,\\\nn \\geq 1,\n$$\nwhere ${\\rm Span}_{\\bfk}{({\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n)}$ denotes the $\\bfk$-vector space\nspanned by ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n$ with the induced $\\Sigma_n$-action and where\n$\\Sigma_n$ acts on $\\otexp Vn$ by permuting the\nfactors. Therefore ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n(V)$ is the set of trees with $n$\nvertices decorated by elements of $V$.\n\n\n\\begin{example}\nThe set ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_1$ consists of a single tree \n\\hskip .3em $\\bullet$ \\hskip .3em with one vertex (which is also the\nroot) and no edges, thus ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_1(V) \\cong V$. The set ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_2$\nconsists of labelled trees\n$$\n\\stromI{\\sigma_1}{\\sigma_2}\n$$\nwhere $\\hskip .1em \\rule{.4em}{.4em} \\hskip .1em $ \ndenotes the root and $\\sigma \\in \\Sigma_2$. This means\nthat $V$-decorated trees from ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_2(V)$ look~as\n$$\n\\stromI uv\n$$\nwhere $u,v \\in V$, therefore ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_2(V) \\cong \\otexp V2$. Similarly,\n${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_3(V) \\cong \\otexp V3 \\oplus (V \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL S^2(V))$, where $S^2(V)$\ndenotes the second symmetric power of $V$. The corresponding decorated\ntrees are\n$$\n\\rule{0em}{4em}\n\\mbox{\\stromII uvw \\hskip 3em \\raisebox{1.5em}{ and }\\hskip 3em \\stromIII uvw}\n$$\nfor $u,v,w \\in V$. Finally,\n$$\n{\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_4(V) \\cong \\otexp V4 \\oplus \\otexp V4 \\oplus (\\otexp V2 \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL S^2(V))\n\\oplus (V \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL S^3(V)),\n$$\nwith the summands corresponding to the decorated trees\n\\begin{equation}\n\\label{snad_nenachladnu}\n\\raisebox{-2.7em}{\\rule{0em}{5em}}\n\\stromIV tuvw \\hskip 3em \\stromV tuvw \\hskip 3em \\stromVI tuvw \n \\hskip 3em \\stromVII tuvw \n\\end{equation}\nwith $t,u,v,w \\in V$.\n\\end{example}\n\n\\begin{theorem}\n{\\rm (Chapoton-Livernet~\\protect\\cite{chapoton-livernet:pre-lie})}\n\\label{jsem_zvedav_jak_povezu_tu_VOSu_v_desti}\nLet ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}(V) := \\bigoplus_{n \\geq 1}{\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n(V).$ Then there is a natural\nisomorphism\n\\begin{equation}\n\\label{zitra_odjizdim_na_zavody}\n\\pL(V) \\cong {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}(V).\n\\end{equation}\n\\end{theorem}\n \nThe pre-Lie multiplication in the left hand side\nof~(\\ref{zitra_odjizdim_na_zavody}) translates to the vertex\ninsertion of decorated trees in the right hand side,\nsee~\\cite{chapoton-livernet:pre-lie} for details.\n\n\n\\begin{example}\nThe most efficient way to identify decorated trees with elements\nof free pre-Lie algebras is to use the formalism of symmetric brace\nalgebras~\\cite{lm:sb}. The trees in~(\\ref{snad_nenachladnu})\nthen represent the following elements of $\\pL(V)$:\n$$\nt \\langle u \\langle v \\langle w \\rangle \\rangle \\rangle,\\\nt \\langle u,v \\langle w \\rangle \\rangle, \\\nt \\langle u \\langle v,w \\rangle \\rangle\\\n\\mbox { and } \\ t \\langle u,v,w \\rangle.\n$$\n\\end{example}\n\nUsing the same tree description~\\cite{chapoton-livernet:pre-lie} of the\nfree graded pre-Lie algebra $\\pL(V,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$, one can easily get\na natural isomorphism\n\\begin{equation}\n\\label{zitra_odjizdim_na_zavody-uz_tam_sem}\n\\pL^1(V) \\cong {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}^1(V) := \\bigoplus_{n \\geq 0} {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n^1(V),\n\\end{equation}\nwhere ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n^1(V)$ is the set of all trees with $n$ vertices decorated\nby elements of $V$ and one vertex decorated by the dummy variable\n${\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to$. We call the vertex decorated by ${\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to$ the {\\em special\nvertex\\\/}.\n\n\n\\begin{example}\nClearly ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_0^1(V) \\cong \\bfk$ while ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_1^1(V) \\cong V \\oplus V$ with\nthe corresponding decorated trees\n$$\n\\stromVIII {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to\\bullet {}u \n\\hskip 3em \\raisebox{.8em}{ and }\\hskip 3em \n\\stromVIII \\bullet{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to u{} \n$$\nwhere $u \\in V$. Similarly, \n$$\n{\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_2^1(V) \\cong \\otexp V2 \\oplus \\otexp V2 \\oplus \\otexp V2 \\oplus \n\\otexp V2 \\oplus S^2(V)\n$$\nwith the corresponding trees\n$$\n\\stromIX {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to\\bullet\\bullet{}uv\n\\hskip 3em \n\\stromIX \\bullet{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to\\bullet u{}v\n\\hskip 3em \n\\stromIX \\bullet\\bullet{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to uv{}\n\\hskip 3em \n\\stromX \\bullet\\bullet{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to uv{}\n\\hskip 3em \n\\stromX \\bullet{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to\\bullet u{}v\n$$\nfor $u,v \\in V$.\nIn the above pictures we always placed the root\non the top. Some examples of decorated trees from\n${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n^1(V)$, $n\\geq 3$, can also be found in\nExamples~\\ref{abych_jeste_neonemocnel},\n\\ref{uz_ctvrty_neletovy_den_v_rade}, \\ref{mmm} and~\\ref{yyy}.\n\\end{example}\n\n\nLet us describe the map $d : \\pL(V) \\to \\pL^1(V)$ of Theorem~\\ref{.}\nin terms of decorated trees. We say that an edge $e$ of a decorated\ntree $S\\in {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n^1(V)$ is {\\em special\\\/} if it is adjacent to the\nspecial vertex of $S$. Given such an edge $e$, we define\nthe quotient $S\/e \\in {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n(V)$ by contracting the special edge of $S$\ninto a vertex and decorating this vertex by the label of the (unique)\nendpoint of $e$ different from the special vertex. In the following\nexamples, the special edge will be marked by the double line.\n\n\\begin{example}\n\\label{abych_jeste_neonemocnel}\nIf $S \\in {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}^1_3(V)$ is the tree\n$$\n\\raisebox{.5em}{S \\hskip .3em = \\hskip .3em \\stromXIVe \\hskip .3em ,}\n$$\n$u,v,w \\in V$, then\n\\vskip .3em \n$$\nS\/e \\hskip .3em = \\hskip .6em \\raisebox{-1em}{\\stromIII vuw}\\hskip .3em .\n$$\n\\end{example}\n\nLet $T \\in {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n(V)$. We call a couple $(S,e)$, where $S \\in\n{\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n^1(V)$ and $e$ a special edge of $S$, a {\\em blow-up\\\/} of\\\/\n$T$ if $S\/e \\cong T$ and if the arity (= the number of incoming\nedges) of the special vertex of $S$ is $\\geq 2$.\nWe denote by ${\\it bl\\\/}}\\def\\Span{{\\it Span\\\/}(T)$ the set of all blow-ups of $T$.\n\n\\begin{example}\n\\label{uz_ctvrty_neletovy_den_v_rade}\nThe set ${\\it bl\\\/}}\\def\\Span{{\\it Span\\\/}(\\hskip .1em \\raisebox{.1em}{\\rule{.4em}{.4em}}\\hskip .1em)$ is\nempty. The simplest nontrivial example of a blow-up is\n\\begin{eqnarray*}\n{\\it bl\\\/}}\\def\\Span{{\\it Span\\\/} \\left(\\hskip -2.5em \n\\raisebox {-.8em}{\\stromI uv} \\hskip .3em\\right) &=& \n\\left\\{ \n\\stromXI \\bullet{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to\\bullet u{}v \n\\right\\}, \n\\end{eqnarray*}\n\\vskip .2em\n\\noindent \nwhere the double line denotes, as in\nExample~\\ref{abych_jeste_neonemocnel}, the special edge. Let us give\ntwo more examples where $u$, $v$ and $w$ are elements of $V$:\n\\begin{eqnarray*}\n{\\it bl\\\/}}\\def\\Span{{\\it Span\\\/} \\left(\\rule{0pt}{2.7em}\\hskip 1em \\raisebox {-1.3em}{\\stromII uvw} \\hskip\n.6em\\right) &=&\n\\left\\{ \\rule{0em}{3em} \\hskip .4em\n\\stromXII \\hskip .4em , \\stromXIII\\hskip .4em\n\\right\\} \\hskip .4em \\mbox { and }\n\\\\\n{\\it bl\\\/}}\\def\\Span{{\\it Span\\\/} \\left(\\hskip .4em \\raisebox {-1.3em}{\\stromIII uvw} \\hskip\n.4em\\right) &=&\n\\left\\{ \\rule{0em}{3em} \\hskip .4em\n\\stromXIV \\hskip .4em , \\hskip .4em\n\\stromXV \\hskip .4em , \\stromXVI\\hskip .4em, \\stromeXtra\\hskip .4em\n\\right\\}.\n\\end{eqnarray*}\n\\end{example}\n\nThe last thing we need is to introduce, for $(S,e) \\in {\\it bl\\\/}}\\def\\Span{{\\it Span\\\/}(T)$, the \nsign $\\epsilon_{(S,e)} \\in \\{-1,+1\\}$ as\n$$\n\\epsilon_{(S,e)} := \\cases {+1}{if $e$ is an\n incoming edge of the special vertex, and}\n{-1}{if $e$ is the\noutgoing edge of the special vertex.}\n$$\nFinally, define the map\n\\begin{equation}\n\\label{Nunyk}\n\\delta : {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}(V) \\to {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}^1(V)\n\\end{equation}\nby\n$$\n\\delta(T) : = \\sum_{(S,e) \\in {\\it bl\\\/}}\\def\\Span{{\\it Span\\\/}(T)} \\epsilon_{(S,e)} S.\n$$\n\n\\begin{example}\n\\label{mmm}\nIn this example, $t,u,v$ and $w$ are arbitrary elements of $V$. We\nstick to our convention that the root is placed on the top. \nLet us give first some examples of the map $\\delta : {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}(V) \\to\n{\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}^1(V)$ that follow immediately from the calculations in\nExample~\\ref{uz_ctvrty_neletovy_den_v_rade}. We keep the \ndouble lines indicating\nwhich edges has been blown-up:\n\\begin{eqnarray*}\n\\delta(\\hskip .1em \\raisebox{.1em}{\\rule{.4em}{.4em}}\\hskip .1em ) &=& 0,\n\\\\\n\\delta\\left(\\hskip -2.5em \n\\raisebox {-.8em}{\\stromI uv} \\hskip .4em\n\\right) \n&=& \\stromXI \\bullet{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to\\bullet u{}v, \\rule{0em}{2.5em}\n\\\\\n\\rule{0em}{3.3em}\\delta\\left( \\hskip -.4em \\rule{0em}{2.3em} \\hskip 1em \n\\raisebox{-1.3em}{\\stromII uvw}\\hskip .8em \\right) &=& \n\\stromXII \\hskip .4em + \\stromXIII,\n\\\\\n\\rule{0em}{3.3em}\\delta \\left(\\hskip .4em \n\\raisebox {-1.3em}{\\stromIII uvw} \\hskip .4em\\right) \n&=& \n\\stromXV \\hskip .4em + \\hskip .4em \\stromXIV \\hskip .4em\n+ \\hskip .4em \\stromeXtra \\hskip .4em -\n\\stromXVI.\n\\end{eqnarray*}\n\n\\vskip .5em\n\\noindent \nLet us give some more formulas, this time without indicating the\nblown-up edges:\n\\begin{eqnarray*}\n\\rule{0em}{3.7em}\\delta \n\\left(\\hskip 1.3em\\raisebox{.4em}{\\stromIV tuvw} \\right) &=&\n\\stromXX + \\stromXXI + \\stromXXII,\n\\\\\n\\rule{0em}{3.7em}\\delta \n\\left(\\hskip .3em\\raisebox{.4em}{\\stromV tuvw} \\hskip .1em \\right) &=&\n\\stromXXIII + \\stromXXIV + \\stromXXV \n\\\\\n&& \\hskip 2em \\raisebox{1.5em}{ + \\stromXXVIextra - \\stromXXVI,}\n\\\\\n\\rule{0em}{3.3em}\\delta \\left(\\hskip .4em \n\\raisebox {.3em}{\\stromVI tuvw} \\hskip .4em\\right) &=&\n\\stromXXVII + \\stromXXVIII - \\stromXXIX\n\\\\\n&& \\hskip 2em \\raisebox{1.5em}{+ \\stromXXX +\\stromExtra.}\n\\end{eqnarray*}\n\\end{example}\n\n\nThe proof of the following proposition is a direct verification based\non the induction on the number of vertices and formula~(\\ref{0}).\n\n\n\\begin{proposition}\n\\label{abych_neonemocnel}\nThe diagram\n$$\n\\unitlength 1em\n\\thicklines\n\\begin{picture}(10,7)(7,10)\n\\put(6,16){\\makebox(0,0)[cc]{$\\pL(V)$}}\n\\put(16,16){\\makebox(0,0)[cc]{$\\pL^1(V)$}}\n\\put(6,10){\\makebox(0,0)[cc]{${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}(V)$}}\n\\put(16,10){\\makebox(0,0)[cc]{${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}^1(V)$}}\n\\put(8,16){\\vector(1,0){6}}\n\\put(8,10){\\vector(1,0){6}}\n\\put(6,15){\\vector(0,-1){4}}\n\\put(16,15){\\vector(0,-1){4}}\n\\put(11,17){\\makebox(0,0)[cc]{$d$}}\n\\put(11,11){\\makebox(0,0)[cc]{$\\delta$}}\n\\put(7,13){\\makebox(0,0)[cc]{$\\cong$}}\n\\put(17,13){\\makebox(0,0)[cc]{$\\cong$}}\n\\end{picture}\n$$\nin which the vertical maps are\nisomorphism~(\\ref{zitra_odjizdim_na_zavody})\nand~(\\ref{zitra_odjizdim_na_zavody-uz_tam_sem}), is commutative.\n\\end{proposition}\n\n\n\\begin{corollary}\n\\label{,}\nThere is a natural isomorphism\n$$\n\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V) \\cong \\Ker ( \\delta : {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}(V) \\to {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}^1(V)).\n$$\n\\end{corollary}\n\n\n\\begin{example}\n\\label{yyy}\nIt follows from the formulas given in Example~\\ref{mmm} that, for each\n$u,v,w \\in V$,\n\\def\\po#1{{\\hskip 1em \\raisebox{1.5em}{#1} \\hskip .7em }}\n\\def\\poo#1{{\\hskip .7em \\raisebox{1.5em}{#1} \\hskip .5em }}\n\\begin{eqnarray*}\n\\lefteqn{\n\\rule{0em}{3.3em}\\delta\\left( \\hskip -.4em \\rule{0em}{2.3em} \\hskip 1em \n\\raisebox{-1.3em}{\n\\stromII uvw \\po- \\stromII uwv \\po- \\stromII vwu \\po+ \\stromII wvu\n}\\hskip .8em \\right) =}\n\\\\\n&=& \\rule{0em}{2.5em} \\hskip .5em \n\\stromXii uvw \\hskip .5em-\\hskip .5em \\stromXii uwv\\hskip .5em \n-\\hskip .5em \n\\stromXii vwu \\hskip .5em+ \\hskip .5em\\stromXii wvu\n\\\\\\rule{0em}{4em}\n&&+ \\stromXvi uvw - \\stromXvi uwv - \\stromXvi vwu + \\stromXvi wvu \n\\\\\n&=& \n\\rule{0em}{5em}\\delta \\left(\\hskip .4em \n\\raisebox {-1.3em}{\\stromIII vuw\\poo- \\stromIII wuv} \\hskip .4em\\right),\n\\end{eqnarray*}\ntherefore the combination\n$$\n\\xi_{u,v,w} := \n\\raisebox{-1.3em}{\n\\stromII uvw \\po- \\stromII uwv \\po- \\stromII vwu \\po+ \\stromII wvu\n\\po- \\stromIII vuw \\poo+ \\stromIII wuv}\n$$\nbelongs to the kernel of $\\delta : {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_3(V) \\to {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_3^1(V)$. It is\neasy to see that elements of this form in fact span this kernel and\nthat the correspondence $\\xi_{u,v,w} \\mapsto [u,[v,w]]$ defines an\nisomorphism\n$$\n\\Ker \\left({\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_3(V) \\to {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_3^1(V)\\right) \\cong \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}_3(V),\n$$\nwhere $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}_3(V) \\subset \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)$ denotes the subspace of\nelements of monomial length~$3$.\n\\end{example}\n\n\n\\section{Cohomology operations}\n\\label{tt}\n\n\nIn this section we show how an object closely related to the cochain\ncomplex $\\rpL^*(V) = (\\rpL^*(V),d)$\nof~(\\ref{koupali-jsme-se-na-Hradistku}), considered in\nProposition~\\ref{a}, naturally acts on the Chevalley-Eilenberg complex\nof a Lie algebra with coefficients in itself. For $n \\geq 1$, let\n$\\bfk^n := \\Span_{\\bfk}(\\Rada e1n)$ and let $\\rpL^*(n)$ denote the subspace\nof the graded vector space $\\rpL^*(\\bfk^n)$ spanned by monomials which\ncontain each basic element $\\Rada e1n$ exactly once.\n\nMore formally, given an $n$-tuple $\\Rada t1n \\in \\bfk$,\nconsider the map $\\varphi_{\\Rada t1n} : \\bfk^n \\to \\bfk^n$ defined~by\n$$\n\\varphi_{\\Rada t1n}(e_i) := t_i e_i,\\ 1 \\leq i \\leq n.\n$$\nLet us denote by the same symbol also the induced map $\\varphi_{\\Rada\nt1n} : \\rpL^*(\\bfk^n) \\to \\rpL^*(\\bfk^n)$. Then\n$$\n\\rpL^*(n) := \\left\\{\\rule{0pt}{.9em} x \\in \\rpL^*(\\bfk^n);\\\n\\varphi_{\\Rada t1n}(x) = t_1\\cdots t_n x \\ \\mbox { for each $\\Rada t1n\n \\in \\bfk$}\\right\\}. \n$$\nThe above description immediately implies that $\\rpL^*(n)$ is a $d$-stable\nsubspace of $\\rpL^*(\\bfk^n)$, therefore $\\rpL^*(n) = (\\rpL^*(n),d)$ is\na chain complex for each $n\\geq 1$. Clearly $\\pL(n) \\cong\n\\Span_{\\bfk}({\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n)$ and $\\pL^1(n) \\cong \\Span_{\\bfk}({\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n^1)$.\nObserve that the above reduction does not erase any information,\nbecause $\\rpL^*(V)$ can be reconstructed as\n$$\n\\rpL^*(V) \\cong \\bigoplus_{n \\geq 1}\\rpL^*(n) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL_{\\Sigma_n} \\otexp Vn.\n$$\n\nLet us explain how each $U \\in \\rpL^d(n)$ determines an\n$n$-multilinear degree $d$ operation on the Chevalley-Eilenberg\ncomplex $\\CE*$ of a Lie algebra $L$ with coefficients in\nitself~\\cite{chevalley-eilenberg}. \nWe will use the standard\nidentification~\\cite[Definition~II.3.99]{markl-shnider-stasheff:book}\n\\begin{equation}\n\\label{potim_se}\n\\CE * \\cong {\\it Coder\\\/}}\\def\\desusp{{\\downarrow \\hskip .1em}^*(\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}^c(\\desusp L))\n\\end{equation}\nwhere $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}^c(\\desusp L)$ denotes the cofree conilpotent Lie\ncoalgebra~\\cite{markl-shnider-stasheff:book} \ncogenerated by the desuspension $\\desusp L$ of\nthe vector space $L$. Let $\\lambda \\in {\\it Coder\\\/}}\\def\\desusp{{\\downarrow \\hskip .1em}^1(\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}^c(\\desusp L))$\nbe the co-extension of the desuspended Lie algebra bracket \n$$\n\\desusp \\circ [-,-]\\circ (\\uparrow \\land \\uparrow) : \\desusp L \\land \\desusp\nL \\to \\desusp L\n$$ \ninto a coderivation. Then $\\lambda^2=0$ and~(\\ref{potim_se})\ntranslates the Chevalley-Eilenberg differential ${d_{\\it CE}}$ into the\ncommutator with $\\lambda$. \n\nThe above construction can be easily generalized to the case when $L$\nis an $L_\\infty$-algebra, $L = (L,l_1,l_2,l_3,\\ldots)$~\\cite{lada-markl:CommAlg95}. The\nstructure operations $(l_1,l_2,l_3,\\ldots)$ assemble again into a\ncoderivation $\\lambda \\in {\\it Coder\\\/}}\\def\\desusp{{\\downarrow \\hskip .1em}^1(\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}^c(\\desusp L))$ with\n$\\lambda^2=0$~\\cite[Theorem~2.3]{lada-markl:CommAlg95}, and~(\\ref{potim_se}) can be taken for a\ndefinition of the (Chevalley-Eilenberg) cohomology of \n$L_\\infty$-algebras with coefficients in itself.\n\n\nThe last fact we need to recall here is\nthat ${\\it Coder\\\/}}\\def\\desusp{{\\downarrow \\hskip .1em}^*(\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}^c(\\desusp L))$ is a natural pre-Lie algebra, with\nthe product $\\star$ defined as\nfollows~\\cite[Section~II.3.9]{markl-shnider-stasheff:book}. Let\n$\\Theta,\\Omega \\in {\\it Coder\\\/}}\\def\\desusp{{\\downarrow \\hskip .1em}^*(\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}^c(\\desusp L))$ and denote by\n$\\overline{\\Omega} : \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}^c(\\desusp L) \\to \\desusp L$ the corestriction\nof $\\Omega$. The pre-Lie product $\\Theta \\star \\Omega$ is then defined\nas the coextension of the composition\n$$\n(-1)^{|\\Theta||\\Omega|}\\cdot\n\\overline{\\Omega}\\circ \\Theta : \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}^c(\\desusp L) \\to \\desusp L,\n$$\nsee~\\cite[Section~II.3.9]{markl-shnider-stasheff:book} for details.\n\nBy the freeness of the pre-Lie algebra $\\pL^*(\\bfk^n,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$,\neach choice $\\Rada f1n \\in {\\it Coder\\\/}}\\def\\desusp{{\\downarrow \\hskip .1em}(\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}^c(\\desusp L))$ determines a\nunique pre-Lie algebra homomorphism\n$$\n\\Psi_{\\Rada f1n} : \\pL^*(\\bfk^n,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) \\to {\\it Coder\\\/}}\\def\\desusp{{\\downarrow \\hskip .1em}^*(\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}^c(\\desusp L))\n$$ \nsuch that $\\Psi_{\\Rada f1n}(e_i) := f_i$ for each $1 \\leq i \\leq n$,\nand $\\Psi_{\\Rada f1n}({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) := \\lambda$. Because $\\Psi_{\\Rada f1n}({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to\n\\star {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)= \\lambda^2 = 0$, the map $\\Psi_{\\Rada f1n}$ induces a map\nof the quotient $\\rpL^*(\\bfk^n) =\\pL^*(\\bfk^n,{\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)\/({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star {\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)$ \n$$\n{\\rm r}\\Psi_{\\Rada f1n} : \\rpL^*(\\bfk^n) \\to {\\it Coder\\\/}}\\def\\desusp{{\\downarrow \\hskip .1em}^*(\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}^c(\\desusp L))\n$$\nDefine finally $U(\\Rada f1n) \\in \\CE {d + |f_1| + \\cdots +|f_n|}$ by\n\\begin{equation}\n\\label{20}\nU(\\Rada f1n) := {\\rm r}\\Psi_{\\Rada f1n}(U).\n\\end{equation}\nOne can easily verify the following formula that relates the\nChevalley-Eilenberg differential ${d_{\\it CE}}$ with the differential $d$ in\n$\\rpL^*(n)$:\n\\begin{eqnarray*}\n\\lefteqn{\nd(U)(f_1,\\ldots,f_n) = {d_{\\it CE}}( U(f_1,\\ldots,f_n))} \\rule{1em}{0em}\n\\\\\n&& \\rule{1em}{0em} -\n(-1)^{|U|} \\sum_{1 \\leq i \\leq n} \n(-1)^{|f_1| + \\cdots + |f_{i-1}|}\\cdot\nU(f_1,\\ldots,{d_{\\it CE}}( f_i),\\ldots, f_n),\n\\end{eqnarray*}\nfor each $U \\in \\rpL^*(n)$ and $\\Rada f1n \\in \\CE*$.\n\n\\begin{proposition}\n\\label{to_jsem_zvedav_jestli_za_Jitkou_pojedu}\nThe collection \n$\\rpL^* := \\{\\rpL^*(n)\\}_{n \\geq 1}$ forms an operad in the category of\ndg-vector spaces. Formula~(\\ref{20}) determines\nan action that makes $\\CE *$ a differential graded \n$\\rpL^*$-algebra. Consequently, the\ncohomology operad $H^*(\\rpL)$ naturally acts on the Chevalley-Eilenberg\ncohomology $H^*_{CE}(L;L)$ of an arbitrary Lie or $L_\\infty$ algebra. \n\\end{proposition}\n\n\\begin{Proof}\nThe symmetric group $\\Sigma_n$ acts on $\\rpL^*(n)$ by\npermuting the basis $\\Rada e1n$ of $\\bfk^n$. The operadic\ncomposition, induced by the vertex\ninsertion of decorated trees representing elements of $\\pL^*(\\bfk^n)$, is\nconstructed by exactly the same method as the one used in the proof\nof~\\cite[Proposition~II.1.27]{markl-shnider-stasheff:book}. \nThe verification that $U$ defines an operadic action is easy. \n\\end{Proof}\n\n\nLet ${\\cal B}_{\\it Lie}}\\def\\Bshl{{\\cal B}_{L_\\infty}^*$ denote, as in Section~1, the dg-operad of\nnatural operations on the Chevalley-Eilenberg complex of a Lie algebra\nwith coefficients in itself and $\\Bshl^*$ an analog of this operad for\n$L_\\infty$-algebras. Because each Lie algebra is also an\n$L_\\infty$-algebra, there exists an obvious\n`forgetful' homomorphism $c: \\Bshl^* \\to {\\cal B}_{\\it Lie}}\\def\\Bshl{{\\cal B}_{L_\\infty}^*$. By\nProposition~\\ref{to_jsem_zvedav_jestli_za_Jitkou_pojedu},\nformula~(\\ref{20}) defines maps $t : \\rpL^* \\to {\\cal B}_{\\it Lie}}\\def\\Bshl{{\\cal B}_{L_\\infty}^*$ and\n$\\widetilde t : \\rpL^* \\to \\Bshl^*$. The diagram\n$$\n\n\\unitlength=.07em\n\\begin{picture}(130.00,78.00)(0.00,0.00)\n\\put(70.00,0.00){\\makebox(0.00,0.00){$t$}}\n\\put(58.00,52.00){\\makebox(0.00,0.00){$\\widetilde t$}}\n\\put(130.00,40.00){\\makebox(0.00,0.00)[l]{$c$}}\n\\put(0.00,10.00){\\makebox(0.00,0.00){$\\rpL^*$}}\n\\put(120.00,10.00){\\makebox(0.00,0.00){${\\cal B}_{\\it Lie}}\\def\\Bshl{{\\cal B}_{L_\\infty}^*$}}\n\\put(120.00,70.00){\\makebox(0.00,0.00){$\\Bshl^*$}}\n\\put(20.00,10.00){\\vector(1,0){80.00}}\n\\put(120.00,60.00){\\vector(0,-1){40.00}}\n\\put(20.00,20.00){\\vector(2,1){80.00}}\n\\end{picture}}\n$$\nis clearly commutative and $\\widetilde t : \\rpL^*\n\\to \\Bshl^*$ is in fact an {\\em inclusion\\\/} of dg-operads, compare the\nremarks in Subsection~\\ref{mot}.\n\n\n\\section{Proof of Theorem~\\protect\\ref{.}} \n\\label{zitra_se_mozna_poleti}\n\nFor the purposes of the proof of Theorem~\\ref{.}, it will be\nconvenient to reduce the complex $\\rpL^*(V) = (\\rpL^*(V),d)$\nconstructed in Proposition~\\ref{a} as follows. Since the construction\nof $\\rpL^*(V)$\nis functorial in $V$, one may consider the map $\\rpL^*(V) \\to\n\\rpL^*(0)$ induced by the map $V \\to 0$ from $V$ to the trivial vector\nspace $0$. The kernel $\\overline{\\rpL}^*(V) := \\Ker(\\rpL^*(V) \\to\n\\rpL^*(0))$ is clearly a subcomplex of $\\rpL^*(V)$. Since \n$$\n\\rpL^n(0) = \\cases {\\Span_\\bfk({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to)}{for $n = 1$ and}0{otherwise,}\n$$\nthe complexes $\\overline{\\rpL}^*(V)$ and $\\rpL^*(V)$ differ only at\nthe second term, and, under the\nisomorphism~(\\ref{zitra_odjizdim_na_zavody-uz_tam_sem}), \n$$\n\\overline{\\rpL}^1(V) \\cong \\bigoplus_{n \\geq 1} {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}_n^1(V).\n$$\nIt is also obvious that\n$$\n\\Ker \\left(\\rule{0em}{1em} d : \\pL(V) \\to \\pL^1(V) \\right)\n= \\Ker \\left(\\rule{0em}{1em} d : \\pL(V) \\to \\overline{\\rpL}^1(V) \\right).\n$$\n\nThe central object of this section is the commutative\ndiagram:\n\\begin{equation}\n\\label{d}\n\\unitlength .8em\n\\thicklines\n\\begin{picture}(21,7)(4,17)\n\\put(8,17){\\makebox(0,0)[cc]{$\\pL(V)$}}\n\\put(8,11){\\makebox(0,0)[cc]{$\\bT(V)$}}\n\\put(20,17){\\makebox(0,0)[cc]{$\\overline{\\rpL}^1(V)$}}\n\\put(20,11){\\makebox(0,0)[cc]{$\\bT(V) \\otimes \\bT(V)$}}\n\\put(8,23){\\makebox(0,0)[cc]{$\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)$}}\n\\put(8,16){\\vector(0,-1){4}}\n\\put(20,16){\\vector(0,-1){4}}\n\\put(0,-1){\n\\put(1,0){\n\\put(-.5,0){\\bezier{50}(8,22)(8,22.25)(7.75,22.25)}\n\\bezier{50}(7.25,22.25)(7,22.25)(7,22)\n}\n\\put(8,22){\\vector(0,-1){3}}\n}\n\\put(9.8,11){\\vector(1,0){6.1}}\n\\put(10,17){\\vector(1,0){7.5}}\n\\put(7,20){\\makebox(0,0)[cc]{$i$}}\n\\put(7,14){\\makebox(0,0)[cr]{$p$}}\n\\put(14,18){\\makebox(0,0)[cc]{$d$}}\n\\put(14,12){\\makebox(0,0)[cc]{$\\bDelta$}}\n\\put(21,14){\\makebox(0,0)[lc]{$p^1$}}\n\\end{picture}\n\\end{equation}\n\\vskip 5em\n\\noindent \nin which $i :\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V) \\hookrightarrow \\pL(V)$ is the inclusion and $p :\n\\pL(V) \\to \\bT(V)_{\\it pL} = \\bT(V)$ the canonical map of pre-Lie\nalgebras induced by the inclusion $V \\hookrightarrow \\bT(V)$. The\ndefinition of $p^1 : \\overline{\\rpL}^1(V) \\to \\bT(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL\n\\bT(V)$ will use the following simple facts.\n\n\\vskip .5em\n{\\em Fact~1\\\/}. The graded pre-Lie algebra structure of $\\rpL^*(V)$\ninduces on $\\rpL^1(V)= \\pL^1(V)$ a~structure of\na $\\pL(V)$-bimodule. \n\n{\\em Fact~2\\\/}. With the structure above, $\\rpL^1(V)$ is the free\n$\\pL(V)$-bimodule generated by the dummy~${\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to$.\n\n{\\em Fact~3\\\/}. The $*$-action~(\\ref{2}) makes ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$ a\nbimodule over the pre-Lie algebra ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)_{pL}$. Therefore ${\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL\n{\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$ is a $\\pL(V)$-bimodule, via the canonical map $p : \\pL(V) \\to\n{\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)_{pL}$.\n\n\\vskip .5em\nThe above facts imply that one can define a map $\\widehat p^1 : \\rpL(V)^1\n\\to {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V)$ by requiring that it is a $\\pL(V)$-bimodule\nhomomorphism satisfying\n$$\n\\widehat p^1({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) := 1 \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL 1 \\in {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL {\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V). \n$$\nIt is clear that this $\\widehat p^1$ restricts to the requisite map\n$p^1 : {\\overline {\\rpL}}^1(V) \\to \\bT(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\bT(V)$.\n\nTo prove that the bottom square of~(\\ref{d}) commutes, we notice that\nboth compositions $\\bDelta p$ and $p^1 d$ behave in the\nsame way with respect to the pre-Lie multiplication $\\star$ on\n$\\pL(V)$. \nIndeed, for $a,b \\in \\pL(V))$, by~(\\ref{4})\n$$\n\\bDelta p(a \\star b) = \\bDelta(p(a) \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em p(b)) = \\bDelta p(a) * p(b)\n+ p(a)* \\bDelta p(b) + R(p(a),p(b)).\n$$\nSimilarly, by~(\\ref{0}) and the definition of $p^1$,\n\\begin{eqnarray*}\np^1 d(a \\star b) &=& p^1(d(a) \\star b) + p^1(a \\star d(b)) +\np^1(Q(a,b))\n\\\\\n&=& p^1 d(a)* p(b) + p(a) * p^1 d(b) + p^1(Q(a,b)).\n\\end{eqnarray*}\nIt remains to verify that $p^1(Q(a,b)) = R(p(a),p(b))$. By the definitions\nof $p^1$, $Q$, $R$ and the $*$-action~(\\ref{2}),\n\\begin{eqnarray*}\np^1(Q(a,b)) &=& p^1(({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star a) \\star b) - p^1({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star (a \\star\nb)) = p^1({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to \\star a) * p(b) - p^1({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) * p(a \\star b)\n\\\\\n&=&\n(p^1({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) * p(a)) * p(b) - p^1({\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to) * (p(a) \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em p(b))\n\\\\\n&=&\n((1 \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL 1) * p(a)) * p(b) - (1\\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL 1) * (p(a) \\hskip .2em{\\mbox {\\small $\\bullet$}}\\hskip .2em p(b))\n\\\\\n&=&\np(a) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL p(b) + p(b) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL p(a) = R(p(a),p(b)).\n\\end{eqnarray*}\nObserve finally that $\\bDelta p(v) = p^1 dv = 0$ for $v \\in V$. The\ncommutativity $\\bDelta p = p^1 d$ of the bottom square\nof~(\\ref{d}) then follows from the following lemma.\n\n\\begin{lemma}\nLet $S : \\pL(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\pL(V) \\to \\bT(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\bT(V)$ be a symmetric\nlinear map such that the expression\n\\begin{equation}\n\\label{jsem_zvedav_jestli_budou_Pastviny_fungovat}\nS(a,b) * p(c) + S(a \\star b,c) - S(a,b\\star c),\\ a,b,c \\in \\pL(V),\n\\end{equation}\nis symmetric in $b$ and $c$. Then there exists precisely one linear\nmap $F : \\pL(V) \\to \\bT(V) \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\bT(V)$ such that\n\\begin{itemize}\n\\item[(i)]\n$F(a \\star b) = F(a)* p(b) + p(a) * F(b) + S(a,b)$ for each $a,b \\in\n\\pL(V)$, and\n\\item[(ii)]\n$F(v) = 0$ for each $v \\in V$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{Proof}\nThe map $F$ is constructed by the induction on the\nmonomial length of elements of $\\pL(V)$, its uniqueness is\nobvious. The symmetry of the form\nin~(\\ref{jsem_zvedav_jestli_budou_Pastviny_fungovat}) in $b$ and $c$\nis necessary for the compatibility of the rule~(i) with the\naxiom~(\\ref{aby_to_nezkoncilo_prusvihem}). \n\\end{Proof}\n\n\nWe claim that Proposition~\\ref{.} follows from the following\n\n\\begin{lemma}\n\\label{aby_mne_nerozbolel_zub}\nIn diagram~(\\ref{d}),\n\\begin{itemize}\n\\item[(i)]\n$di = 0$,\n\\item[(ii)]\n$\\Ker(d) \\cap \\Ker(p) = 0$ and\n\\item[(iii)]\n$p(\\Ker(d)) \\subset \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)$.\n\\end{itemize}\n\\end{lemma}\n\n\nIndeed, (i)~implies that $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V) \\subset \\Ker(d)$ while (ii) and (iii)\ntogether imply that $p$ maps $\\Ker(d)$ monomorphically to\n$\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)$. Since all these spaces are graded of finite type and\ntheir maps preserve the gradings, one concludes that $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)\n= \\Ker(d)$.\n\n\\noindent \n{\\bf Proof of Lemma~\\ref{aby_mne_nerozbolel_zub}.} \\hglue 1.8em \nThe symmetry of $Q$ in~(\\ref{0}) implies that $d$ is a derivation of\nthe Lie algebra $\\pL(V)_L$ associated to $\\pL(V)$. This fact, together\nwith $d(V) = 0$, readily implies that $d$ annihilates Lie elements in\n$\\pL(V)$, which is~(i).\n\nOur proof of~(ii) relies on the tree language introduced in\nSection~4. We will use\nthe following terminology.\nA decorated tree $T \\in {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}(V)$ is {\\em linear\\\/} if all its vertices are of\narity $\\leq 1$. Such a tree $T$ is of the form\n\\begin{equation}\n\\label{18}\n\n\\unitlength=1.5pt\n\\begin{picture}(0.00,27.00)(0.00,30.00)\n\\thicklines\n\\put(5.00,0.00){\\makebox(0.00,0.00)[l]{$v_{i}$}}\n\\put(5.00,10.00){\\makebox(0.00,0.00)[l]{$v_{i-1}$}}\n\\put(5.00,30.00){\\makebox(0.00,0.00)[l]{$v_3$}}\n\\put(5.00,40.00){\\makebox(0.00,0.00)[l]{$v_2$}}\n\\put(5.00,50.00){\\makebox(0.00,0.00)[l]{$v_1$}}\n\\put(0.00,20.00){\\makebox(0.00,0.00){$\\vdots$}}\n\\put(0.00,0.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(0.00,10.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(0.00,30.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(0.00,40.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(0.00,50.00){\\makebox(0.00,0.00){$\\rule{.4em}{.4em}$}}\n\\put(0.00,10.00){\\line(0,-1){10.00}}\n\\put(0.00,50.00){\\line(0,-1){20.00}}\n\\end{picture}}\n\\end{equation}\n\\vskip 40pt\n\\noindent \nwith some $\\Rada v1i \\in V$, $i \\geq 1$. Each non-linear tree $T \\in {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}(V)$\nnecessarily looks as\n\\begin{equation}\n\\label{19}\n\n\\unitlength=1.5pt\n\\begin{picture}(40.00,40.00)(0.00,0.00)\n\\thicklines\n\\put(0,-30){\n\\thicklines\n\\put(25.00,20.00){\\makebox(0.00,0.00)[l]{$v_{i}$}}\n\\put(25.00,30.00){\\makebox(0.00,0.00)[l]{$v_{i-1}$}}\n\\put(25.00,50.00){\\makebox(0.00,0.00)[l]{$v_3$}}\n\\put(25.00,60.00){\\makebox(0.00,0.00)[l]{$v_2$}}\n\\put(25.00,70.00){\\makebox(0.00,0.00)[l]{$v_1$}}\n\\put(20.00,10.00){\\makebox(0.00,0.00){$S$}}\n\\put(20.00,40.00){\\makebox(0.00,0.00){$\\vdots$}}\n\\put(20.00,20.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(20.00,30.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(20.00,50.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(20.00,60.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(20.00,70.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(40.00,0.00){\\line(-1,1){20.00}}\n\\put(0.00,0.00){\\line(1,0){40.00}}\n\\put(20.00,20.00){\\line(-1,-1){20.00}}\n\\put(20.00,30.00){\\line(0,-1){10.00}}\n\\put(20.00,70.00){\\line(0,-1){20.00}}\n}\n\\end{picture}}\n\\end{equation}\n\\vskip 40pt\n\\noindent \nwhere $S$ is a tree whose root vertex $v_i$ has arity $\\geq 2$. We say\nthat such a decorated tree has {\\em tail of length $i$\\\/}. These\nnotions translate to decorated trees from ${\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}^1(V)$ in the obvious\nmanner.\n\nWe leave to the reader to verify that, under\nidentification~(\\ref{zitra_odjizdim_na_zavody}), the map $p : \\pL(V)\n\\to \\bT(V)$ is described as\n$$\np(T) =\n\\cases{v_1 \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL v_2 \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL \\cdots \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL v_{i-1} \\otimes}\\def\\Sh{{\\it Sh}}\\def\\pl{\\pL v_i}\n {if $T$ is linear as in~(\\ref{18}), and}0\n {if $T$ is non-linear.}\n$$\nTherefore $\\Ker(p)$ consists of linear combinations of non-linear\ntrees. Before going further, we need to inspect how the map $\\delta :\n{\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}(V) \\to {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}^1(V)$ of~(\\ref{Nunyk}), which is the differential $d:\n\\pL(V) \\to \\pL^1(V)$ written in terms of trees, acts on non-linear\ntrees. If $T$ is the decorated tree~(\\ref{19}), then it immediately\nfollows from the definition~(\\ref{Nunyk}) of $\\delta$ that\n\\begin{equation}\n\\label{snad-to-konecne-dopisu}\n\\delta (T) = -T' + \\mbox {trees with tails of length $\\leq i$},\n\\end{equation}\nwhere $T'$ is the following decorated tree with tail of length $i+1$\n$$\n\n\\unitlength=1.5pt\n\\begin{picture}(40.00,75.00)(0.00,0.00)\n\\thicklines\n\\put(25.00,30.00){\\makebox(0.00,0.00)[l]{$v_i$}}\n\\put(25.00,50.00){\\makebox(0.00,0.00)[l]{$v_3$}}\n\\put(25.00,60.00){\\makebox(0.00,0.00)[l]{$v_2$}}\n\\put(25.00,70.00){\\makebox(0.00,0.00)[l]{$v_1$}}\n\\put(20.00,10.00){\\makebox(0.00,0.00){$S'$}}\n\\put(20.00,40.00){\\makebox(0.00,0.00){$\\vdots$}}\n\\put(20.00,20.00){\\makebox(0.00,0.00){${\\circ}}\\def\\td{{\\tilde d}}\\def\\epi{\\to \\hskip -.6em \\to$}}\n\\put(20.00,30.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(20.00,50.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(20.00,60.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(20.00,70.00){\\makebox(0.00,0.00){$\\bullet$}}\n\\put(40.00,0.00){\\line(-1,1){19.00}}\n\\put(0.00,0.00){\\line(1,0){40.00}}\n\\put(19.00,19.00){\\line(-1,-1){19.00}}\n\\put(20.00,30.80){\\line(0,-1){9.00}}\n\\put(20.00,70.00){\\line(0,-1){20.00}}\n\\end{picture}}\n$$\nin which $S'$ is the tree obtained from $S$ by replacing the root\nvertex decorated by $v_i$ by the special one. \nThe map $\\delta' : {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}(V) \\to {\\rm Tr}}\\def\\otexp#1#2{#1^{\\otimes #2}^1(V)$ given by $\\delta'(T) :=\n-T'$ is a {\\em monomorphism\\\/}.\n\n\nLet $x$ be a linear combination of non-linear trees and assume $\\delta\n(x) =0$. We must prove that then $x=0$. Assume $x \\not= 0$ and\ndecompose $x = x_s + x_{s-1} + \\cdots + x_1$, where $x_i$ is, for $1\n\\leq i \\leq s$, a linear combination of decorated trees with tails of\nlength $i$, and $x_s \\not= 0$. By~(\\ref{snad-to-konecne-dopisu}), the\nonly trees with tails of length $s+1$ in $\\delta(x)$ are those spanning\n$\\delta'(x_s)$, therefore $\\delta(x) = 0$ implies $\\delta'(x_s)=0$ which\nin turn implies that $x_s = 0$, because $\\delta'$ is monic. This is a\ncontradiction, therefore $x=0$ which proves~(ii).\n\nTo verify~(iii), notice that, by the commutativity of the bottom\nsquare of~(\\ref{19}), $p (\\Ker(\\delta)) \\subset \\Ker(\\bDelta))$ while\n$\\Ker(\\bDelta) = \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)$ by Theorem~\\ref{psano_v_aute}. This finishes\nthe proof of the lemma.\n\\qed\n \n\n\\section{Some open questions and ramifications}\n\n\\begin{odstavec}\n\\label{trip}\n{\\rm\n{\\it Triplettes of operads (after J.-L.~Loday)\\\/}.\nThe following notion was introduced in~\\cite{loday:slides}. \n\n\\begin{Definition}\n\\label{jeste_ji_musim_napsat_SMS}\nThe data $(\\calC,\\spin,\\mbox {${\\cal A}$-{\\tt alg}} \\stackrel{F}\\to \\mbox {${\\cal P}$-{\\tt alg}})$, where\n\\begin{itemize}\n\\item[(i)]\n$\\calC$ and $\\calA$ are operads,\n\\item[(ii)]\n$\\spin$ are `spin' relations intertwining $\\calC$-co-operations and\n$\\calA$-operations, so that $(\\calC,\\spin,\\calA)$ determines a class of\nbialgebras, \n\\item[(iii)]\nthe operad ${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}$ governs the algebra structure of the primitive\npart ${\\it Prim\\\/}}\\def\\calH{{\\cal H}(\\calH)$ of $(\\calC,\\spin,\\calA)$-bialgebras, and\n\\item[(iv)]\n$F$ is a forgetful functor functor from the category of\n$\\calA$-algebras to the category of ${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}$-algebras such that the\ninclusion ${\\it Prim\\\/}}\\def\\calH{{\\cal H}(\\calH) \\subset F(\\calH)$ is a morphism of ${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}$-algebras,\n\\end{itemize}\nis called a {\\em triplette\\\/} of operads.\n\\end{Definition}\n\n\nAn example is $(\\Com,\\spin, {\\cal A{\\it ss\\\/}}} \\def\\Com{{\\cal C{\\it om\\\/}},{\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A})$, with $\\spin$\nthe usual bialgebra relation recalled\nin~(\\ref{to_jsem_zvedav_jesli_budou_pastviny_fungovat}). \nLet $U$ be a left adjoint to $F$.\nA triplette in Definition~\\ref{jeste_ji_musim_napsat_SMS} is \n{\\em good\\\/}~\\cite{loday:slides}, \nif the following three conditions are equivalent:\n\n\\begin{itemize}\n\\item[(i)]\na $(\\calC,\\spin,\\calA)$-bialgebra $\\calH$ is connected,\n\\item[(ii)]\n$\\calH \\cong U({\\it Prim\\\/}}\\def\\calH{{\\cal H}(\\calH))$, and\n\\item[(iii)]\n$\\calH$ is cofree among connected\n$\\calC$-coalgebras.\n\\end{itemize}\nLet $\\calA(V)$ (resp.~${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}(V)$) denote the free $\\calA$-\n(resp.~${\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}$-)algebra on $V$.\nAs observed in~\\cite{loday:slides}, for good triplettes\n\\begin{equation}\n\\label{`}\n{\\it Prim\\\/}}\\def\\calH{{\\cal H} (\\calA (V)) \\cong {\\cal P}}\\def\\calB{{\\cal B}}\\def\\calS{{\\cal S}(V).\n\\end{equation}\n \nThe classical Theorem~\\ref{psano_v_aute} in Section~2 is a\nconsequence of the goodness of the triplette $(\\Com,\\spin, {\\cal A{\\it ss\\\/}}} \\def\\Com{{\\cal C{\\it om\\\/}},{\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A})$\nmentioned above, because~(\\ref{`}) in this case says that ${\\it Prim\\\/}}\\def\\calH{{\\cal H}(T(V))\n\\cong \\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)$. Other, in some cases very surprising, good triplettes\ncan be found in~\\cite{loday:slides}. The following problem was\nsuggested by J.-L.~Loday:\n\n\\begin{problem}\n\\label{kk}\nAre there an\noperad $\\calC$ and spin relations $\\spin$ with the property that\n$(\\calC,\\spin,p{\\cal L{\\it ie\\\/}}}\\def\\Brace{{\\cal B{\\it race\\\/}},{\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A})$ is a good triplette?\n\\end{problem}\n\nAs we remarked in Subsection~\\ref{gen}, the affirmative answer to the\nDeligne conjecture given in~\\cite{kontsevich-soibelman} implies that\nthere exist a characterization of Lie elements in brace\nalgebras~\\cite{gerstenhaber-voronov:FAP95} similar to our\nTheorem~\\ref{.}. This suggests formulating the following version of\nProblem~\\ref{kk} in which $\\Brace$ is the operad for brace\nalgebras.\n\n\\begin{problem}\nAre there an\noperad $\\calC$ and spin relations $\\spin$ with the property that\n$(\\calC,\\spin,\\Brace,{\\cal L{\\it ie\\\/}}}\\def\\calC{{\\cal C}}\\def\\calA{{\\cal A})$ is a~good triplette?\n\\end{problem}\n\n}\\end{odstavec}\n\n\n\\begin{odstavec}\n{\\rm \n{\\it Lie elements and cobar constructions.\\\/} \nIn Section~2 we calculated the cohomology\nof the cobar construction~(\\ref{1a}) of the\nshuffle coalgebra and observed that $H^0({\\sf T}}\\def\\fLie{{\\sf L}}\\def\\fpLie{p{\\sf L}}\\def\\bfk{{\\bf k}(V),\\Delta)$ is\nisomorphic to the free Lie algebra $\\fLie}\\def\\bT{\\bfA}\\def\\pL{{\\sf pL}} \\def\\rpL{{\\rm r\\pL}(V)$. In our characterization of\nLie elements in pre-Lie algebras, the role of~(\\ref{1a}) is played by\ncomplex~(\\ref{za_chvili_tam_musim_volat_a_hrozne_se_mi_nechce!}).\nThis leads to the following problem, which may or may not be related\nto Problem~\\ref{kk},\n\n\n\\begin{problem}\nCalculate the cohomology\nof~(\\ref{za_chvili_tam_musim_volat_a_hrozne_se_mi_nechce!}). Is \nthis complex the cobar construction of some coalgebra?\n\\end{problem}\n\n\nAs D.~Tamarkin recently informed us, methods proposed in an enlarged\nunfinished, \nunpublished version of~\\cite{tamarkin:def_of_chiral_algebras} may\nimply that the\ncomplex~(\\ref{za_chvili_tam_musim_volat_a_hrozne_se_mi_nechce!}) is\nacyclic in positive dimensions, as envisaged also by some\nconjectures formulated in~\\cite{markl:de}. }\\end{odstavec}\n\n\n\\references\n\n\n\\nextref{chapoton-livernet:pre-lie}\n {Chapoton, F. and Livernet, M.}\n {\\em Pre-{Lie} algebras and the rooted trees operad}\n {Internat. Math. Res. Notices {\\bf 8}(2001), 395--408}\n\n\\nextref{chevalley-eilenberg}\n {Chevalley, C. and Eilenberg, S.}\n {\\em Cohomology theory of {Lie} groups and {Lie} algebras}\n {Trans. Amer. Math. Soc., {\\bf 63}(1948), 85--124}\n\n\\nextref{dzhu-lof:HHA02}\n {Dzhumadil'daev, A. and L\\\"ofwall, C.}\n {\\em Trees, free right-symmetric algebras, free {Novikov} algebras and\n identities}\n {Homotopy, Homology and Applications, {\\bf 4(2)}(2002),\n 165--190}\n\n\\nextref{gerstenhaber:AM63}\n {Gerstenhaber, M.}\n {\\em The cohomology structure of an associative ring}\n {Ann. of Math. {\\bf 78(2)}(1963), 267--288}\n\n\\nextref{gerstenhaber-voronov:FAP95}\n {Gerstenhaber, M. and Voronov, A.A.}\n {\\em Higher operations on the {Hochschild} complex}\n {Functional Anal. Appl. {\\bf 29(1)}(1995), 1--6 (in Russian)}\n\n\\nextref{guin-oudom}\n {Guin, D. and Oudom, J.-M.}\n {\\em Sur l'alg\\'ebre enveloppante d'une alg\\'ebre pr\\'e-{Lie}}\n {C. R. Acad. Sci. Paris S{\\'e}r. I Math., {\\bf 340(5)}{2005}, 331--336}\n\n\\nextref{kontsevich-soibelman}\n {Kontsevich, M. and Soibelman, Y.}\n {\\em Deformations of algebras over operads the {Deligne} conjecture}\n {In {Dito, G. et al.}, editor, {Conf\\'erence {Mosh\\'e Flato} 1999:\n Quantization, deformation, and symmetries}, number~21 in \n Math. Phys. Stud.,\n pages 255--307. Kluwer Academic Publishers, 2000}\n\n\\nextref{lada-markl:CommAlg95}\n {Lada, T. and Markl, M.}\n {\\em Strongly homotopy {Lie} algebras}\n {Comm. Algebra {\\bf 23(6)}(1995), 2147--2161}\n\n\\nextref{lm:sb}\n {Lada, T. and Markl, M.}\n {\\em Symmetric brace algebras}\n {Applied Categorical Structures {\\bf 13(4)}, 351--370}\n\n\\nextref{loday:slides}\n {Loday, J.-L.}\n {\\em Generalized bialgebras and triples of operads}\n {A slide show. Available from the {\\tt www} home page of\n J.-L.~Loday}\n\n\\nextref{maclane:homology}\n {Mac~Lane, S.}\n {\\em Homology}\n {Springer-Verlag, 1963}\n\n\\nextref{markl:de}\n {Markl, M.}\n {\\em Cohomology operations and the {Deligne} conjecture}\n {preprint {\\tt math.AT\/0506170}, June 2005}\n\n\n\\nextref{markl-remm}\n {Markl, M. and Remm, E.}\n {Algebras with one operation including {Poisson} and other\n {Lie}-admissible algebras}\n {J. Algebra, {\\bf 299}(2006), 171--189}\n\n\\nextref{markl-shnider-stasheff:book}\n {Markl, M., Shnider, S. and Stasheff, J.D.}\n {\\em Operads in Algebra, Topology and Physics}\n {volume~96 of Mathematical Surveys and Monographs,\n American Mathematical Society, Providence, Rhode Island,\n 2002}\n\n\\nextref{ree:AnM69}\n {Ree, R.}\n {\\em Lie elements and an algebra associated with shuffles}\n {Ann. of Math. {\\bf 68}, 210--220}\n\n\\nextref{serre:65}\n {Serre, J.-P.}\n {\\em Lie Algebras and {Lie} Groups}\n {Benjamin, 1965}\n\n\\nextref{tamarkin:def_of_chiral_algebras}\n {Tamarkin, D.}\n {\\em Deformations of chiral algebras}\n {In {Ta Tsien et al.}, editor, {Proceedings of ICM 2002, Beijing, \n China, August 20-28},\n vol. II: Invited lectures, pages 105--116. \n Beijing: Higher Education Press;\n Singapore: World Scientific\/distributor, 2002}\n\n\n\\lastpage\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nLet $\\mathcal{T}_{q+1}$ be a $(q+1)$-regular tree and let $\\textrm{Aut}(\\mathcal{T}_{q+1})$ be the group of isometries of $\\mathcal{T}_{q+1}$. Denote by $V\\mathcal{T}_{q+1}$ the set of all the vertices of $\\mathcal{T}_{q+1}$. For a discrete subgroup $\\Gamma$ of $\\textrm{Aut}(\\mathcal{T}_{q+1})$, the \\emph{critical exponent} of $\\Gamma$ is defined by\n$$\\delta_\\Gamma=\\underset{n\\to\\infty}{\\overline{\\lim}}\\frac{\\log\\#\\{\\gamma\\in\\Gamma\\colon d(v,\\gamma v)\\le n\\}}{n}$$ for a fixed vertex $v\\in V\\mathcal{T}_{q+1}$. This does not depend on the choice of $v\\in V\\mathcal{T}_{q+1}$.\n\nEquivalently, it can be defined as the infimum of $s>0$ such that the \\emph{Poincar\\'e} \\emph{series} $\\sum_{\\gamma\\in\\Gamma}e^{-sd(v,\\gamma v)}$ converges.\n\n\n\nIn some cases, it captures the dimension at ``infinity\" of $\\Gamma$ as well as measures the exponential growth rate of the size of $\\Gamma$-orbits. It is shown in (\\cite{Pa}) that if $\\Gamma$ acts on $\\mathcal{T}_{q+1}$ without a global fixed point on $\\partial_\\infty\\mathcal{T}_{q+1}$, then $\\delta_\\Gamma$ is equal to the Hausdorff dimension of the \\emph{conical limit set}. Here, $\\partial_\\infty\\mathcal{T}_{q+1}$ is the \\emph{Gromov boundary at infinity} of $\\mathcal{T}_{q+1}$ defined as the set of equivalence class $[\\ell]$ of geodesic ray $\\ell$ starting from a fixed vertex $v\\in V\\mathcal{T}_{q+1}$, where two geodesic rays $\\ell$ and $\\ell'$ are equivalent if and only if $\\{d_{\\mathcal{T}}(\\ell(n),\\ell'(n))\\colon n\\in\\mathbb{Z}_{>0}\\}$ is bounded above. A conical limit set is the set of points $\\omega$ in $\\partial_\\infty\\mathcal{T}_{q+1}$ such that there is a sequence of points in $\\Gamma v$ converging towards $\\omega$ while staying at bounded distance from a geodesic ray ending at $\\omega$.\n\n\n\n\n\n\nIf the quotient graph $\\Gamma\\backslash\\mathcal{T}_{q+1}$ is finite, then we say that $\\Gamma$ is a \\emph{uniform lattice} in $\\textrm{Aut}(\\mathcal{T}_{q+1})$ and in this case $\\delta_\\Gamma=\\log q$. If $\\Gamma$ is either itself finite or has a cyclic group as a finite index subgroup, then $\\delta_\\Gamma$ is equal to zero. In general, we always have $0\\le\\delta_\\Gamma\\le\\log q$. Recently, it is proved in \\cite{CDS} that if $\\Gamma$ is convex co-compact (that is, every limit point of $\\Gamma$ is a conical limit point) acting isometrically on a $CAT(-1)$ space and $\\Lambda$ is a subgroup of $\\Gamma$, then $\\delta_\\Gamma=\\delta_{\\Lambda}$ if and only if $\\Lambda$ is \\emph{co-amenable} in $\\Gamma$. Recall that $\\Lambda$ is said to be co-amenable in $\\Gamma$ if the left coset space $\\Lambda\\backslash\\Gamma$ has $\\Gamma$-invariant mean. See also \\cite{No} and \\cite{No2} for the similar results in regular and arbitrary graphs.\n\nIn this article, we ask the following question of existence: For a given number $\\delta\\in [0,\\log q]$, is there a discrete subgroup $\\Gamma$ of $\\textrm{Aut}(\\mathcal{T}_{q+1})$ such that $\\delta_\\Gamma=\\delta$? We give a partial answer to this question. Namely, for $\\delta\\in [0,\\frac{1}{2}\\log q]$, there exists a discrete subgroup $\\Gamma$ of $\\textrm{Aut}(\\mathcal{T}_{q+1})$ for which $\\delta_\\Gamma=\\delta$.\n\nLet $E=\\{\\delta\\in[0,\\log q]\\colon \\delta=\\delta_\\Gamma\\textrm{ for some discrete }\\Gamma<\\textrm{Aut}(\\mathcal{T}_{q+1})\\}$.\n\n\\begin{thm}\\label{main1}\nIf $\\delta\\in[0,\\frac{1}{2}\\log q]$, then $\\delta\\in E$.\n\\end{thm}\n\nIn Section 2, we summarize the connection between graph of groups and its edge-indexed graphs and their covering theory, which is the main idea of the proof. The proof of Theorem~\\ref{main1} is given in Section 3 in which we construct an edge-indexed graph corresponding to a real number $\\delta$ in $[0,\\frac{1}{2}\\log q]$, for which the fundamental group of its finite grouping has the critical exponent $\\delta$. In Section 4, we give some remarks involving two questions: Which real numbers larger than $\\frac{1}{2}\\log q$ are realized? Which algebraic integers are realized as a critical exponents of free groups?\n\n\\subsubsection*{Acknowledgement} \n\n\n\n\\section{Edge-indexed graphs and graph of groups}\n\nWe briefly review the essential features of the covering theory for graph of groups which is the main idea of the proof. Being developed by Serre \\cite{Se} and the substantial contribution of the subsequent work of Bass \\cite{Ba}, the theory is currently called as Bass-Serre theory. We mainly follows \\cite{Se} and refer to \\cite{Ba} for further details.\n\nLet $A$ be an undirected graph which is allowed to have loops and multiple edges. We denote by $VA$ the set of vertices of $A$ and by $EA$ the set of edges of $A$. Given an undirected graph $A$, we get a \\emph{symmetric} directed graph corresponding to $A$, whose set of vertices are the same as $VA$ and every edge of $A$ are bidirected. Denote by $E^{ori}\\!\\!A$ the set of all oriented edges of the symmetric directed graph corresponding to $A$, hence $|E^{ori}A|=2|EA|$.\n\nFor $e\\in E^{ori}\\!\\!A$, let $\\overline{e}\\in E^{ori}\\!\\!A$ be the opposite edge of $e$ and let $\\partial_0e$ and $\\partial_1e$ be the initial vertex and the terminal vertex of $e$, respectively. \n\n\\begin{de}Let $i\\colon E^{ori}\\!\\!A\\to \\mathbb{Z}_{>0}$ be a map assigning a positive integer to each oriented edge. We say $(A,i)$ an \\emph{edge-indexed graph}.\n\\end{de}\n\n\n\n\n\\begin{de}By a \\emph{graph of groups} $\\mathbf{A}=(A,\\mathcal{A})$, we mean a connected graph $A$ together with attached groups $\\mathcal{A}_a$ $(a\\in VA)$, $\\mathcal{A}_e=\\mathcal{A}_{\\overline{e}}$ $(e\\in E^{ori}\\!\\!A)$, and monomorphisms $\\alpha_e\\colon \\mathcal{A}_e\\to\\mathcal{A}_{\\partial_1 e}$ $(e\\in E^{ori}\\!\\!A)$. \n\\end{de}\n\nAn \\emph{isomorphism} between two graph of groups $\\mathbf{A}=(A,\\mathcal{A})$ and $\\mathbf{A'}=(A',\\mathcal{A'})$ is an isomomorphism $\\phi\\colon A\\to A'$ between two underlying graphs together with the set of isomomorphisms $\\phi_a\\colon \\mathcal{A}_a\\to \\mathcal{A'}_{\\phi(a)}$ and $\\phi_e\\colon \\mathcal{A}_e\\to \\mathcal{A'}_{\\phi(e)}$ satisfying the following property: for each $e\\in EA$, there is an element $h_e\\in \\mathcal{A'}_{\\phi(\\partial_1e))}$ such that \n$$\\phi_{\\partial_1e}(\\alpha_e(g))=h_e\\cdot(\\alpha'_{\\phi(e)}(\\phi_e(g)))\\cdot h_e^{-1}$$\nfor all $g\\in\\mathcal{A}_e$ (\\cite{Ba}, Section 2).\n\nGiven an edge-indexed graph $(A,i)$, a graph of groups $(A,\\mathcal{A})$ is called a \\emph{grouping} of $(A,i)$ if $i(e)=[\\mathcal{A}_{\\partial_1e}\\colon\\alpha_e\\mathcal{A}_e]$ and called a \\emph{finite grouping} if all $\\mathcal{A}_a$ $(a\\in VA)$ are finite.\n\n\nSuppose that we have a graph of groups $\\mathbf{A}$. Choosing a base point $a_0\\in VA$, we can define a \\emph{fundamental group} $\\pi_1(\\mathbf{A},a_0)$ (\\cite{Se} Section 5.1), a \\emph{universal covering tree} $(\\widetilde{\\mathbf{A},a_0})$ and an action without inversion of $\\pi_1(\\mathbf{A},a_0)$ on $(\\widetilde{\\mathbf{A},a_0})$ with a morphism $p\\colon (\\widetilde{\\mathbf{A},a_0}) \\to A$ which can be identified with the quotient projection (\\cite{Se}, Section 5.3).\n\n\\begin{de} Given a graph of groups $\\mathbf{A}=(A,\\mathcal{A})$, we denote by $F(\\mathcal{A}, E^{ori})$ the group generated by the groups $\\mathcal{A}_a$, $(a\\in VA)$ and the elements $e\\in E^{ori}$, subject to the relations\n$$\\overline{e}=e^{-1}\\textrm{ and }e\\alpha_e(g)e^{-1}=\\alpha_{\\overline{e}}(g)\\textrm{ for }e\\in E^{ori}A\\textrm{ and }a\\in \\mathcal{A}_e.$$\nLet $\\pi_1(\\mathbf{A},a_0)$ be the set of elements of $F(\\mathcal{A},E^{ori})$ of the form $$g_0e_1g_1e_2\\cdots e_{n-1}g_{n-1},$$ where $o(e_{i+1})=t(e_i)$ (mod $n)$ and $g_i\\in o(e_{i+1})$. It is a subgroup of $F(\\mathcal{A},E^{ori})$, called \\emph{fundamental group} of $\\mathbf{A}$ based at $a_0$.\n\\end{de}\n\nGiven a graph of groups $\\mathbf{A}$, the graph $\\widetilde{X}=(\\widetilde{\\mathbf{A},a_0})$ is defined as\n$$V\\widetilde{X}=\\bigcup_{a\\in VA}\\pi_1(\\mathbf{A},a_0)\/\\textrm{Stab}_{\\pi_1(\\mathbf{A},a_0)}(a)$$ and $$E\\widetilde{X}=\\bigcup_{e\\in E^{ori}A}\\pi_1(\\mathbf{A},a_0)\/\\textrm{Stab}_{\\pi_1(\\mathbf{A},a_0)}(e).$$\n\n\\begin{thm}[\\cite{Se}, Theorem 12] The graph $\\widetilde{X}$ defined as above is a tree.\n\\end{thm}\nThe tree $\\widetilde{X}=(\\widetilde{\\mathbf{A},a_0})$ is called a \\emph{universal covering tree} of the graph of groups $\\mathbf{A}$.\n\nSuppose that a group $\\Gamma$ acts on a graph $A$. We call $s\\in\\Gamma$ an \\emph{inversion} on $A$ if $se=\\overline{e}$ for some $e\\in E^{ori}\\!\\!A$. If $\\Gamma$ acts without inversions, then the quotient graph $\\Gamma\\backslash A$ is well-defined and we have a natural projection $p\\colon A\\to\\Gamma\\backslash A$.\n\nWhen a group $\\Gamma$ acts without inversion on a tree $\\mathcal{T}_{q+1}$, we can naturally identify $\\Gamma$ and $\\mathcal{T}_{q+1}$ with the fundamental group and universal covering tree, respectively, of a graph of groups called \\emph{quotient graph of groups}.\n\n\n\\begin{de}\nSuppose that $\\Gamma$ acts without inversion on a tree $\\mathcal{T}_{q+1}$. For each $v\\in V(\\Gamma\\backslash\\mathcal{T})$ and $e\\in E^{ori}(\\Gamma\\backslash\\mathcal{T})$, we choose any corresponding vertex $\\widetilde{v}\\in V\\mathcal{T}$ and edge $\\widetilde{e}\\in E\\mathcal{T}$. Let $\\overline{\\widetilde{e}}=\\widetilde{\\overline{e}}$ and fix an element $\\gamma_e\\in\\Gamma$ which satisfies $\\gamma_e\\widetilde{\\partial_1(e)}=\\partial_1(\\widetilde{e})$. Define $\\mathcal{A}_v$ and $\\mathcal{A}_e$ be the stabilizer of $\\widetilde{v}$ and $\\widetilde{e}$ in $\\Gamma$, respectively, and let $\\alpha_e\\colon\\mathcal{A}_e\\to\\mathcal{A}_{\\partial_1e}$ be the monomorphism $h\\mapsto \\gamma_e^{-1}h\\gamma_e$. Then the graph of groups $(\\Gamma\\backslash\\mathcal{T}_{q+1},\\mathcal{A})$ is called the \\emph{quotient graph of groups}.\n\\end{de}\n\nIt does not depend on the choice of $\\widetilde{v},\\widetilde{e}$ and $\\gamma_e$ up to isomorphism of graph of groups (\\cite{Ba}, Section 3).\n\nConversely, if $(A,\\mathcal{A})$ is a graph of groups and $(A,i)$ is the corresponding edge-indexed graph, then fixing a basepoint $a_0\\in VA$, the universal covering tree $\\mathcal{T}=\\widetilde{(A,a_0)}$ and the natural projection $\\pi\\colon \\mathcal{T}\\to A$ depend only on the edge indexed graph $(A,i)$ (\\cite{Ba}, Remark 1.18).\n\n\n\\section{Proof of Theorem}\n\nOur goal is to construct an edge-indexed graph $(A,i)$ which admits a finite grouping $\\mathbf{A}$ of which the fundamental group has critical exponent $\\delta\\in [0,\\frac{1}{2}\\log q]$.\n\nWe recall the following criterion for an edge-indexed graph to have a finite grouping.\n\n\\begin{thm}[\\cite{bk90}, Corollary 2.4] An edge-indexed graph $(A,i)$ admits a finite grouping if and only if there is a positive integral valued function $N\\colon VA\\to\\mathbb{Z}_{>0}$ on $VA$ satisfying \\begin{equation}\\label{ivo}\\frac{i(e)}{i(\\overline{e})}=\\frac{N(\\partial_0e)}{N(\\partial_1e).}\\end{equation}\n\\end{thm}\n\nWe say such a function $N$ an \\emph{integral vertex ordering}.\n\n For convenience to describe some graphs constructed in the proof, we introduce a graph operation between two graphs which we call \\emph{graph merging}.\n\\begin{de} Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be two (undirected or directed) graphs. The \\emph{graph merging} $G_1\\underset{x,y}{\\star} G_2$ with respect to $x\\in V_1$ and $y\\in V_2$ is a graph $G=(V_1\\cup V_2\/{x\\sim y}, E_1\\cup E_2)$, obtained by identifying two vertices $x$ and $y$ in $G_1\\cup G_2$.\n\\end{de}\n\n\\vspace{.1 in}\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tikzpicture}[every loop\/.style={}]\n \\tikzstyle{every node}=[inner sep=0pt]\n \\node (0) {$\\bullet$} node [left=22pt] at (0,0) {$G_1$}; \n \\node (0) {}node [above=4pt] at (0,0) {$x\\sim y$};\n \\node (1) {}node [right=24pt] at (0,0) {$G_2$};\n \n \\draw[dashed] (-1cm,0cm) circle (1cm) ;\n \\draw[dashed] (1cm,0cm) circle (1cm) ;\n\\end{tikzpicture}\n\\vspace{0.5em}\n\\caption{The graph $G_1\\underset{x,y}{\\star} G_2$}\n\\end{center}\n\\end{figure}\n\nWe give a proof of Theorem~1.1. Let $A_0$ be a ray given as follows:\n\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tikzpicture}[every loop\/.style={}]\n \\tikzstyle{every node}=[inner sep=0pt]\n \\node (0) {$\\bullet$} node [below=4pt] at (0,0) {$x_0$};\n \\node (2) at (1.5,0) {$\\bullet$} node [below=4pt] at (1.5,0) {$x_1$}; \n \\node (4) at (3,0) {$\\bullet$}node [below=4pt] at (3,0) {$x_2$}; \n \\node (6) at (4.5,0) {$\\bullet$}node [below=4pt] at (4.5,0) {$x_3$}; \n \\node (8) at (6,0) {$\\bullet$}node [below=4pt] at (6,0) {$x_4$}; \n \\node (10) at (7.5,0) {$\\bullet$}node [below=4pt] at (7.5,0) {$x_5$}; \n \\node (11) at (9.3,0) {$\\bullet \\cdots$}node [below=4pt] at (9,0) {$x_6$}; \n \n \n \n \n\n\n \\path[-] (0) edge node [above=4pt] {} (2)\n (2) edge node [above=4pt] {} (4)\n (4) edge node [above=4pt] {} (6)\n (6) edge node [above=4pt] {} (8)\n (8) edge node [above=3.2pt] {} (10)\n (10) edge node [above=4pt] {} (11);\n\\end{tikzpicture}\n\\vspace{0.5em}\n\\caption{A ray $A_0$}\n\\end{center}\n\\end{figure}Let $I\\cup J=\\mathbb{Z}_{>0}$ be an arbitrary partition of $\\mathbb{Z}_{>0}$. If $n\\in I$, then we assign $i(\\overrightarrow{x_nx_{n-1}})=q$ and $i(e)=1$ otherwise. We adjoin a tree $\\mathcal{T}_{q+1}'$ which is $(q+1)$-regular except one $y$ vertex of degree $q$ to $x_0$. If $n\\in J$, then for each $n$ we adjoin a copy of a tree $\\mathcal{T}_{q+1}''$ which is $(q+1)$-regular except one vertex $z_n$ of degree $q-1$ to $x_n$. In other words, $(A,i)=(A_0\\underset{x_0,y}{\\star}\\mathcal{T}_{q+1}'\\underset{x_j,z_j,j\\in J}{\\star}\\mathcal{T}_{q+1}'',i)$, where $i(\\overrightarrow{x_nx_{n-1}})=q$ for each $n\\in I$ and $i(e)=1$ otherwise.\n\n\nThen $(A,i)$ admits an integral vertex ordering. Indeed, we can define $N(x_0)=1$ and $N(x_j)=i(\\overrightarrow{x_jx_{j-1}})N(x_{j-1})$ for all $j\\ge 1$. Then $N$ satisfies the equation~(\\ref{ivo}). \n\n\n\nLet $\\Gamma$ be the subgroup of $\\textrm{Aut}(\\mathcal{T}_{q+1})$ such that the edge-indexed graph of the quotient graph of groups $(\\Gamma\\backslash \\mathcal{T}_{q+1},\\mathcal{A})$ is $(A,i)$. As we discussed in Section 2, such a group $\\Gamma$ exists and is isomorphic to the fundamental group of a graph of groups obtained by certain finite grouping of $(A,i)$. In addition, for any finite grouping, the group $\\Gamma$ is discrete since $\\Gamma_x$ is finite for every $x\\in VA$.\n\n\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tikzpicture}[every loop\/.style={}]\n \\tikzstyle{every node}=[inner sep=0pt]\n \\node (0) {$\\bullet$} node [below=4pt] at (0,0) {$x_0$};\n \\node (2) at (1.5,0) {$\\bullet$} node [below=4pt] at (1.5,0) {$x_1$}; \n \\node (4) at (3,0) {$\\bullet$}node [above=4pt] at (3,0) {$x_2$}; \n \\node (6) at (4.5,0) {$\\bullet$}node [above=4pt] at (4.5,0) {$x_3$}; \n \\node (8) at (6,0) {$\\bullet$}node [below=4pt] at (6,0) {$x_4$}; \n \\node (10) at (7.5,0) {$\\bullet$}node [below=4pt] at (7.5,0) {$x_5$}; \n \\node (11) at (9,0) {$\\bullet \\cdots$}node [above=4pt] at (9,0) {$x_6$}; \n \\node (-1) at (-1,0) {$\\mathcal{T}_{q+1}'$};\n \\node (12) at (3,-0.5) {$\\mathcal{T}_{q+1}''$};\n \\node (13) at (4.5,-0.5) {$\\mathcal{T}_{q+1}''$};\n \\node (14) at (8.7, -0.5) {$\\mathcal{T}_{q+1}''$};\n \\draw[dashed] (-1cm,0cm) circle (1cm);\n \\draw[dashed] (3cm,-0.5cm) circle (0.5cm);\n \\draw[dashed] (4.5cm,-0.5cm) circle (0.5cm);\n \\draw[dashed] (8.7cm,-0.5cm) circle (0.5cm);\n\n \\path[-] (0) edge node [above=4pt] {\\quad \\quad $q$} (2)\n (2) edge node [above=4pt] {$1$ \\quad \\quad } (4)\n (4) edge node [above=4pt] {} (6)\n (6) edge node [above=4pt] {\\quad\\quad $q$} (8)\n (8) edge node [above=3.2pt] {$1$ \\quad \\,\\,$q$} (10)\n (10) edge node [above=4pt] {$1$ \\quad \\quad} (11);\n\\end{tikzpicture}\n\\vspace{0.5em}\n\\caption{$(A,i)$, Example: $1,4,5\\in I$ and $2,3,6\\in J$}\n\\end{center}\n\\end{figure}\n\nFor each $n\\in\\mathbb{Z}_{>0}$, if we denote by $\\Gamma_{x}$ the stabilizer of $x$ in $\\Gamma$ and by $\\widetilde{x_0}$ a lift of $x_0$ in the universal covering tree $\\mathcal{T}_{q+1}$, then \n \\[\n \\#\\{\\gamma\\in\\Gamma\\colon d(\\widetilde{x_0},\\gamma \\widetilde{x_0}) =2n\\}= \\left\\{\\begin{array}{cl}\n \\displaystyle 0& \\textrm{if }N(x_n)=N(x_{n-1})\\\\\n \\displaystyle \\frac{q-1}{q}N(x_n)|\\Gamma_{x_0}|& \\textrm{if }N(x_n)=qN(x_{n-1})\n \\end{array}\\right.\n .\\]\n\nLet $s_n=\\#\\{1\\le k\\le n\\colon i(\\overrightarrow{x_kx_{k-1}})=q\\}$.\nThen, \\begin{align*}\\displaystyle\\delta_\\Gamma=\\underset{n\\to\\infty}{\\overline{\\lim}}\\frac{1}{2n}\\log\\frac{(q-1)q^{s_n}}{q}=\\underset{n\\to\\infty}{\\overline{\\lim}}\\frac{s_n}{2n}\\log q.\n\\end{align*}\nBy taking a suitable partition $I$ and $J$, we can make $\\displaystyle\\underset{n\\to\\infty}{\\overline{\\lim}}\\frac{s_n}{n}$ be any real number in $[0,1]$. This completes the proof of Theorem~\\ref{main1}.\n\n\n\\section{Further discussion: Ihara zeta function and generating functions}\n\nWe give some remarks towards two problems in this section. First question is to characterize explicitly the set $$E=\\{\\delta\\in[0,\\log q]\\colon \\delta=\\delta_\\Gamma\\textrm{ for some discrete }\\Gamma<\\textrm{Aut}(\\mathcal{T}_{q+1})\\}.$$\n\nWe are interested in constructing a discrete group for a real number $\\delta\\in\\left(\\frac{1}{2}\\log q,\\log q\\right)$ whose critical exponent is eqaul to $\\delta$. We consider an edge-indexed graph obtained by the graph merging of those appeared in Section 3. The critical exponent of the discrete group corresponding to the one is related to those of smaller edge-indexed graphs, being the radius convergence of certain Laurent series.\n\nGiven an edge-indexed graph $X=(A,i)$, let $N_X(m)$ be the number of closed paths of length $m$ without backtracking or tails in the graph $A$ with weights $i$. More precisely, given a closed cycle $C=e_1\\cdots e_m$ of length $m$, let $w(C)=i(e_1)+\\cdots i(e_m)$ and\n$$N_X(m)=\\sum_{C=e_1\\cdots c_m}w(C)$$\nwhere the summation runs over the closed cycle of length $m$ without backtracking or tails.\n\nLet us denote by $F_X(u)$ the generating function given by $$\\displaystyle \\sum_{m=1}^{\\infty} N_X(m)u^m.$$\nSuppose that every closed cycle of $X$ and $Y$ passes through the vertex $x$ and $y$, respectively.\n\nThen, we have\n\\begin{align*}N_{X\\underset{x,y}{\\star}Y}(m)=&\\left(N_X(m)+\\sum N_{X}(k_1)N_{Y}(k_2)+\\sum N_{X}(k_1)N_{Y}(k_2)N_{X}(k_3)+\\cdots\\right)\\\\+&\\left(N_Y(m)+\\sum N_{Y}(k_1)N_{X}(k_2)+\\sum N_{Y}(k_1)N_{X}(k_2)N_{Y}(k_3)+\\cdots\\right) \n\\end{align*}\nwhere all the summations run over $k_1+k_2+\\cdots k_n=m$. Hence, it follows that\n\\begin{align*}&\\sum_{m=1}^{\\infty}N_{X\\underset{x,y}{\\star}Y}(m)u^m\\\\=&\\left(\\sum_{m=1}^{\\infty}N_X(m)u^m+\\sum_{m=1}^{\\infty}N_X(m)u^m\\sum_{m=1}^{\\infty}N_Y(m)u^m+\\cdots\\right)\\\\\n+&\\left(\\sum_{m=1}^{\\infty}N_Y(m)u^m+\\sum_{m=1}^{\\infty}N_Y(m)u^m\\sum_{m=1}^{\\infty}N_X(m)u^m+\\cdots\\right)\n\\end{align*}\nwhich yields the following proposition. \n\n\\begin{prop} Let $X$ and $Y$ be an edge-indexed graph such that every closed cycle of $X$ and $Y$ passes through the vertex $x$ and $y$, respectively. Then we have\n$$F_{X\\star Y}(u)=\\sum_{m=1}^{\\infty}N_{X\\underset{x,y}{\\star}Y}(m)u^m=\\frac{F_X(u)+F_Y(u)+2F_X(u)F_Y(u)}{1-F_X(u)F_Y(u)}.$$\n\\end{prop}\nFor instance, if both $X$ and $Y$ are edge-indexed graphs given in Section 3, then $N_X(m)$ is either $q^k$ for some $k<m$ or is equal to $0$. Thus, given any $0$-$1$ sequences $a_n$ and $b_n$, if we denote by $s_n$ and $t_n$ the partial sum of each, then any real numbers realized as a solution of the equation $u\\in\\mathbb{R}$ such that $$\\sum_{m=1}^{\\infty} q^{s_m}u^m\\sum_{m=1}^{\\infty} q^{t_m}u^m=1$$ are contained in the set $E$.\n\n\\vspace{.1 in}\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tikzpicture}[every loop\/.style={}]\n \\tikzstyle{every node}=[inner sep=0pt]\n \\node (0) {$\\bullet$} node [above=4pt] at (0,0) {$x_3$};\n \\node (2) at (1.5,0) {$\\bullet$} node [below=4pt] at (1.5,0) {$x_2$}; \n \\node (4) at (3,0) {$\\bullet$}node [above=4pt] at (3,0) {$x_1$}; \n \\node (6) at (4.5,0) {$\\bullet$}node [above=4pt] at (4.5,0) {$x_0\\sim y_0$}; \n \\node (8) at (6,0) {$\\bullet$}node [below=4pt] at (6,0) {$y_1$}; \n \\node (10) at (7.5,0) {$\\bullet$}node [below=4pt] at (7.5,0) {$y_2$}; \n \\node (11) at (9,0) {$\\bullet \\cdots$}node [above=4pt] at (9,0) {$y_3$}; \n \\node (-1) at (0,-0.5) {$\\mathcal{T}_{q+1}''$};\n \\node (12) at (3,-0.5) {$\\mathcal{T}_{q+1}''$};\n \\node (13) at (4.5,-0.5) {$\\mathcal{T}_{q+1}''$};\n \\node (14) at (8.7, -0.5) {$\\mathcal{T}_{q+1}''$};\n \\draw[dashed] (0cm,-0.5cm) circle (0.5cm);\n \\draw[dashed] (3cm,-0.5cm) circle (0.5cm);\n \\draw[dashed] (4.5cm,-0.5cm) circle (0.5cm);\n \\draw[dashed] (8.7cm,-0.5cm) circle (0.5cm);\n\n \\path[-] (0) edge node [above=4pt] {\\quad \\quad $1$} (2)\n (2) edge node [above=4pt] {$q$ \\quad \\quad \\, } (4)\n (4) edge node [above=4pt] {} (6)\n (6) edge node [above=4pt] {\\quad\\quad $q$} (8)\n (8) edge node [above=4pt] {$1$ \\quad\\quad $q$} (10)\n (10) edge node [above=4pt] {$1$ \\quad \\quad} (11);\n\\end{tikzpicture}\n\\vspace{0.5em}\n\\caption{$X\\underset{x_0,y_0}{\\star} Y\\underset{x_0}{\\star}\\mathcal{T}_{q+1}''$}\n\\end{center}\n\\end{figure}\n\nThe other question concerns the set of critical exponents of \\emph{free groups}. Using Ihara zeta function (the definition will be given below), we see that if $\\Gamma$ is a free subgroup of $\\textrm{Aut}(\\mathcal{T}_{q+1})$, then the critical exponent $\\delta_\\Gamma$ is equal to $\\log \\alpha$ for some algebraic integer $\\alpha$. We are interested in characterizing the irreducible polynomials of such $\\alpha$. For example, in \\cite{Kw} the critical exponents of certain dumbbell graphs, consisting of two vertex-disjoint cycles joining them having only its end-vertices in common with the two cycles (see Figure~\\ref{dumbbell}), are described.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tikzpicture}[every loop\/.style={}]\n \\tikzstyle{every node}=[inner sep=0pt]\n \\draw[thick] (0cm,0cm) circle (1cm);\n \\draw (-1,0) node[fill=white]{$\\bullet$};\n \\draw (0,1) node[fill=white]{$\\bullet$};\n \\draw (0,-1) node[fill=white]{$\\bullet$};\n \\draw (0,0) node{\\Large{$\\underset{a}{\\circlearrowleft}$}};\n \\draw (1,0) node[fill=white]{$\\bullet$}; \n \\draw (1cm,0cm) --(2cm,0cm);\n \\draw (2,0) node[fill=white]{$\\bullet$};\n \\draw (2,0)--(2.7,0);\n \\draw (3.5,0) node{$\\cdots\\cdots$};\n \\draw (4.3,0) --(5cm,0cm); \n \\draw (1.4cm,0cm) node[above=3pt]{$x_1$};\n \\draw (2cm,0cm) node[above=3pt]{$x_2$};\n \\draw (5cm,0cm) node[above=3pt]{$x_{n-1}$};\n \\draw (5.7cm,0cm) node[above=3pt]{$x_{n}$};\n \\draw (5,0) node[fill=white]{$\\bullet$}; \n \\draw (6,0) node[fill=white]{$\\bullet$};\n \\draw (5,0)--(6,0);\n \\draw[thick] (7cm,0cm) circle (1cm);\n \\draw (7,1) node[fill=white]{$\\bullet$};\n \\draw (8,0) node[fill=white]{$\\bullet$};\n \\draw (7,-1) node[fill=white]{$\\bullet$};\n \\draw (7,0) node{\\Large{$\\underset{b}{\\circlearrowright}$}};\n\\end{tikzpicture}\\label{dumbbell}\n\\caption{Dumbbell graph $D_{a,b,n}$}\n\\end{center}\n\\end{figure}\n\n\nWe briefly explain how to calculate the irreducible polynomial of the critical exponent of the free group $\\Gamma$. This is related to the radius of convergence of the zeta function of the graph $\\Gamma\\backslash\\mathcal{T}_{q+1}^\\textrm{min}$. Here, $\\mathcal{T}_{q+1}^\\textrm{min}$ is the minimal $\\Gamma$-invariant subtree of $\\mathcal{T}_{q+1}$.\n\nSuppose that $A$ is a finite connected undirected graph with no degree 1 vertices. A closed cycle is called \\emph{primitive} if it is not a finite multiple of a smaller closed cycle. Let $P=(e_1,e_2,\\ldots,e_{l(P)-1},e_{l(P)})$ be a primitive closed cycle without backtracking. That is, $i(e_1)=t(e_{l(P)})$, $e_{i+1}\\ne e_i^{-1}$ $(\\textrm{mod }l(P))$ for all $i$ and $P\\ne D^m$ for any integer $m\\ge 2$ and a closed cycle $D$ in $A$. If a closed cycle $Q$ is obtained by changing the cyclic order of $P$, then we say $P$ and $Q$ are \\emph{equivalent}. A \\emph{prime} $[P]$ in $A$ are equivalence classes of primitive closed cycle without backtracking. \n\n\\begin{de} The \\emph{Ihara zeta function} of a finite graph $A$ is defined at $u\\in\\mathbb{C}$, for which $|u|$ is sufficiently small, by\n$$Z_A(u)=\\prod_{[P]}(1-u^{l(P)})^{-1}$$ where $[P]$ runs over the primes of $A$.\n\\end{de}\n\nThe Ihara determinant formula (\\cite{Ih} which was established also by \\cite{Ba2} and \\cite{KS}) says that \n$$Z_A(u)=\\frac{1}{(1-u^2)^{\\chi(A)-1}\\det(I-Adj(u)+Qu^2)}$$\nwhere $\\chi=|EA|-|VA|+1$, $Adj$ is the vertex adjacency matrix of $A$ and $Q$ is diagonal matrix whose $j$-th diagonal entry is $\\textrm{deg}(v_j)-1$.\n\nLet $\\Delta_A$ be the g.c.d. of $\\{l(P)\\colon P\\textrm{ prime in }A\\}$ and $\\pi_A(n)$ be $$\\#\\{[P]\\colon P\\textrm{ prime in }A, l(P)=n\\}.$$ The following is the graph version of the prime geodesic theorem.\n\n\\begin{thm}[\\cite{Te}, Theorem 10.1] Let $R_A$ be the radius of convergence of $Z_A(u)$. If $\\Delta_A=1$, then $$\\lim_{n\\to\\infty}n\\pi(n)R_A^{n}=1.$$ If $\\Delta_A>1$, then $\\pi(m)=0$ unless $\\Delta_A|m$ and $$\\lim_{n\\to\\infty}n\\Delta_A\\pi(n\\Delta_A) R_A^{n\\Delta_A}=1.$$ \n\\end{thm}\n\n\nIn the proof of the above theorem (\\cite{Te}, Theorem 10.1), it is shown that if we denote by $W$ the adjacency matrix of the oriented line graph of $A$, then we have \n$$\\pi_A(n)\\sim\\frac{1}{n}\\sum_{\\underset{\\lambda\\in \\textrm{Spec}W}{|\\lambda|\\,\\textrm{max}}}\\lambda^n\\quad\\textrm{and}\\quad N_A(m)=\\sum_{\\lambda\\in\\textrm{Spec}W}\\lambda^m$$ which implies that $$\\underset{n\\to\\infty}{\\overline{\\lim}}\\frac{\\log \\pi_A(n)}{n}=\\underset{n\\to\\infty}{\\overline{\\lim}}\\frac{\\log N_A(n)}{n}=\\log\\lambda_A^{\\textrm{max}},$$\nwhere $\\lambda_A^{\\textrm{max}}\\in \\textrm{Spec}\\,W$ is the eiegenvalue of $W$ with maximum modulus. In \\cite{KS}, we also have $$Z_A(u)=\\frac{1}{\\det(I-uW)}.$$\n\nSuppose that $\\Gamma$ is a free subgroup of $\\textrm{Aut}(\\mathcal{T}_{q+1})$. Then, we have \n\\begin{align*}\\delta_\\Gamma=&\\underset{n\\to\\infty}{\\overline{\\lim}}\\frac{\\log\\#\\{\\gamma\\in\\Gamma\\colon d(v,\\gamma v)\\le n\\}}{n}\\\\=&\\underset{n\\to\\infty}{\\overline{\\lim}}\\frac{\\log|\\textrm{Stab}_\\Gamma(\\widetilde{v})|\\sum_{k=1}^{n}N_A(k)}{n} \\\\=&\\underset{n\\to\\infty}{\\overline{\\lim}}\\frac{\\log N_A(n)}{n}\\\\=&\n \\underset{n\\to\\infty}{\\overline{\\lim}}\\frac{\\log \\pi_A(n)}{n}=\\lambda_A^{\\textrm{max}}=\\log R_A^{-1}\n\\end{align*}\nwhere we take $A$ by $\\Gamma\\backslash\\mathcal{T}_{q+1}^\\textrm{min}$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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