diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmopc" "b/data_all_eng_slimpj/shuffled/split2/finalzzmopc" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmopc" @@ -0,0 +1,5 @@ +{"text":"\\section{\\label{sec:level1}Introduction\\protect\\\\ } \nThe Skyrme-Faddeev-Niemi (SFN) model which is an O(3) $\\sigma$ model \nin three dimensional space upto fourth-order in the first derivative \nhas topological soliton solutions with torus or knot-like structure. \nThe model was initiated in 70's~\\cite{faddeev75} and its interest has \nbeen extensively growing. The numerical simulations were performed in \nRefs.~\\cite{faddeev97,gladikowski97,sutcliffe98,hietarinta99,hietarinta00}, \nthe integrability was shown in Ref.~\\cite{aratyn99}, and the application to \nthe condensed matter physics~\\cite{babaev02} and the Weinberg-Salam \nmodel~\\cite{fayzullaev} were also considered. \nThe recent research especially focuses on the consistency between \nthe SFN and fundamental theories such as QCD~\\cite{faddeev99,langmann99,\nshabanov99,cho02}. \nIn those references, it is claimed that the SFN action should be deduced \nfrom the SU(2) Yang-Mills (YM) action at low energies. \nOne can also show from the Wilsonian renormalization group argument that \nthe effective action of Yang-Mills theory recovers the SFN in the infrared \nregion~\\cite{gies01}. \nHowever, the derivative expansion for slowly varying \nfields $\\bm{n}$ upto quartic order brings an additional fourth-order term \nin the SFN model to destabilize the soliton solution. \n\nSimilar situations can be seen also in various topological soliton \nmodels. In the Skyrme model, the chirally invariant lagrangian \nwith quarks produces fourth order terms after the derivative expansion \nand they destabilize the soliton solution~\\cite{dhar85,aitchison85}.\nTo recover the stability of the skyrmion, the author of Ref.\\cite{marleau01} \nintroduced a large number of higher order terms in the first derivative \nwhose coefficients were determined from the coefficients of the \nSkyrme model by using the recursion relations. \nAlternatively, in Ref.~\\cite{gies01} Gies pointed out the possibility that \nthe second derivative order term can work as a stabilizer for the soliton. \n\nIn this paper, we examine the Gies's supposition by numerical analysis.\nIn section \\ref{sec:level2}, we give an introduction to the Skyrme-Faddeev-Niemi \nmodel with its topological property. \nIn section \\ref{sec:level3}, we show how to derive the SFN model action \nfrom the SU(2) Yang-Mills theory. In section \\ref{sec:level4}, \nsoliton solutions of this truncated YM action are studied. \nIn order to find stable soliton solutions, we introduce \na second derivative term which can be derived in a perturbative \nmanner. \nThe naive extremization scheme, however, produce the fourth\norder differential equation and the model has no stable soliton solution. \nFailure of finding the soliton is caused by the \nbasic feature of the second derivative field theory. \nIn section \\ref{sec:level5}, the higher derivative theory and \nOstrogradski's formulation are reviewed. We show the absence \nof bound state in the second derivative theory using an example in \nquantum mechanics and introduce the perturbative treatment for the \nsecond derivative theory. \nIn section \\ref{sec:level6}, we present our numerical results. \nIn section \\ref{sec:level7} are concluding remarks. \n\n\\section{\\label{sec:level2}Skyrme-Faddeev-Niemi model\\protect\\\\}\nThe Faddeev-Niemi conjecture for the low-energy model of SU(2) \nYang-Mills theory is expressed by following effective action:\n\\begin{eqnarray}\nS_{\\rm SFN}=\\Lambda \\int d^4x\\Bigl[\\frac{1}{2}(\\partial_\\mu \\bm{n})^2\n+\\frac{g_1}{8}(\\bm{n}\\cdot \\partial_\\mu \\bm{n}\\times \\partial_\\nu \\bm{n})^2 \\Bigr] \n\\label{fsn_ac}\n\\end{eqnarray}\nwhere $\\bm{n}(\\bm{x})$ is a three component vector field normalized as \n$\\bm{n}\\cdot\\bm{n}=1$. The mass scale $\\Lambda$ is a free parameter \nand in this paper we set $\\Lambda=1$. \nStable soliton solutions exist when $g_1 > 0$. \n\nThe static field $\\bm{n}(\\bm{x})$ maps $\\bm{n}:R^3\\mapsto S^2$ and \nthe configurations are classified by the topological maps characterized \nby a topological invariant $H$ called Hopf charge\n\\begin{eqnarray}\nH=\\frac{1}{32\\pi^2}\\int A \\wedge F,~~F=dA\n\\label{hopf}\n\\end{eqnarray}\nwhere $F$ is the field strength and can be written as \n$F=(\\bm{n}\\cdot d\\bm{n}\\wedge d\\bm{n})$.\n\nThe static energy $E_{\\rm stt}$ from the action (\\ref{fsn_ac}) has \na topological lower bound~\\cite{ward98}, \n\\begin{eqnarray}\nE_{\\rm stt}\\ge K H^{3\/4}\n\\label{lowerbound}\n\\end{eqnarray}\nwhere $K=16\\pi^2\\sqrt{g_1}$.\n\n\nPerforming numerical simulation, one can find that the static configurations \nfor $H=1,2$ have axial symmetry~\\cite{sutcliffe98}. \nThus ``the toroidal ansatz'' which was studied in Ref.\\cite{gladikowski97} \nis suitable to be imposed on these configurations. The ansatz is given by \n\\begin{eqnarray}\n&&n_1=\\sqrt{1-w^2(\\eta,\\beta)}\\cos(N\\alpha+v(\\eta,\\beta))\\,, \\nonumber \\\\\n&&n_2=\\sqrt{1-w^2(\\eta,\\beta)}\\sin(N\\alpha+v(\\eta,\\beta)\\,, \n\\label{toroidal} \\\\\n&&n_3=w(\\eta,\\beta)\\,, \\nonumber \n\\end{eqnarray}\nwhere $(\\eta,\\beta,\\alpha)$ is toroidal coordinates which are related to \nthe $R^3$ as follows:\n\\begin{eqnarray}\nx=\\frac{a\\sinh\\eta\\cos\\alpha}{\\tau},y=\\frac{a\\sinh\\eta\\sin\\alpha}{\\tau},\nz=\\frac{a\\sin\\beta}{\\tau}\n\\end{eqnarray}\nwith $\\tau=\\cosh\\eta-\\cos\\beta$.\n\nThe function $w(\\eta,\\beta)$ is subject to the boundary conditions $w(0,\\beta)=1,w(\\infty,\\beta)=-1$\nand is periodic in $\\beta$. $v(\\eta,\\beta)$ is set to be $v(\\eta,\\beta)=M\\beta+v_0(\\eta,\\beta)$ and \n$v_0(,\\beta)$ is considered as a constant map. \nEquation (\\ref{hopf}) then gives $H=NM$.\n\nIn this paper we adopt a simpler ansatz than (\\ref{toroidal}), which \nis defined by\n\\begin{eqnarray}\n&&n_1=\\sqrt{1-w^2(\\eta)}\\cos(N\\alpha+M\\beta)\\,,\\nonumber \\\\\n&&n_2=\\sqrt{1-w^2(\\eta)}\\sin(N\\alpha+M\\beta)\\,, \n\\label{afz} \\\\\n&&n_3=w(\\eta)\\,, \\nonumber \n\\end{eqnarray}\nwhere $w(\\eta)$ satisfies the boundary conditions $w(0)=1,w(\\infty)=-1$.\nWe numerically study soliton solutions for both ansatz (\\ref{toroidal}) \nand (\\ref{afz}). By comparing those results, we found that this simple \nansatz produces at most 10\\% errors and it does not affect to the \nproperty of the soliton solution. \n\nBy using (\\ref{afz}), the static energy is written in terms of the function \n$w(\\eta)$ as\n\\begin{eqnarray}\n&&E_{\\rm stt}=2\\pi^2a \\int d\\eta\n\\Biggl[\n\\frac{(w')^2}{1-w^2}+(1-w^2) U_{M,N}(\\eta) \\nonumber \\\\\n&&\\hspace{2cm}+\\frac{g_1}{4a^2}\\sinh\\eta\\cosh\\eta (w')^2\nU_{M,N}(\\eta)\\Biggr]\\,,\\nonumber \\\\\n&&\\hspace{1cm}w'\\equiv \\frac{dw}{d\\eta},~~\nU_{M,N}(\\eta)\\equiv \\Bigl(M^2+\\frac{N^2}{\\sinh^2\\eta}\\Bigr)\\,. \\nonumber\n\\end{eqnarray}\nThe Euler-lagrange equation of motion is then derived as \n\\begin{eqnarray}\n&&\\frac{w''}{1-w^2}+\\frac{ww'^2}{(1-w^2)^2}+U_{M,N}(\\eta)w \\nonumber \\\\\n&&+\\frac{g_1}{2a^2}\\Bigl(-2N^2\\coth^2\\eta w'+(\\cosh^2\\eta+\\sinh^2\\eta)\nU_{M,N}(\\eta)w' \\nonumber \\\\ \n&&+\\sinh\\eta\\cosh\\eta U_{M,N}(\\eta)w''\\Bigr)=0\\,.\n\\label{fsn_eq}\n\\end{eqnarray}\nThe variation with respect to $a$ produces the equation for variable $a$.\nSoliton solutions are obtained by solving the equations for $a$ as \nwell as for $w$.\n\n\\section{\\label{sec:level3}Effective action in the Yang-Mills theory\nwith CFNS decomposition\\protect\\\\}\nIn this section, we briefly review how to derive the\nSFN effective action from the action of SU(2) Yang-Mills theory in the \ninfrared limit~\\cite{shabanov99,gies01}. \nFor the gauge fields $\\bm{A}_\\mu$, the Cho-Faddeev-Niemi-Shabanov \ndecomposition is applied~\\cite{faddeev99,langmann99,shabanov99,cho02}\n\\begin{eqnarray}\n\\bm{A}_\\mu=\\bm{n}C_\\mu+(\\partial_\\mu\\bm{n})\\times\\bm{n}+\\bm{W}_\\mu\\,.\n\\label{cfns}\n\\end{eqnarray}\nThe first two terms are the ``electric'' and ``magnetic'' Abelian connection, \nand $\\bm{W}_\\mu$ are chosen so as to orthogonal to $\\bm{n}$, \n$\\bm{W}_\\mu\\cdot\\bm{n}=0$.\nObviously, the degrees of freedom on the left- and right-hand side \nof Eq.(\\ref{cfns}) do not match. While the LHS describes \n$3_{\\rm color}\\times 4_{\\rm Lorentz}=12$, the RHS is comprised of \n$(C_\\mu:)4_{\\rm Lorentz}+(\\bm{n}:)2_{\\rm color}+(\\bm{W}_\\mu:)3_{\\rm color}\n\\times4_{\\rm Lorentz}-4_{\\bm{n}\\cdot\\bm{W}_\\mu=0}=14$ degrees freedom. \nShabanov introduced in his paper \\cite{shabanov99} the following constraint \n\\begin{eqnarray}\n\\bm{\\chi}(\\bm{n},C_\\mu,\\bm{W}_\\mu)=0,~{\\rm with}~~\\bm{\\chi}\\cdot\\bm{n}=0\\,.\n\\end{eqnarray}\nThe generating functional of YM theory can be written by using \nEq.(\\ref{cfns}) as \n\\begin{eqnarray}\n{\\cal Z}=\\int {\\cal D}\\bm{n}{\\cal D}C{\\cal D}\\bm{W}\\delta(\\bm{\\chi})\n\\Delta_{\\rm FP}\\Delta_{\\rm S}e^{-S_{\\rm YM}-S_{\\rm gf}}\\,.\n\\label{vf0}\n\\end{eqnarray}\n$\\Delta_{\\rm FP}$ and $S_{\\rm gf}$ are the Faddeev-Popov determinant and \nthe gauge fixing action respectively, and Shabanov introduced another \ndeterminant $\\Delta_{\\rm S}$ corresponding to the condition $\\bm{\\chi}=0$.\nYM and the gauge fixing action is given by \n\\begin{eqnarray}\n&&S_{\\rm YM}+S_{\\rm gf}=\\int d^4x\\Bigl[\\frac{1}{4g^2}\\bm{F}_{\\mu\\nu}\\cdot\\bm{F}_{\\mu\\nu}\n+\\frac{1}{2\\alpha_{\\rm g} g^2}(\\partial_\\mu \\bm{A}_\\mu)^2\\Bigr] \\,. \\nonumber \n\\end{eqnarray}\nInserting Eq.(\\ref{cfns}) into the action, one obtains the vacuum functional\n\\begin{eqnarray}\n&&{\\cal Z}=\\int {\\cal D}\\bm{n}e^{-{\\cal S}_{\\rm eff}(\\bm{n})} \\nonumber \\\\\n&&~~~=\\int {\\cal D}\\bm{n} e^{-{\\cal S}_{\\rm cl}(\\bm{n})}\\int {\\cal D}\\tilde{C}{\\cal D}\\bm{W}_\\mu\n\\Delta_{\\rm FP}\\Delta_{\\rm S}\\delta({\\bm \\chi}) \\nonumber \\\\\n&&~~~\\times e^{-(1\/2g^2)\\int(\\tilde{C}_\\mu M^C_{\\mu\\nu}\\tilde{C}_\\nu+\\bm{W}_\\mu \\bar{M}^{\\bm{W}}_{\\mu\\nu}\\bm{W}_\\nu\n+2C_\\nu K^C_\\nu+2\\bm{W}_\\mu\\cdot\\bm{K}^{\\bm{W}}_\\mu)}\n\\nonumber \\\\\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n&&M^C_{\\mu\\nu}=-\\partial^2\\delta_{\\mu\\nu}+\\partial_\\mu\\bm{n}\\cdot\\partial_\\nu\\bm{n}\\,, \\nonumber \\\\\n&&M^{\\bm{W}}_{\\mu\\nu}=-\\partial^2\\delta_{\\mu\\nu}-\\partial_\\mu\\bm{n}\\otimes\\partial_\\nu\\bm{n}\n+\\partial_\\nu\\bm{n}\\otimes\\partial_\\mu\\bm{n}\\,, \\nonumber \\\\\n&&\\bm{Q}^C_{\\mu\\nu}=\\partial_\\mu\\bm{n}\\partial_\\nu+\\partial_\\nu\\bm{n}\\partial_\\mu+\\partial_\\mu\\partial_\\nu\\bm{n}\\,,\n\\\\\n&&K^C_{\\mu\\nu}=\\partial_\\nu(\\bm{n}\\cdot\\partial_\\nu\\bm{n}\\times\\partial_\\mu\\bm{n})\n+\\partial_\\mu\\bm{n}\\cdot\\partial^2\\bm{n}\\times\\bm{n}\\,, \\nonumber \\\\\n&&\\bm{K}^{\\bm W}_{\\mu\\nu}=\\partial_\\mu(\\bm{n}\\times\\partial^2\\bm{n})\\,, \n\\hspace{1cm}{\\rm (in~gauge~\\alpha_g=1}) \\nonumber \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n&&\\bar{M}^{\\bm{W}}_{\\mu\\nu}:=M^{\\bm{W}}_{\\mu\\nu}-\\bm{Q}_{\\mu s}{M^C}^{-1}_{s\\lambda}\\bm{Q}_{\\lambda \\nu}\\,, \\nonumber \\\\\n&&\\tilde{C}_\\mu=C_\\mu+\\bm{W}_s\\cdot \\bm{Q}_{s\\lambda}{M^C}^{-1}_{\\lambda\\mu}\\,.\n\\end{eqnarray}\nThe classical action of $\\bm{n}$ including the gauge fixing term is given by\n\\begin{eqnarray}\n{\\cal S}_{\\rm cl}=\\int d^4x\\Bigl[\\frac{1}{4g^2}(\\partial_\\mu\\bm{n}\\times \\partial_\\nu\\bm{n})^2\n+\\frac{1}{2 \\alpha_{\\rm g} g^2}(\\partial^2\\bm{n}\\times\\bm{n})^2\\Bigr]\\,.\\nonumber \\\\\n\\end{eqnarray}\nThe $\\delta$ functional is expressed by its Fourier transform\n\\begin{eqnarray}\n\\delta(\\bm{\\chi})=\\int {\\cal D}\\bm{\\phi}e^{-i\\int (\\bm{\\phi}\\cdot \\partial\\bm{W}_\\mu\n+\\bm{\\phi}\\cdot C_\\mu\\bm{n}\\times \\bm{W}_\\mu+(\\bm{\\phi}\\cdot\\bm{n})(\\partial_\\mu\\bm{n}\\cdot\\bm{W}_\\mu))}\\,.\n\\nonumber \\\\\n\\end{eqnarray} \nIntegrating over $C,\\bm{W},\\bm{\\phi}$, we finally obtain \n\\begin{eqnarray}\n&&e^{-S_{\\rm eff}}=e^{-S_{\\rm cl}}\\Delta_{\\rm FP}\\Delta_{\\rm S}\n(\\det M^C)^{-1\/2} (\\det \\bar{M}^{\\bm{W}})^{-1\/2}\n\\nonumber \\\\\n&&\\hspace{1cm}\\times (\\det -Q^{\\bm{\\phi}}_\\mu (\\bar{M}^{\\bm{W}})^{-1}_{\\mu\\nu}\nQ^{\\bm{\\phi}}_\\nu)^{-1\/2} \\label{determ} \n\\end{eqnarray}\nwhere several nonlocal terms and the higher derivative components have been \nneglected. \n\nWe perform the derivative expansion for the four determinants in \nEq.~(\\ref{determ}) under the following assumptions \n\\begin{itemize}\n\\item [(i)]the theory is valid for the momenta $p$ with $k0$ and $g_2=0$, the action is identical to the FSN effective \naction~(\\ref{fsn_ac}). \n\nIn order to get the stable soliton solutions, $g_2$ must be \npositive~\\cite{gladikowski97}. However, $g_2$ is found to be \nnegative according to the above analysis. \nTherefore we consider higher-derivative terms and investigate \nif the model with the higher-derivatives possess soliton solutions. \n\n\\section{\\label{sec:level4}Search for the stable soliton solutions (1)\\protect\\\\}\nThe static energy is derived from Eq.(\\ref{fsn2}) as \n\\begin{eqnarray}\nE_{\\rm stt}&=& \\int d^3x\\Bigl[\\frac{1}{2}(\\partial_i \\bm{n})^2\n+\\frac{g_1}{8}(\\partial_i \\bm{n}\\times \\partial_j \\bm{n})^2\n+\\frac{g_2}{8}(\\partial_i \\bm{n})^4 \\Bigr] \\nonumber \\\\\n&:=&E_2(\\bm{n})+E_4^{(1)}(\\bm{n})+E_4^{(2)}(\\bm{n})\\,.\n\\end{eqnarray}\nA spatial scaling behaviour of the static energy, so called Derrick's scaling \nargument, can be applied to examine the stability of the soliton~\\cite{sutcliffe05}. \nConsidering the map \n$\\bm{x}\\mapsto \\bm{x}'=\\mu\\bm{x}~(\\mu>0)$, with \n$\\bm{n}^{(\\mu)}\\equiv \\bm{n}(\\mu\\bm{x})$, the static energy scales as \n\\begin{eqnarray}\ne(\\mu)&=&E_{\\rm stt}(\\bm{n}^{(\\mu)}) \\nonumber \\\\\n&=&E_2(\\bm{n}^{(\\mu)})+E_4^{(1)}(\\bm{n}^{(\\mu)})+E_4^{(2)}(\\bm{n}^{(\\mu)}) \\nonumber \\\\\n&=&\\frac{1}{\\mu}E_2(\\bm{n})+\\mu(E_4^{(1)}(\\bm{n})+E_4^{(2)}(\\bm{n}))\\,.\n\\label{derrick}\n\\end{eqnarray}\nDerrick's theorem states that if the function $e(\\mu)$ has no stationary point, \nthe theory has no static solutions of the field equation with finite density, \nother than the vacuum. \nConversely, if $e(\\mu)$ has stationary point, the possibility of \nhaving finite energy soliton solutions is not excluded. \nEq.(\\ref{derrick}) is stationary at $\\mu=\\sqrt{E_2\/(E_4^{(1)}+E_4^{(2)})}$. \nThen, the following inequality\n\\begin{eqnarray}\n&&g_1(\\partial_i\\bm{n}\\times\\partial_j\\bm{n})^2+g_2(\\partial_i\\bm{n})^2(\\partial_j\\bm{n})^2 \\nonumber \\\\\n&&=g_1(\\partial_i\\bm{n})^2(\\partial_j\\bm{n})^2-g_1(\\partial_i\\bm{n}\\cdot\\partial_j\\bm{n})^2\n+g_2(\\partial_i\\bm{n})^2(\\partial_j\\bm{n})^2 \\nonumber \\\\\n&&\\geqq g_2(\\partial_i\\bm{n}\\cdot\\partial_j\\bm{n})^2\n~~~~(\\because (\\partial_i\\bm{n})^2(\\partial_j\\bm{n})^2\\geqq (\\partial_i\\bm{n}\\cdot\\partial_j\\bm{n})^2) \\nonumber \\\\\n\\end{eqnarray}\nensures the possibility of existence of the stable soliton solutions for $g_2\\geqq 0$. \nAs mentioned in the section \\ref{sec:level3}, $g_2$ should be negative at least within \nour derivative expansion analysis of YM theory. \n\nA promising idea to tackle the problem was suggested by Gies~\\cite{gies01}.\nHe considered the following type of effective action, accompanying second \nderivative term \n\\begin{eqnarray}\n&&S_{\\rm eff}=\\int d^4x\\Bigl[\\frac{1}{2}(\\partial_\\mu \\bm{n})^2\n+\\frac{g_1}{8}(\\partial_\\mu \\bm{n}\\times \\partial_\\nu \\bm{n})^2 \n\\nonumber \\\\\n&&\\hspace{2cm}-\\frac{g_2}{8}(\\partial_\\mu \\bm{n})^4\n+\\frac{g_2}{8}(\\partial^2 \\bm{n}\\cdot\\partial^2 \\bm{n})\n \\Bigr]\\,.\n\\label{fsn_ac2}\n\\end{eqnarray}\nHere we choose positive value of $g_2$ and assign the explicit negative sign \nto the third term. In principle, it is possible to estimate the second derivative term \nby the derivative expansion without neglecting \nthroughout the calculation.\nThe calculation is, however, very laborious and hence we show only \none simple example of the $C$ determinant.\nThe determinant is real and thus it is expanded as follows\n\\begin{eqnarray}\n&&\\log(\\det M^C)^{-1\/2}\n=-\\frac{1}{2}{\\rm Tr}\\log(-\\partial^2+\\partial_\\mu\\bm{n}\\cdot\\partial_\\mu\\bm{n}) \\nonumber \\\\ \n&&~~\\to-\\frac{1}{4}{\\rm Tr}\\log[\\partial^4\n-2(\\partial\\bm{n})^2\\partial^2+(\\partial\\bm{n})^4-\\partial^2(\\partial\\bm{n})^2]\n\\nonumber \\\\\n&&~~=-\\frac{1}{4}{\\rm Tr}\\log(\\partial^4) \\nonumber \\\\\n&&~~~~~-\\frac{1}{4}{\\rm Tr}\\log\\Bigl[1-2\\frac{(\\partial\\bm{n}^2)}{\\partial^2}\n+\\frac{(\\partial\\bm{n})^4}{\\partial^4}\n-\\frac{\\partial^2(\\partial\\bm{n})^2}{\\partial^4}\\Bigr] \\nonumber \\\\\n&&~~=-\\frac{1}{4}{\\rm Tr}\\log(\\partial^4) \\nonumber \\\\\n&&~~~~~-\\frac{1}{4}{\\rm Tr}\\Bigl[-2\\frac{(\\partial\\bm{n}^2)}{\\partial^2}\n+\\frac{(\\partial\\bm{n})^4}{\\partial^4}\n-\\frac{\\partial^2(\\partial\\bm{n})^2}{\\partial^4}\\Bigr] \\nonumber \\\\\n&&~~~~~+\\frac{1}{8}{\\rm Tr}\\Bigl[-2\\frac{(\\partial\\bm{n})^2}{\\partial^2}\\Bigr]^2\n+O((\\partial\\bm{n})^6)\n\\end{eqnarray}\nwhere we have defined $\\partial_\\mu\\bm{n}\\cdot\\partial_\\mu\\bm{n}\\to(\\partial\\bm{n})^2,\n(\\partial_\\mu\\bm{n}\\cdot\\partial_\\nu\\bm{n})^2\\to(\\partial\\bm{n})^4$.\nEmploying the integral formulas~\\cite{gies01}\n\\begin{eqnarray}\n&&\\int_{[k,\\Lambda]} \\frac{d^4p}{(2\\pi)^4}\\frac{1}{p^2}=\\frac{1}{16\\pi^2}(\\Lambda^2-k^2)\\,, \\nonumber \\\\\n&&\\int_{[k,\\Lambda]} \\frac{d^4p}{(2\\pi)^4}\\frac{1}{p^4}=\\frac{1}{8\\pi^2}\\log\\frac{\\Lambda}{k} \\nonumber \n\\end{eqnarray}\ntogether with the equality (up to second derivative)\n\\begin{eqnarray}\n\\partial^2(\\partial_\\mu\\bm{n}\\cdot\\partial_\\mu\\bm{n})=-\\partial^2(\\bm{n}\\cdot\\partial^2\\bm{n})\n= -\\partial^2\\bm{n}\\cdot\\partial^2\\bm{n}\\,,\n\\end{eqnarray}\nwe obtain the form for the $C$ determinant\n\\begin{eqnarray}\n&&\\log(\\det M^C)^{-1\/2}= \\nonumber \\\\\n&&-\\frac{1}{32\\pi^2}\\int d^4x\\Bigl[(\\Lambda^2-k^2) (\\partial_\\mu\\bm{n})^2\n-\\log\\frac{\\Lambda}{k}(\\partial_\\mu\\bm{n}\\times\\partial_\\nu\\bm{n})^2 \\nonumber \\\\ \n&&\\hspace{1.5cm}+\\log\\frac{\\Lambda}{k}(\\partial_\\mu\\bm{n})^4\n-\\log\\frac{\\Lambda}{k}(\\partial^2\\bm{n}\\cdot\\partial^2\\bm{n})\\Bigr]\\,.\n\\end{eqnarray}\nThe other determinants can be estimated in a similar manner. \n\nThe static energy of Eq.(\\ref{fsn_ac2}) with the ansatz (\\ref{afz}) is written as \n\\begin{eqnarray}\n&&E_{\\rm stt}=2\\pi^2a \\int d\\eta\n\\Biggl[\n\\frac{(w')^2}{1-w^2}+(1-w^2) U_{M,N}(\\eta) \\nonumber \\\\\n&&+\\frac{g_1}{4a^2}\\sinh\\eta\\cosh\\eta (w')^2\nU_{M,N}(\\eta)\\nonumber \\\\\n&&-\\frac{g_2}{4a^2}\\sinh\\eta\\cosh\\eta\\biggl[\\frac{(w')^2}{1-w^2}\n+(1-w^2)U_{M,N}(\\eta)\\biggr]^2 \\nonumber \\\\\n&&+\\frac{g_2}{4a^2}\\biggl[\\Bigl(\\coth\\eta+\\sinh^2\\eta-\\sinh\\eta\\cosh\\eta\\Bigr)\n\\frac{(w')^2}{1-w^2} \\nonumber \\\\\n&&\\hspace{1cm}+(\\sinh\\eta\\cosh\\eta-\\sinh^2\\eta)(1-w^2)M^2 \\nonumber \\\\\n&&\\hspace{1cm}+2\\Bigl\\{ \\frac{w(w')^3}{(1-w^2)^2}\n+\\frac{w' w''}{1-w^2}\n+w w'U_{M,N}(\\eta) \\Bigr\\}\\nonumber \\\\\n&&\\hspace{1cm}+\\sinh\\eta\\cosh\\eta\\Bigl\\{ \\frac{1}{1-w^2}\\Bigl[\\frac{(w')^2}{1-w^2}\n+w w'' \\nonumber \\\\\n&&\\hspace{1.5cm}+(1-w^2)U_{M,N}(\\eta)\\Bigr]^2 \n+(w'')^2\\Bigr\\}\\biggr]\n\\Biggl], \\nonumber \\\\\n&&\\hspace{4cm}w''\\equiv \\frac{d^2w}{d\\eta^2}.\\nonumber \n\\nonumber \n\\end{eqnarray} \nThe Euler-Lagrange equation of motion is derived by \n\\begin{eqnarray}\n-\\frac{d^2}{d\\eta^2}\\Bigl(\\frac{\\partial E_{\\rm stt}}{\\partial w''}\\Bigr)\n+\\frac{d}{d\\eta}\\Bigl(\\frac{\\partial E_{\\rm stt}}{\\partial w'}\\Bigr)\n-\\frac{\\partial E_{\\rm stt}}{\\partial w}=0 \\,,\n\\end{eqnarray}\nwhich is too complicated and thus we adopt the following notation \n\\begin{eqnarray}\n&&f_0(w,w',w'')+g_1f_1(w,w',w'') \\nonumber \\\\\n&&\\hspace{1.5cm}+g_2f_2(w,w',w'',w^{(3)},w^{(4)})=0\\,.\n\\label{fsn_eq2}\n\\end{eqnarray}\nHere $w^{(3)},w^{(4)}$ represent the third and the fourth derivative with \nrespect to $\\eta$. The first two terms of Eq.(\\ref{fsn_eq2}) are \nidentical to those in Eq.(\\ref{fsn_eq}).\n\nUnfortunately, we could not find out stable soliton solutions from \nEq.(\\ref{fsn_eq2}) for any value of $g_2$. \n\nFrom the relation\n\\begin{eqnarray}\n\\int d^4x[(\\partial^2 \\bm{n}\\cdot\\partial^2 \\bm{n})-(\\partial_\\mu \\bm{n})^4]\n=\\int d^4x(\\partial^2 \\bm{n}\\times\\bm{n})^2\\,,\n\\end{eqnarray} \none easily finds that the static energy obtained from the last two terms \nin Eq.(\\ref{fsn_ac2}) \n\\begin{eqnarray}\n\\tilde{E}^{(2)}_4=\\int d^3x(\\partial^2 \\bm{n}\\times\\bm{n})^2\n\\label{energy2}\n\\end{eqnarray}\ngives the positive contribution. The total static energy is stationary \nat $\\mu=\\sqrt{E_2\/(E_4^{(1)}+\\tilde{E}_4^{(2)})}$ and hence the possibility \nof existence of soliton solutions is not excluded. \nAnd also, the positivity of Eq.(\\ref{energy2}) does not spoil the lower \nbound (\\ref{lowerbound}) of original SFN and the possibility is still \nnot excluded, too. \n\nTherefore, we suspect that the absence of the stable soliton is caused \nby the fact that {\\it higher derivative theory has no lower bound state}. \nWe shall investigate the lower bound in the higher derivative theory \nin detail in the next section. \n\n\\section{\\label{sec:level5}Higher derivative theory\\protect\\\\}\nIn this section, we address the basic problems in the higher derivative \ntheory~\\cite{pais,smilga,eliezer89,jaen,simon} which essentially falls \ninto two categories. The first problem concerns the increase in the \nnumber of degrees of freedom. For example, if the theory contains \nsecond derivative terms, the equation of motion becomes up to the order \nin the fourth derivative. Thus, four parameters are required for \nthe initial conditions. If one considers more higher order terms, \nthe situation gets worse. However, this is not serious problem for \nour study because our concern is the existence of static soliton solutions. \nThe second problem is that the actions of the theory are not bounded from \nbelow. This feature makes the higher derivative theories unstable.\n\nThe lagrangian and the hamiltonian formalism with higher derivative was firstly developed\nby Ostrogradski~\\cite{ostrogradski}. We consider the lagrangian containing up to $n$th order derivatives\n\\begin{eqnarray}\nS=\\int dt {\\cal L}(q,\\dot{q},\\cdots,q^{(n)})\\,.\n\\end{eqnarray} \nTaking the variation of the action $\\delta S=0$ leads the Euler-lagrange equation of motion\n\\begin{eqnarray}\n\\sum_{i=0}^n (-1)^i\\frac{d^i}{dt^i}\\Bigl(\\frac{\\partial {\\cal L}}{\\partial q^{(i)}}\\Bigr)=0\\,.\n\\end{eqnarray}\nThe hamiltonian is obtained by introducing $n$ generalized momenta\n\\begin{eqnarray}\np_i=\\sum_{j=i+1}^n (-1)^{j-i-1}\\frac{d^{j-i-1}}{dt^{j-i-1}}\\Bigl(\\frac{\\partial {\\cal L}}{\\partial q^{(j)}}\\Bigr)\\,,\ni=1,\\cdots,n,\n\\end{eqnarray}\nor\n\\begin{eqnarray}\n&&p_n=\\frac{\\partial {\\cal L}}{\\partial q^{(n)}}\\,, \\nonumber \\\\\n&&p_i=\\frac{\\partial {\\cal L}}{\\partial q^{(i)}}-\\frac{d}{dt}p_{i+1}\\,,~~i=1,\\cdots,n-1,\n\\label{canonical momenta}\n\\end{eqnarray}\nand $n$ independent variables\n\\begin{eqnarray}\n&&q_1\\equiv q\\,,\\nonumber \\\\\n&&q_i\\equiv q^{(i-1)}\\,,~~(i=2,\\cdots,n)\\,.\n\\end{eqnarray}\nThe lagrangian now depends on the $n$ coordinates $q_i$ and on the first derivative $\\dot{q}_n=q^{(n)}$.\nThe hamiltonian is defined as\n\\begin{eqnarray}\n{\\cal H}(q_i,p_i)=\\sum^n_{i=1}p_i\\dot{q}_i-{\\cal L}=\\sum^{n-1}_{i=1} p_i q_{i+1}+p_n \\dot{q}_n-{\\cal L}\\,.\n\\end{eqnarray}\nThe canonical equations of motion turn out to be\n\\begin{eqnarray}\n\\dot{q}_i=\\frac{\\partial {\\cal H}}{\\partial p_i}\\,,~~\\dot{p}_i=-\\frac{\\partial {\\cal H}}{\\partial q_i}\\,.\n\\end{eqnarray}\n\nWe consider a simple example including second derivative term \\cite{simon}, defined as \n\\begin{eqnarray}\n{\\cal L}=\\frac{1}{2}(1+\\varepsilon^2\\omega^2)\\dot{q}^2-\\frac{1}{2}\\omega^2q^2-\\frac{1}{2}\\varepsilon^2\\ddot{q}^2\\,,\n\\end{eqnarray}\nwhere constant $\\epsilon$ works as a coupling constant of second derivative term. \nThe equation of motion is\n\\begin{eqnarray}\n(1+\\varepsilon^2 \\omega^2)\\ddot{q}+\\omega^2 q+\\varepsilon^2 q^{(4)}=0\\,.\n\\label{eq_quanta}\n\\end{eqnarray}\nFrom Eq.~(\\ref{canonical momenta}), one gets \n\\begin{eqnarray}\n&&\\pi_{\\dot{q}}=\\frac{\\partial{\\cal L}}{\\partial \\ddot{q}}=-\\varepsilon^2\\ddot{q}\\,, \\nonumber \\\\\n&&\\pi_q=\\frac{\\partial{\\cal L}}{\\partial \\dot{q}}-\\frac{d}{dt}\\Bigl(\\frac{\\partial{\\cal L}}{\\partial \\ddot{q}}\\Bigr)\n=(1+\\varepsilon^2\\omega^2)\\dot{q}+\\varepsilon^2\\dddot{q}\\,.\n\\end{eqnarray}\nThus the hamiltonian becomes \n\\begin{eqnarray}\n{\\cal H}&=&\\dot{x}\\pi_q+\\ddot{q}\\pi_{\\dot{q}}-{\\cal L} \\nonumber \\\\\n&=&\\dot{q}\\pi_q-\\frac{1}{2\\varepsilon^2}\\pi_{\\dot{q}}^2\n-\\frac{1}{2}(1+\\varepsilon^2\\omega^2)\\dot{q}^2+\\frac{1}{2}\\omega^2q^2\\,.\n\\end{eqnarray}\nWe introduce the new canonical variables\n\\begin{eqnarray}\n&&q_+=\\frac{1}{\\omega\\sqrt{1-\\varepsilon^2\\omega^2}}(\\varepsilon^2\\omega^2\\dot{q}-\\pi_q)~,\\nonumber \\\\\n&&p_+=\\frac{w}{\\sqrt{1-\\varepsilon^2\\omega^2}}(q-\\pi_{\\dot{q}})\\,, \\nonumber \\\\\n&&q_-=\\frac{\\varepsilon}{\\sqrt{1-\\varepsilon^2\\omega^2}}(\\dot{q}-\\pi_q)~, \\nonumber \\\\\n&&p_-=\\frac{1}{\\varepsilon\\sqrt{1-\\varepsilon^2\\omega^2}}(\\varepsilon^2\\omega^2 q-\\pi_{\\dot{q}})\\,,\n\\nonumber \n\\end{eqnarray}\nand the hamiltonian has of the form by using these variables\n\\begin{eqnarray}\n{\\cal H}\\to \\frac{1}{2}(p_+^2+\\omega^2 q_+^2)-\\frac{1}{2}(p_-^2+\\frac{1}{\\varepsilon^2} q_-^2)\\,. \\nonumber \n\\end{eqnarray}\nThe corresponding energy spectra is then given by\n\\begin{eqnarray}\nE=(n+\\frac{1}{2})\\omega-(m+\\frac{1}{2})\\frac{1}{\\varepsilon}~,~~n,m=0,1,2,\\cdots\n\\end{eqnarray}\nOne can see that in the limit $\\epsilon\\to 0$ the energy goes to \nnegative infinity rather than approaching to the harmonic oscillator \nenergy eigenstates. \n\nTo obtain physically meaningful solutions, we employ the perturbative \nanalysis where the solution is expanded in terms of the small coupling \nconstant and the Euler-Lagrange equation of motion is replaced with the \ncorresponding perturbative equation. \nThe solutions of the equations of motion that are ill behaved in the limit \n$\\epsilon\\to 0$ are excluded from the very beginning~\\cite{eliezer89,jaen,simon}.\n\nWe assume that the solution of Eq.(\\ref{eq_quanta}) can be written as \n\\begin{eqnarray}\nq_{\\rm pert}(t)=\\sum^{\\infty}_{n=0} \\epsilon^n q(t)\\,. \\label{q}\n\\end{eqnarray}\nSubstituting Eq.(\\ref{q}) into Eq.(\\ref{eq_quanta}) and taking time \nderivatives of these equations, we obtain the constraints for higher \nderivative terms\n\\begin{eqnarray}\n&&O(\\epsilon^0) \\nonumber \\\\\n&&\\hspace{3mm}equation :~~\\ddot{q}_0+\\omega^2q_0=0\\,, \\label{cnst00} \\\\\n&&\\hspace{3mm}constraints :~~\\dddot{q}_0=-\\omega^2\\dot{q}_0, \\ddddot{q}_0=\\omega^4 q_0\\,. \\label{cnst01} \\\\\n&&O(\\epsilon^2) \\nonumber \\\\\n&&\\hspace{3mm}equation :~~\\ddot{q}_2+\\omega^2\\ddot{q}_0+\\omega^2 q_2+\\ddddot{q}_0=0\\,, \\nonumber \\\\\n&&\\hspace{13mm}\\Rightarrow \\ddot{q}_2+\\omega^2q_2=0\\,,~~({\\rm using}~(\\ref{cnst00}),(\\ref{cnst01}))\\,, \\label{cnst20} \\\\\n&&\\hspace{3mm}constraints :~~\\dddot{q}_2=-\\omega^2\\dot{q}_2, \\ddddot{q}_2=\\omega^4 q_2\\,. \\label{cnst21} \\\\\n&&O(\\epsilon^4) \\nonumber \\\\\n&&\\hspace{3mm}equation :~~\\ddot{q}_4+\\omega^2\\ddot{q}_2+\\omega^2 q_4+\\ddddot{q}_2=0\\,, \\nonumber \\\\\n&&\\hspace{13mm}\\Rightarrow \\ddot{q}_4+\\omega^2q_4=0\\,,~~({\\rm using}~(\\ref{cnst20}),(\\ref{cnst21}))\\,, \\\\\n&&\\hspace{3mm}constraints :~~\\dddot{q}_4=-\\omega^2\\dot{q}_4, \\ddddot{q}_4=\\omega^4 q_4\\,. \\label{cnst4}\n\\end{eqnarray}\nCombining these results, we find the perturbative equation of motion up to $O(\\epsilon^4)$\n\\begin{eqnarray}\n\\ddot{q}_{\\rm pert}+\\omega^2q_{\\rm pert}=O(\\epsilon^6)\\,.\n\\end{eqnarray}\nwhich is the equation for harmonic oscillator. \n\n\\begin{figure}\n\\includegraphics[height=7cm, width=9cm]{Fig1}\n\\caption{\\label{fig:Fig1} The function $w(\\eta)$ for $g_1=0.4, g_2=0,0.05,0.1$\n(the rescaling radial coordinate $x=\\eta\/(1-\\eta)$ is used). }\n\\end{figure}\n\\begin{figure}\n\\includegraphics[height=7cm, width=9cm]{Fig2}\n\\caption{\\label{fig:Fig2} The energy density for $g_1=0.4, g_2=0,0.05,0.1$.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[height=7cm, width=9cm]{Fig3}\n\\caption{\\label{fig:Fig3} The energy as a function of $g_2$($g_1=0.4$).}\n\\end{figure}\n\n\\section{\\label{sec:level6}Search for the stable soliton solutions (2) -- perturbative analysis --\\protect\\\\}\nLet us employ the perturbative method introduced in the last section to our \nproblem. We assume that $g_2$ is relatively small and can be considered as \na perturbative coupling constant. Thus, the perturbative solution is written \nby a power series in $g_2$\n\\begin{eqnarray}\nw(\\eta)=\\sum^\\infty_{n=0}g_2^n w_n(\\eta)\\,.\n\\label{sol_expand}\n\\end{eqnarray}\nSubstituting Eq.(\\ref{sol_expand}) into Eq.(\\ref{fsn_eq2}), we obtain \nthe classical field equation in $O(g^0_2)$\n\\begin{eqnarray}\nf_0(w_0,w_0',w_0'')+g_1f_1(w_0,w_0',w_0'')=0\\,. \\label{classical}\n\\end{eqnarray}\nTaking derivatives for both sides in Eq.(\\ref{classical}) and solving \nfor $w_0'',w_0^{(3)},w_0^{(4)}$ read the following form\nof the constraint equations for higher derivatives\n\\begin{eqnarray}\nw_0^{(i)}=F^{(i)}(w_0,w_0')\\,,~~i=2,3,4\\,.\n\\label{constraint}\n\\end{eqnarray}\nThe equation in $O(g^1_2)$ can be written as \n\\begin{eqnarray}\n(f_0+g_2 f_1)_{O(g^1_2)}+f_2(w_0,w_0',w_0'',w_0^{(3)},w_0^{(4)})=0\\,.\n\\label{fsn_eq21}\n\\end{eqnarray}\nSubstituting the constraint equations (\\ref{constraint}) into Eq.(\\ref{fsn_eq21})\nand eliminate the higher derivative terms, one can obtain the perturbative \nequation of motion\n\\begin{eqnarray}\nf_0(w,w',w'')+g_1 f_1(w,w',w'')+g_2 \\tilde{f}_2(w,w')=O(g^2_2)\\,.\\nonumber \\\\\n\\label{fsn_eq2_p}\n\\end{eqnarray}\n\nNow Eq.(\\ref{fsn_eq2_p}) has topological soliton solutions. \nOur results of the estimated function $w(\\eta)$ and the energy density are displayed \nin Figs.\\ref{fig:Fig1},\\ref{fig:Fig2}.(In all figures, we show the results for the \ncase of Hopf charge $H=2;N=2,M=1$). We have small changes for varying the \ncoupling constant $g_2$. The dependence of the $g_2$ for the total energy \nis shown in Fig.\\ref{fig:Fig3}. \nIt can be seen that the change is moderate with respect to $g_2$.\n\n\\section{\\label{sec:level7}Summary\\protect\\\\}\nIn this paper we have studied the Skyrme-Faddeev-Niemi\nmodel and its extensions by introducing the reduction scheme of the SU(2) \nYang-Mills theory to the corresponding low-energy effective model. \nThe requirement of consistency between the low-energy effecive action \nof the YM and the SFN type model lead us to take into account second \nderivative terms in the action. \nHowever, we found that such an action including the second derivative terms \ndoes not have stable soliton solutions.\nThis is due to the absence of the energy bound in higher \nderivative theory. \nThis fact inspired us to employ the perturbative analysis to our effective \naction. Within the perturbative analysis, we were able to obtain the topological \nsoliton solutions. \n \nOur analysis is based on perturbation and the coupling constant $g_2$ is \nassumed to be small. \nHowever, Wilsonian renormalization analysis of YM theory~\\cite{gies01} \nsuggest that the coupling constants $g_1,g_2$ (and the mass scale parameter \n$\\Lambda$) depend on the renormalization group time $t=\\log k\/\\Lambda$ \n($k,\\Lambda$ are infrared, ultraviolet cutoff parameter) and those are \nalmost comparable. To improve the analysis, we could perform the \nnext order of perturbation, but it is tedius and spoils the simplicity \nof the FSN model unfortunately. \n\nIt should be noted that our solutions do not much differ from \nthe solution of original SFN model, at least in the perturbative regime. \nWe suspect that some appropriate truncation ({\\it like} ``extra fourth order term \n+ second derivative term'') always supply the stable solutions that are \nclose to the original SFN model. Thus we conclude that the topological \nsoliton model comprised of the ``kinetic term + a special fourth order \nterm'' like SFN model is a good approximation. \n\n\\begin{acknowledgments}\nThe authors thank to Kei-Ichi Kondo for drawing our attention to the \ncoefficient problem of this model.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{#1}\\setcounter{lemma}{0}}\n\n\n\n\n\n\n\\title{\\textbf{Quantum Edge Correspondences and Quantum Cuntz--Krieger Algebras}}\n\\author{ Michael Brannan \n \\footnote{Department of Pure Mathematics and the Institute for Quantum Computing, University of Waterloo \\hfill \\url{michael.brannan@uwaterloo.ca}}\n \n \\and\n \n Mitch Hamidi\n \\footnote{Department of Mathematics, Embry-Riddle Aeronautical University \\hfill \\url{hamidim@erau.edu}}\n \n \\and\n \n Lara Ismert\n \\footnote{Department of Mathematics, Embry-Riddle Aeronautical University \\hfill \\url{ismertl@erau.edu}}\n \n \n \n \\and\n \n Brent Nelson\n \\footnote{Department of Mathematics, Michigan State University \\hfill \\url{brent@math.msu.edu}}\n \n \\and\n \n Mateusz Wasilewski\n \\footnote{Institute of Mathematics, Polish Academy of Sciences \\hfill\n \\url{mwasilewski@impan.pl}}\n }\n\n\n\n\\begin{document}\n\n\\date{}\n\n\\maketitle\n\n\\begin{abstract}\nGiven a quantum graph $\\cG=(B,\\psi,A)$, we define a C*-correspondence $E_\\cG$ over the noncommutative vertex C*-algebra $B$, called the \\emph{quantum edge correspondence}. For a classical graph $\\cG$, $E_\\cG$ is the usual graph correspondence spanned by the edges of $\\cG$. When the quantum adjacency matrix $A\\colon B\\to B$ is completely positive, we show that $E_\\cG$ is faithful if and only if $\\ker(A)$ does not contain a central summand of $B$. In this case, we show that the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$ is isomorphic to a quotient of the quantum Cuntz--Krieger algebra $\\cO(\\cG)$ defined in \\cite{BEVW20}. Moreover, the kernel of the quotient map is shown to be generated by ``localized'' versions of the quantum Cuntz--Krieger relations, and $\\cO_{E_\\cG}$ is shown to be the universal object associated to these local relations. We study in detail some concrete examples and make connections with the theory of Exel crossed products.\n\\end{abstract}\n\n\n\n\\section*{Introduction}\nThe notion of a quantum graph goes back to the work of Erdos-Katavolos-Shulman \\cite{ErKaSh98} and Weaver \\cite{We12}, and was subsequently developed further by Duan-Severini-Winter \\cite{DuSeWi13} and Musto-Reutter-Verdon \\cite{MuReVe19}. Quantum graphs play an intriguing role in the study of the graph isomorphism game in quantum information via their connections with quantum symmetries of graphs (see \\cite{MR4097284} and \\cite{BCEHPSWCMP19}). Moreover, based on the use of quantum symmetries, fascinating results on the graph theoretic interpretation of quantum isomorphisms between finite graphs were recently obtained by Man\\v{c}inska-Roberson \\cite{MaRo19}.\nIn this paper, we take a finite directed quantum graph to mean a triple $(B,\\psi, A)$ consisting of a finite-dimensional C*-algebra $B$, a state $\\psi$ on $B$, and a linear map $A$ on $B$ satisfying a quantum Schur idempotent condition. Finite directed quantum graphs generalize classical finite directed graphs (without multiple edges) by encoding a classical graph $\\cG$ in the triple $(B,\\psi, A)$, where $B = C(V)$ is the C*-algebra of continuous functions on the vertex set $V$ of $\\mathcal G$, $\\psi$ is integration with respect to the uniform probability on $V$, and $A$ is the adjacency matrix of $\\cG$. \n\n\nIn \\cite{BEVW20}, given a finite directed quantum graph $(B,\\psi,A)$ Eifler, Voigt, Weber and first author introduced a novel C*-algebra $\\bF\\cO(B, \\psi, A)$, called the \\textit{free quantum Cuntz--Krieger algebra}. This generalizes the well-studied Cuntz--Krieger algebra $\\cO_A$ arising from classical graphs \n(or rather a free version of it), where the standard generators are replaced by matrix-valued valued partial isometries whose matrix sizes are determined by the quantum graph, and the Cuntz--Krieger relations are expressed using the quantum adjacency matrix of the quantum graph in analogy to the scalar case.\nIntroduced in \\cite{ck1}, Cuntz--Krieger algebras have intimate connections with symbolic dynamics, and have been studied intensively in the framework of graph C*-algebras over the past decades, thus providing a rich supply of interesting examples \\cite{Raeburn, T17, ERRS18, ERRS21}. The structure of graph C*-algebras is understood to an impressive level of detail, and many algebraic properties can be interpreted in terms of the combinatorial properties of the underlying graphs. Motivated by this success, the original constructions and results have been generalized in several directions, including higher rank graphs \\cite{KuPa00}, Exel-Laca algebras \\cite{ExLa99}, and ultragraph algebras \\cite{To03}, among others.\nRecall that, under mild assumptions, the Cuntz--Krieger algebra corresponding to a classical graph is isomorphic to the Cuntz--Pimsner algebra associated to the graph's edge correspondence \\cite[Example 2, p. 193]{Pimsner}. It is worth mentioning that in the more general setting of quantum graphs, the free quantum Cuntz--Krieger algebras seem to be difficult to compute in general, and their isomorphism classes are only known under very strict assumptions (e.g., when $(B,\\psi,A)$ is \\emph{complete} and $\\psi$ is an $n^2$-form for $n\\in \\mathbb{N}$; see \\cite[Theorem 4.5]{BEVW20}).\n\nIn the present paper we consider a natural unital version of $\\mathbb{F}\\mathcal{O}(B,\\psi,A)$, which we denote $\\mathcal{O}(B,\\psi,A)$, and under the assumption that $A$ is completely positive we show that $\\mathcal{O}(B,\\psi,A)$ quotients onto the Cuntz--Pimsner algebra associated to a C*-correspondence over $B$ which can be viewed as the quantum analogue of the edge correspondence for a classical graph. This is accomplished by showing that this Cuntz--Pimsner algebra is the universal C*-algebra associated to ``local'' versions of the quantum Cuntz--Krieger relations introduced in \\cite{BEVW20}.\n\nIn Section~\\ref{sec:quantumedgecorrespondences}, we define the quantum edge correspondence $E_\\cG$ for a given quantum graph $\\cG = (B,\\psi,A)$ whose associated state is a $\\delta$-form. This C*-correspondence is generated by a generalized version of the Choi--Jamio\\l kowski matrix $\\epsilon_\\cG$ associated to the quantum adjacency operator $A$, and $E_\\cG$ generalizes the usual edge correspondence for a classical graph (see Example~\\ref{exmp:commutativecase}). Given the role of the element $\\epsilon_\\cG$ in generating this analogue of an edge correspondence, we think of it as the quantum analogue of a classical graph's edge matrix. Theorem~\\ref{thm:faithful_full_correspondence} identifies conditions on the quantum adjacency matrix which result in faithfulness and fullness of the quantum edge correspondence, and Proposition \\ref{prop:recognition} provides a recognition theorem for when a cyclic C*-correspondence is the quantum edge correspondence for a quantum graph. \n \nIn Section~\\ref{sec:QCK_and_local_relations}, we introduce a natural quotient of the quantum Cuntz--Krieger algebra $\\cO(\\cG)$ by introducing certain``local\" relations on the generators, and we call this quotient a {\\em local quantum Cuntz--Krieger algebra}. Given a quantum graph $\\cG$, we show in Theorem \\ref{thm:universal_local_quantum_Cuntz--Krieger_algebra} that the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$ constructed from the quantum edge correspondence is precisely the local quantum Cuntz--Krieger algebra for that same quantum graph. As mentioned previously, the free quantum Cuntz--Krieger algebras (and their unital cousins) seem to be very difficult to describe concretely, except in the most basic cases. This sentiment is emphasized by the fact that we are unable to find an example of a quantum graph $\\cG$ whose local quantum Cuntz--Krieger algebra is a {\\it proper} quotient of $\\cO(\\cG)$.\n\nIn Section~\\ref{sec:simpler_examples}, we focus on examples, beginning with a very special class of quantum graphs called the {\\it complete quantum graphs} $\\cG = K(B,\\psi)$. Here we study the associated quantum Cuntz--Krieger algebras and Cuntz--Pimsner algebras, and make some connections to Exel's theory of crossed products of C*-algebras by endomorphisms \\cite{Exel03}. \nIn the classical setting, any square $\\{0,1\\}$-matrix $A = [A(x,y)]$ gives rise to the Markov subshift $(X_A, \\sigma)$, which is the topological dynamical system given by the infinite compact path space $X_A = \\{x = (x_i) \\in \\{1, \\ldots, n\\}^{\\mathbb N}\\ \\big| \\ A(x_i,x_{i+1}) = 1 \\ \\forall i \\ge 1 \\}$ together with the left shift action $\\sigma:X_A \\to X_A$ given by $\\sigma(x)_i = x_{i+1}$. From this dynamical system $(X_A, \\sigma)$, one can associate an {\\it Exel system} $(C(X_A),\\alpha,\\mathcal{L})$, where $C(X_A)$ is the unital C*-algebra of continuous functions on $X_A$, $\\alpha: C(X_A) \\to C(X_A)$ is the $*$-endomorphism defined by $\\alpha(f) = f \\circ \\sigma$, and $\\mathcal{L}:C(X_A) \\to C(X_A)$ is a {\\it transfer operator} for $\\alpha$. In \\cite{Exel03}, Exel builds from this data a crossed product C$^\\ast$-algebra $C(X_A)\\rtimes_{\\alpha, \\mathcal{L}} \\mathbb{N}$, and shows that it is isomorphic to the usual Cuntz--Krieger algebra $\\mathcal{O}_A$. For any complete quantum graph $K = K(B,\\psi)$, we associate a natural choice of (non-commutative) Exel system, and show in Proposition \\ref{thm:crossed-product-is-a-Cuntz-algebra}, that this crossed product is isomorphic to the Cuntz algebra $\\cO_{n}$ on $n = \\dim B$ generators. We also show in Proposition \\ref{prop:Cuntz--Pimsner-is-a-Cuntz-algebra} that the Cuntz--Pimsner algebra $\\cO_{E_K}$ is isomorphic to $\\cO_n$. These results combined generalize the well known identifications of Cuntz--Krieger algebras, Cuntz--Pimsner algebras, and Exel crossed products associated to complete graphs.\n\n \nFinally, in the other subsections of Section~\\ref{sec:simpler_examples} of the paper, we study trivial (edgeless) quantum graphs and their two natural generalizations: rank-one quantum graphs, and quantum graphs associated to $\\ast$-automorphic quantum adjacency matrices. We are able associate a natural choice of (non-commutative) Exel system in these cases as well, except for rank-one quantum graphs. Given one of these types of quantum graphs, we show that the Exel crossed product is isomorphic to the Cuntz--Pimsner algebra for that quantum graph's quantum edge correspondence. See Corollary \\ref{cor:triv}, Proposition \\ref{prop:triv}, and Corollary \\ref{cor:triv2}. These examples also mimic the classical setting, where a graph's associated Cuntz--Krieger algebra, Exel crossed product, and Cuntz--Pimsner algebra are all isomorphic.\n\nLet us end this introduction with a remark: For more general quantum graphs $\\cG = (B,\\psi, A)$, it is an interesting and natural problem to associate to $\\cG$ a quantum analogue of (functions on) the path space $X_A$. The construction of an appropriate noncommutative version of $C(X_A)$ seems to be highly non-trivial, and we plan to investigate this in more detail in a followup work.\n\n\n\\subsection*{Acknowledgments} Michael Brannan was partially supported by NSF Grant DMS-2000331. Brent Nelson was partially supported by NSF Grant DMS-1856683. Lara Ismert was partially supported by the NSF-AWM Mentoring Travel Grant. Mateusz Wasilewski was partially supported by the Research Foundation \u2014 Flanders (FWO) through a Postdoctoral Fellowship, by long term structural funding \u2013 Methusalem grant of the Flemish\nGovernment \u2013 and by the European Research Council Starting Grant 677120 INDEX.\n\n\n\n\n\n\n\\section{Preliminaries}\n\\subsection{Quantum graphs}\nIn this paper, we consider \\emph{finite quantum spaces} $(B,\\psi)$, consisting of a finite dimensional C*-algebra $B$ and a distinguished faithful state $\\psi\\colon B\\to \\bC$ satisfying\n \\[\n mm^* = \\delta^2\\text{id},\n \\]\nwhere $m\\colon B\\otimes B\\to B$ is the multiplication map, $m^*$ is its adjoint with respect to the inner product given by $\\psi$, and $\\delta>0$. States $\\psi:B \\to \\bC$ satisfying the above identity are called \\emph{$\\delta$-forms}. Since $B$ is finite dimensional, we have\n \\[\n B\\cong \\bigoplus_{a=1}^d M_{N_a}(\\bC).\n \\]\nThe restriction of $\\psi$ to the $a^{th}$ summand $M_{N_a}(\\mathbb{C})$ appearing in the direct sum decomposition of $B$ will be given by a density matrix with respect to the usual trace, denoted $\\rho_a$. We may and will assume that each $\\rho_a$ is diagonal with the $i^{th}$ diagonal entry being $\\psi(e_{ii}^{a})$. The condition that $\\psi$ is a $\\delta$-form is then equivalent to $\\text{Tr}(\\rho_a^{-1})=\\sum_{i=1}^{N_a} \\psi(e_{ii}^{a})^{-1}=\\delta^2$ for each $a=1,\\ldots, d$.\n\nWe utilize the diagonal entries of these density matrices to define \\emph{adapted matrix units} for $(B,\\psi)$, given by\n \\[\n f_{ij}^{(a)}:= \\frac{1}{\\left[\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})\\right]^{1\/2}} e_{ij}^{(a)} \\qquad \\qquad 1\\leq a\\leq d,\\ 1\\leq i,j\\leq N_a,\n \\]\nand by \\cite[Lemma 3.2]{BEVW20} they satisfy\n \\begin{equation}\\label{Eq:adaptedunits}\n m^*(f_{ij}^{(a)}) = \\sum_{k=1}^{N_a} f_{ik}^{(a)} \\otimes f_{kj}^{(a)}.\n \\end{equation}\n\\begin{equation}\\label{Eq:prodadaptedunits}\n f_{ij}^{(a)}f_{rs}^{(b)} = \\delta_{\\substack{a=b\\\\ j=r}} \\frac{1}{\\psi(e_{jj}^{(a)})} f_{is}^{(a)}. \n\\end{equation}\n\n\n\n\nGiven a finite quantum space $(B,\\psi)$, a linear map $A\\colon B\\to B$ is said to be a \\emph{quantum adjacency matrix} if\n \\[\n m(A\\otimes A)m^* = \\delta^2 A.\n \\]\nIn this case, the triple $(B,\\psi,A)$ is called a \\emph{quantum graph}. At times, it will be convenient to express $A$ as an actual matrix $[A_{ija}^{rsb}]_{\\substack{1 \\le a,b \\le m \\\\\n1 \\le i,j \\le N_a \\\\\n1 \\le r,s \\le N_b}}$ with respect to the basis of adapted matrix units: \\[\n A(f_{ij}^{(a)}) = \\sum_{b=1}^d \\sum_{r,s=1}^{N_b} A_{ija}^{rsb} f_{rs}^{(b)}.\n \\]\nThese coefficients can be used to directly check whether a linear map $A\\colon B\\to B$ is a quantum adjacency matrix (see \\cite[Lemma 3.4]{BEVW20}), but this will not be necessary in the present paper.\n\n\n\n\n\n\\subsection{C*-correspondences and Cuntz--Pimsner algebras}\\label{subsec:correspondences_and_CP_algebras}\n\nGiven a C*-algebra $B$, a \\emph{C*-correspondence} over $B$ is a right Hilbert $B$-module $X$ (with right $B$-valued innner product $\\langle \\cdot, \\cdot \\rangle_B$) which admits a $*$-homomorphism $\\varphi_X\\colon B\\to \\cL(X)$. Here $\\cL(X)$ denotes the right $B$-linear adjointable operators on $X$. The $*$-homomorphism $\\varphi_X$ induces a left $B$-action that commutes with the right $B$-action:\n \\[\n x\\cdot \\xi:=\\varphi_X(x)\\xi \\qquad \\qquad x\\in B,\\ \\xi\\in X.\n \\]\nOne says $X$ is \\emph{faithful} if $\\varphi_X$ is faithful, and \\emph{full} if $\\overline{\\text{span}}\\_B=B$. The \\emph{compact operators} on $X$, denoted $\\cK(X)$, are generated by $\\theta_{\\xi,\\eta}\\in \\cL(X)$, $\\xi,\\eta\\in X$, where\n \\[\n \\theta_{\\xi,\\eta}(\\zeta) := \\xi\\cdot \\<\\eta,\\zeta\\>_B \\qquad \\zeta\\in X.\n \\]\nIn this paper, we will exclusively consider finite dimensional C*-correspondences, for which $\\cK(X)=\\cL(X)$. The following example is particularly relevant for our purposes.\n\n\\begin{example}\\label{example:correspondence_from_cp_map}\nLet $B$ be a finite dimensional C*-algebra and $A\\colon B\\to B$ a completely positive map. Define a $B$-valued inner product on $B\\otimes B$ by\n \\[\n \\< a\\otimes b, c\\otimes d\\>_B := b^* A(a^* c)d,\n \\]\nwhere the positivity follows from the complete positivity of $A$. Then after taking a separation,\n \\[\n B\\otimes_A B:= B\\otimes B\/\\{\\xi\\in B\\otimes B\\colon \\<\\xi,\\xi\\>_B=0\\}\n \\]\ndefines a C*-correspondence over $B$. (Since everything is finite-dimensional, we do not need to take a completion.) The usual left and right actions of $B$ on $B\\otimes B$, \n \\[\n x\\cdot (a\\otimes b)\\cdot y = (xa)\\otimes (by),\\qquad a,b,x,y\\in B\n \\]\nextend to left and right actions on $B\\otimes_A B$.\n\\end{example}\n\n\n\nAfter \\cite{Kat04}, a \\emph{representation} of a C*-correspondence $X$ over $B$ on a C*-algebra $D$ is a pair $(\\pi,t)$ consisting of a $*$-homomorphism $\\pi\\colon B\\to D$ and a linear map $t\\colon X\\to D$ satisfying:\n \\begin{enumerate}[label=(\\roman*)]\n \\item $\\pi(x)t(\\xi)=t(x\\cdot \\xi)$ for $x\\in B$ and $\\xi\\in X$,\n \n \\item $t(\\xi)^*t(\\eta)=\\pi(\\<\\xi,\\eta\\>_B)$ for $\\xi,\\eta\\in X$.\n \\end{enumerate}\nUsing (i), one can also show $t(\\xi)\\pi(x)= t(\\xi\\cdot x)$ for $x\\in B$ and $\\xi\\in X$. One can also define a $*$-homomorphism $\\psi_t\\colon \\cK(X)\\to D$ by $\\psi_t(\\theta_{\\xi,\\eta})=t(\\xi)t(\\eta)^*$. A representation is said to be \\emph{covariant} if $\\pi(x) = \\psi_t(\\varphi_X(x))$ for all $x$ in the \\emph{Katsura ideal} $J_X$, which is defined by\n \\[\n J_X:=\\{x\\in B\\colon \\varphi_X(x)\\in \\cK(X) \\text{ and } xy=0 \\text{ for all }y\\in \\ker{\\varphi_X}\\}.\n \\]\nNote that $J_X=B$ when $X$ is finite dimensional and faithful, which is the situation we will primarily consider. \n\nThe \\emph{Cuntz--Pimsner algebra} for a C*-correspondence $X$ over $B$ is the C*-algebra $\\cO_X=C^*(\\pi_X(B),t_X(X))$ where $(\\pi_X,t_X)$ is the universal covariant representation of $X$. That is, given any covariant representation $(\\pi,t)$ of $X$ on a C*-algebra $D$, there exists a $*$-homomorphism $\\rho \\colon \\cO_X\\to D$ satisfying $\\pi=\\rho\\circ \\pi_X$ and $t=\\rho\\circ t_X$. \n\n\n\n\n\n\n\n\\section{Quantum edge correspondences}\\label{sec:quantumedgecorrespondences}\n\nLet $(B,\\psi)$ be a finite quantum space. Since the state $\\psi:B \\to \\bC \\subset B$ is completely positive we can consider the C*-correspondence $B\\otimes_\\psi B$ over $B$ from Example~\\ref{example:correspondence_from_cp_map}. Note that for $\\xi\\in B\\otimes B$, $\\psi(\\<\\xi,\\xi\\>_B)=\\|\\xi\\|^2_{\\psi\\otimes \\psi}$, and so $B\\otimes_\\psi B=B\\otimes B$ as a vector space. That is, we do not need to take a quotient of $B\\otimes B$.\n\n\\begin{definition}\\label{def:edgecorrespondence}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form. We define the {\\it quantum edge indicator} to be the element \\[\\epsilon_\\cG:=\\frac{1}{\\delta^2}(1\\otimes A)m^*(1) \\in B\\otimes_\\psi B.\\] The \\textit{quantum edge correspondence} of $\\cG$ is the C*-correspondence over $B$ defined by\n \\[\n E_\\cG:=B\\cdot \\epsilon_\\cG \\cdot B =\\text{span}\\{x\\cdot \\epsilon_\\cG\\cdot y \\colon x,y\\in B\\} \\subset B\\otimes_\\psi B.\n \\]\nThat is, $E_\\cG$ is the C*-subcorrespondence of $B\\otimes_\\psi B$ generated by the quantum edge indicator $\\epsilon_\\cG$.\n\\end{definition}\n\n\\begin{example}\\label{exmp:commutativecase}\n In the case of a classical directed graph $\\mathcal{G}=( \\bC(V), \\frac{1}{|V|}, A)$, one has\n \\[\n \\epsilon_\\cG=\\frac{1}{\\delta^2}(1\\otimes A)m^*(1) = \\frac{1}{|V|}(1\\otimes A)\\sum_{v\\in V} |V| p_v\\otimes p_v = \\sum_{w\\to v} p_v\\otimes p_w.\n \\]\nHence $\\epsilon_\\cG\\in \\bC(V\\times V)$ is the indicator function for the set $\\{(v,w)\\colon (w,v) \\text{ is an edge}\\}$, and the quantum edge correspondence $E_\\cG$ is the space of functions supported on this set.\n\\end{example} \n\n\n\n\\subsection{Properties of quantum edge indicators and correspondences}\n\nBefore studying the edge correspondence, we note some important properties of the quantum edge indicator $\\epsilon_\\cG$ which further justify our terminology. \n\n\\begin{proposition}\\label{prop:properties_of_edge_generator}\nLet $\\mathcal{G}=(B,\\psi,A)$ be a directed quantum graph with $\\delta$-form $\\psi$, and let $\\epsilon_\\cG:= \\frac{1}{\\delta^2}(1\\otimes A)m^*(1)$ be the quantum edge indicator. Then:\n \\begin{enumerate}[label=(\\arabic*)]\n \\item $A(x) = \\delta^2 (\\psi\\otimes 1)(x\\cdot \\epsilon_\\cG)$ for all $x\\in B$.\n \n \\item $\\epsilon_\\cG\\# \\epsilon_\\cG=\\epsilon_\\cG$ where $(a\\otimes b)\\#(c\\otimes d)=(ac)\\otimes (db)$ for $a,b,c,d\\in B$.\n \n \\item $A$ is completely positive if and only if $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\epsilon_\\cG)$ is self-adjoint, where $(\\sigma_t^\\psi)_{t \\in \\mathbb R}$ denotes the modular automorphism group of $\\psi$.\n \\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n\\begin{enumerate}\n \\item[\\textbf{(1):}] For $x,y\\in B$, we have\n \\begin{align*}\n \\< \\delta^2 (\\psi\\otimes 1)(x\\cdot \\epsilon_\\cG), y\\>_\\psi &= \\delta^2 \\< \\epsilon_\\cG, x^*\\otimes y\\>_{\\psi\\otimes \\psi}\\\\\n &= \\< m^*(1), x^* \\otimes A^*(y)\\>_{\\psi\\otimes \\psi}\\\\\n &= \\< 1, x^* A^*(y)\\>_\\psi\\\\\n &= \\< A(x), y\\>_\\psi.\n \\end{align*}\nHence $\\delta^2 (\\psi\\otimes 1)(x\\cdot \\epsilon_\\cG)=A(x)$ as claimed.\n \n \\item[\\textbf{(2):}] Observe that $\\epsilon_\\cG$ is the image of $1\\otimes 1$ under the following map on $B\\otimes B$:\n \\[\n \\frac{1}{\\delta^2}(1\\otimes m)(1\\otimes A\\otimes 1)(m^*\\otimes 1).\n \\]\n We will show that this map is a left and right $B$-linear and idempotent. (In fact, $E_\\cG$ is precisely the image of $B\\otimes B$ under this map.) \n Bimodularity comes from the fact that both $m$ and $m^*$ are left and right $B$-linear. Using the associativity of the multiplication operation (i.e. $m(m\\otimes 1) = m(1\\otimes m)$, hence also $(m^*\\otimes 1)m^* = (1\\otimes m^*)m^*$) and that $A$ is a quantum adjacency matrix (i.e. $\\frac{1}{\\delta^2} m(A\\otimes A)m^*=A$), we will show that this map is idempotent. Its square is equal to \n \\[\n \\frac{1}{\\delta^4} \\left(1\\otimes (m(1\\otimes m)\\right) \\left(1\\otimes A\\otimes A \\otimes 1\\right)\\left((m^*\\otimes 1)m^* \\otimes 1\\right).\n \\] \n Using associativity, we get\n \\[\n \\frac{1}{\\delta^4}(1\\otimes m)(1\\otimes m \\otimes 1)(1\\otimes A \\otimes A \\otimes 1)(1\\otimes m^* \\otimes 1)(m^* \\otimes 1).\n \\]\n In the middle, we recognize the expression $1\\otimes (m(A\\otimes A)m^*)\\otimes 1$, which is equal to $\\delta^2 1\\otimes A \\otimes 1$. In the end we obtain\n \\[\n \\frac{1}{\\delta^2}(1\\otimes m)(1\\otimes A \\otimes 1)(m^*\\otimes 1),\n \\]\n which verifies the map is idempotent. Consequently,\n \\begin{align*}\n \\epsilon_\\cG\\# \\epsilon_\\cG &= \\epsilon_\\cG\\# \\frac{1}{\\delta^2} (1\\otimes m)(1\\otimes A\\otimes 1)(m^*\\otimes 1)(1\\otimes 1)\\\\\n &= \\frac{1}{\\delta^2}(1\\otimes m)(1\\otimes A\\otimes 1)(m^*\\otimes 1)(\\epsilon_\\cG)\\\\\n &= \\left[\\frac{1}{\\delta^2}(1\\otimes m)(1\\otimes A\\otimes 1)(m^*\\otimes 1)\\right]^2(1\\otimes 1)\\\\\n &= \\frac{1}{\\delta^2}(1\\otimes m)(1\\otimes A\\otimes 1)(m^*\\otimes 1)(1\\otimes 1) = \\epsilon_\\cG.\n \\end{align*}\n (This identity can also be checked directly using the adapted matrix units of $(B,\\psi)$, but this arduous task is left to the skeptical reader.)\n \n \\item[\\textbf{(3):}] Let $\\eta_\\cG\\in B\\otimes B^\\text{op}$ be the image of $\\epsilon_\\cG$ under the map $a\\otimes b\\mapsto a\\otimes b^\\circ$. Then $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\epsilon_\\cG)$ is self-adjoint if and only if $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\eta_\\cG)$ is self-adjoint. Observe that $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\eta_\\cG)$ is an idempotent by part (2), and so it is self-adjoint if and only if it is a projection, and hence if and only if it is positive. It therefore suffices to show $A$ is completely positive if and only if $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\eta_\\cG)$ is positive. \n \n Recall that if $(M,\\varphi)$ is a von Neumann algebra equipped with a faithful normal linear functional, then $\\overline{\\{\\Delta_\\varphi^{1\/4} x\\colon x\\in M_+\\}}$ is a self-dual cone in $L^2(M,\\varphi)$, where $\\Delta_\\varphi$ is the modular operator with respect to $\\varphi$. Consequently $A$ is completely positive if and only if for all positive elements $X=(x_{ij}), Y=(y_{ij})\\in M_n(B)^+$ one has\n \\[\n \\< \\Delta_{\\psi\\otimes \\text{Tr}}^{1\/4} (A\\otimes I_n)(X), \\Delta_{\\psi\\otimes \\text{Tr}}^{1\/4}Y\\>_{\\psi\\otimes \\text{Tr}}= \\sum_{i,j=1}^n \\< \\Delta_\\psi^{1\/4} A(x_{ij}), \\Delta_\\psi^{1\/4} y_{ij}\\>_\\psi \\geq 0.\n \\]\n Using part (1), we compute\n \\[\n \\< \\Delta_\\psi^{1\/4} A(x_{ij}), \\Delta_\\psi^{1\/4} y_{ij}\\>_\\psi = \\< \\delta^2(\\psi\\otimes 1)(x_{ij}\\cdot \\epsilon_\\cG), \\Delta_\\psi^{1\/2} y_{ij}\\>_\\psi = \\delta^2 \\< \\epsilon_\\cG, x_{ij}^*\\otimes (\\Delta_\\psi^{1\/2} y_{ij})\\>_{\\psi\\otimes \\psi}.\n \\] \nNow, $L^2(B\\otimes B,\\psi\\otimes\\psi)\\ni a\\otimes b\\mapsto a\\otimes [\\Delta_\\psi^{-1\/2} b]^\\circ \\in L^2(B\\otimes B^{\\text{op}},\\psi\\otimes\\psi^{\\text{op}})$ is an isometry, so we can continue the above computation with\n \\begin{align*}\n \\< \\Delta_\\psi^{1\/4} A(x_{ij}), \\Delta_\\psi^{1\/4} y_{ij}\\>_\\psi &= \\delta^2 \\< (1\\otimes \\Delta_\\psi^{-1\/2}) \\eta_\\cG, x_{ij}^*\\otimes y_{ij}^\\circ\\>_{\\psi\\otimes \\psi^{\\text{op}}}\\\\\n &= \\delta^2\\< (\\Delta_\\psi^{1\/4}\\otimes \\Delta_\\psi^{-1\/4})( \\Delta_\\psi^{-1\/2}\\otimes 1)\\eta_\\cG, (\\Delta_\\psi^{1\/4}\\otimes \\Delta_\\psi^{-1\/4})( x_{ij}^*\\otimes y_{ij}^\\circ) \\>_{\\psi\\otimes \\psi^\\text{op}}\\\\\n &= \\delta^2 \\< \\Delta_{\\psi\\otimes \\psi^{\\text{op}}}^{1\/4}\\left[ (\\sigma_{i\/2}^{\\psi}\\otimes 1)(\\eta_\\cG)\\right], \\Delta_{\\psi\\otimes \\psi^{\\text{op}}}^{1\/4}(x_{ij}^*\\otimes y_{ij}^\\circ)\\>_{\\psi\\otimes \\psi^{\\text{op}}}.\n \\end{align*}\nThus\n \\begin{align}\\label{eqn:complete_positivity_equation}\n \\sum_{i,j=1}^n \\< \\Delta_\\psi^{1\/4} A(x_{ij}), \\Delta_\\psi^{1\/4} y_{ij}\\>_\\psi & \\nonumber\\\\\n = \\delta^2 \\sum_{i,j=1}^n &\\< \\Delta_{\\psi\\otimes \\psi^{\\text{op}}}^{1\/4}\\left[ (\\sigma_{i\/2}^{\\psi}\\otimes 1)(\\eta_\\cG)\\right], \\Delta_{\\psi\\otimes \\psi^{\\text{op}}}^{1\/4}(x_{ij}^*\\otimes y_{ij}^\\circ)\\>_{\\psi\\otimes \\psi^{\\text{op}}}.\n \\end{align}\nSuppose $X=W^*W$ and $Y=Z^*Z$ for $W=(w_{ij}), Z=(z_{ij})\\in M_n(B)$. Then\n \\[\n \\sum_{i,j=1}^n x_{ij}^*\\otimes y_{ij}^\\circ = \\sum_{i,j,k,\\ell=1}^n w_{kj}^*w_{ki}\\otimes (z_{\\ell i}^* z_{\\ell j})^\\circ = \\sum_{k,\\ell=1}^n (\\sum_{j=1}^n w_{kj}\\otimes (z_{\\ell j}^*)^\\circ)^* (\\sum_{i=1}^n w_{ki}\\otimes (z_{\\ell i}^*)^\\circ)\n \\]\nis positive. Also note that every positive element of $B\\otimes B^\\text{op}$ can be presented this way. Indeed,\n \\[\n (\\sum_{d=1}^m w_d\\otimes z_d^\\circ)^*(\\sum_{d=1}^m w_d\\otimes z_d^\\circ) = \\sum_{i,j=1}^n x_{ij}^*\\otimes y_{ij}^\\circ\n \\] \nif $X=W^*W, Y=Z^*Z\\in M_m(B)$ where $(W)_{ij} = \\delta_{i=1} w_j$ and $(Z)_{ij}=\\delta_{i=1} z_i^*$. It follows from (\\ref{eqn:complete_positivity_equation}) and these observations that $A$ is completely positive if and only if $(\\sigma_{i\/2}^{\\psi}\\otimes 1)(\\eta_\\cG)$ is positive.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{remark}\nAt this point it is worth remarking on the connection to Weaver's notion of a quantum graph \\cite{We12, We21}. Recall that there is an algebra isomorphism \n \\begin{align*}\n \\pi:B \\otimes B^{{\\text{op}}} &\\to {}_{B'}CB_{B'}(B(L^2(B,\\psi)))\\\\\n (a\\otimes b^\\circ)&\\mapsto (T\\mapsto aTb) \n \\end{align*} \n where ${}_{B'}CB_{B'}(B(L^2(B,\\psi)))$ denotes the space of completely bounded $B'$-$B'$-bimodule maps on $B(L^2(B,\\psi))$. When a quantum adjacency matrix $A$ is completely positive, the projection $p = (\\sigma_{i\/2}^\\psi\\otimes 1)(\\eta_\\cG) \\in B\\otimes B^{\\text{op}}$ induces a $B'$-$B'$-bimodule projection $\\pi(p)$ on $B(L^2(B, \\psi))$ whose range $S \\subseteq B(L^2(B,\\psi))$ is a $B'$-$B'$-bimodule. Such bimodules $S$ are exactly what Weaver refers to as (directed, non-reflexive) quantum graphs on $B \\subseteq B(L^2(B,\\psi))$.\n\\end{remark}\n\n\n\\begin{theorem}\\label{thm:cp_Adjacency_matrices_are_inner_products}\nLet $\\cG=(B,\\psi, A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive. Then for $x,y\\in B$ one has\n \\[\n \\< x \\cdot \\epsilon_\\cG, y\\cdot \\epsilon_\\cG\\>_B = \\frac{1}{\\delta^2}A(x^*y),\n \\]\nwhere $\\epsilon_\\cG = \\frac{1}{\\delta^2} (1\\otimes A)m^*(1)$ is the quantum edge indicator. In particular, $A(x)=\\delta^2\\<\\epsilon_\\cG, x\\cdot \\epsilon_\\cG\\>_B$ for all $x\\in B$.\n\\end{theorem}\n\\begin{proof}\nWrite $\\epsilon_\\cG = \\sum_\\alpha p_\\alpha\\otimes q_\\alpha$ for $p_\\alpha,q_\\alpha\\in B$. Then\n \\begin{align*}\n \\< x \\cdot \\epsilon_\\cG, y \\cdot \\epsilon_\\cG\\>_B &= \\sum_{\\alpha,\\beta} \\< xp_\\alpha\\otimes q_\\alpha, yp_\\beta\\otimes q_\\beta\\>_B\\\\\n &= \\sum_{\\alpha, \\beta} \\psi( p_\\alpha^* x^* y q_\\beta) q_\\alpha^* q_\\beta\\\\\n &= \\sum_{\\alpha, \\beta} \\psi(x^*y p_\\beta \\sigma_{-i}^\\psi(p_\\alpha^*)) q_\\alpha^* q_\\beta \\\\\n &= (\\psi\\otimes 1)[ x^*y\\cdot \\epsilon_\\cG \\# (\\sigma_{-i}^\\psi\\otimes 1)(\\epsilon_\\cG^*)]\\\\\n &=\\ (\\psi\\otimes 1)( x^*y\\cdot \\epsilon_\\cG \\# \\epsilon_\\cG)\\\\\n &= (\\psi\\otimes 1)( x^*y\\cdot \\epsilon_\\cG) = \\frac{1}{\\delta^2} A(x^*y),\n \\end{align*}\nwhere the last three equalities follow from the three parts of Proposition~\\ref{prop:properties_of_edge_generator} (in reverse order).\n\\end{proof} \n\nRecalling the definition of $B\\otimes_A B$ from Example~\\ref{example:correspondence_from_cp_map}, we obtain the following corollary:\n\n\\begin{corollary}\\label{cor:quantum_graph_correspondence_as_cp_correspondence}\nLet $\\cG=(B,\\psi, A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive. Then $E_\\cG\\cong B\\otimes_A B$ as C*-correspondences over $B$ via the map $x\\cdot \\epsilon_\\cG\\cdot y\\mapsto \\frac{1}{\\delta}(x\\otimes y)$.\n\\end{corollary}\n\n\nThe above corollary is somewhat surprising in that a priori $B\\otimes_A B$ appears to be completely independent of $\\psi$. But of course the dependence is hidden in the fact that $A$ is a quantum adjacency matrix for $(B,\\psi)$. We also remark that while $B\\otimes_A B$ is more easily defined, $E_\\cG$ has the advantage of not requiring us to quotient by any null spaces.\n\n\nGiven a finite quantum space $(B,\\psi)$ such that $\\psi$ is a $\\delta$-form, Theorem~\\ref{thm:cp_Adjacency_matrices_are_inner_products} tells us every completely positive adjacency matrix is of the form $A(x)=\\delta^2\\<\\xi, x\\cdot \\xi\\>_B$ for some $\\xi\\in B\\otimes_\\psi B$. Thus it is natural to ask for which elements $\\xi\\in B\\otimes_\\psi B$ is the map $B\\ni x\\mapsto \\delta^2\\< \\xi,x\\cdot \\xi\\>_B$ a (necessarily completely positive) quantum adjacency matrix? Suppose $\\xi\\#\\xi=\\xi$ and $(\\sigma_{i\/2}^\\psi\\otimes 1)(\\xi)$ is self-adjoint, and define $A_\\xi(x):= \\delta^2 \\<\\xi,x\\cdot \\xi\\>_B$, then the same computation as in the above theorem implies\n \\[\n A_\\xi(x)=\\delta^2\\< \\xi, x\\cdot \\xi\\>_B = \\delta^2(\\psi\\otimes 1)(x\\cdot \\xi).\n \\]\nConsequently, for each $1\\leq a \\leq d$ and $1\\leq i,j\\leq N_a$\n \\begin{align*}\n \\frac{1}{\\delta^2} m(A_\\xi\\otimes A_\\xi)m^*(f_{ij}^{(a)}) &= \\frac{1}{\\delta^2}\\sum_{k} A_\\xi( f_{ik}^{(a)}) A_\\xi(f_{kj}^{(a)})\\\\\n &= \\delta^2 \\sum_k (\\psi\\otimes 1)(f_{ik}^{(a)}\\cdot \\xi) (\\psi\\otimes 1)(f_{kj}^{(a)}\\cdot \\xi).\n \\end{align*}\nWriting $\\xi= \\sum_\\alpha p_\\alpha\\otimes q_\\alpha$, we can continue the above with\n \\begin{align*}\n \\frac{1}{\\delta^2} m(A_\\xi\\otimes A_\\xi)m^*(f_{ij}^{(a)}) &= \\delta^2 \\sum_{k,\\alpha,\\beta} \\_\\psi\\_\\psi q_\\alpha q_\\beta\\\\\n &=\\delta^2 \\sum_{k,\\alpha,\\beta} \\_\\psi\\_\\psi q_\\alpha q_\\beta\\\\\n &= \\delta^2 \\sum_{k,\\alpha,\\beta} \\< \\<\\psi(e_{ii}^{(a)}) f_{ij}^{(a)} p_\\beta, f_{ik}^{(a)}\\>_\\psi f_{ki}^{(a)}, p_\\alpha\\>_\\psi q_\\alpha q_\\beta\\\\\n &= \\delta^2 \\sum_{k,\\alpha,\\beta} \\< \\< \\Delta_\\psi f_{ki}^{(a)}, \\psi(e_{ii}^{(a)}) p_\\beta^* f_{ji}^{(a)}\\>_\\psi f_{ki}^{(a)}, p_\\alpha\\>_\\psi q_\\alpha q_\\beta\\\\\n &= \\delta^2 \\sum_{k,\\alpha,\\beta} \\psi(e_{kk}^{(a)}) \\< \\< f_{ki}^{(a)}, p_\\beta^*f_{ji}^{(a)}\\>_\\psi f_{ki}^{(a)}, p_\\alpha\\>_\\psi q_\\alpha q_\\beta.\n \\end{align*}\nNow, using the fact that $\\{ \\psi(e_{rr}^{(b)})^{1\/2}f_{rs}^{(b)}\\colon 1\\leq b \\leq d,\\ 1\\leq r,s\\leq N_b \\}$ is an orthonormal basis for $L^2(B)$, we then have\n \\begin{align*}\n \\frac{1}{\\delta^2} m(A_\\xi\\otimes A_\\xi)m^*(f_{ij}^{(a)}) &= \\delta^2 \\sum_{r,s,b,\\alpha,\\beta} \\< \\< \\psi(e_{rr}^{(b)})^{1\/2}f_{rs}^{(b)}, p_\\beta^*f_{ji}^{(a)}\\>_\\psi \\psi(e_{rr}^{(b)})^{1\/2}f_{rs}^{(b)}, p_\\alpha\\>_\\psi q_\\alpha q_\\beta\\\\\n &= \\delta^2 \\sum_{\\alpha, \\beta} \\_\\psi q_\\alpha q_\\beta\\\\\n &= \\delta^2 \\sum_{\\alpha,\\beta } (\\psi\\otimes 1)( f_{ij}^{(a)}\\cdot (p_\\beta \\otimes q_\\beta)\\# (p_\\alpha\\otimes q_\\alpha))\\\\\n &= \\delta^2 \\sum_\\alpha (\\psi\\otimes 1)( f_{ij}^{(a)}\\cdot (p_\\alpha \\otimes q_\\alpha)) = A_\\xi(f_{ij}^{(a)}).\n \\end{align*}\nThus $A_\\xi$ is a quantum adjacency matrix for $(B,\\psi)$. Moreover, by a similar computation one can show that $\\frac{1}{\\delta^2}(1\\otimes A_\\xi)m^*(1)=\\xi$ and so there is in fact a one-to-one correspondence between completely positive quantum adjacency matrices for $(B,\\psi)$ and the set\n \\[\n \\{\\xi\\in B\\otimes_\\psi B\\colon \\xi\\#\\xi=\\xi,\\ (\\sigma_{i\/2}^\\psi\\otimes 1)(\\xi)^*=(\\sigma_{i\/2}^\\psi\\otimes 1)(\\xi)\\}.\n \\]\nMore generally, we have the following recognition result for the quantum edge correspondence.\n\n\\begin{proposition}\\label{prop:recognition}\nLet $(B,\\psi)$ be a finite quantum space such that $\\psi$ is a $\\delta$-form, and let $X=B\\cdot\\xi\\cdot B$ be a cyclic C*-correspondence over $B$. If the map $A\\colon B\\to B$ defined by $A(x)=\\delta^2\\<\\xi, x\\cdot \\xi\\>_B$ is a quantum adjacency matrix for $(B,\\psi)$, then $x\\cdot \\xi\\cdot y\\mapsto x\\cdot \\epsilon_{(B,\\psi,A)}\\cdot y$ extends to a C*-correspondence isomorphism from $X$ to the quantum edge correspondence $E_{(B,\\psi, A)}$.\n\\end{proposition}\n\\begin{proof}\nIf we denote $\\epsilon:=\\epsilon_{(B,\\psi,A)}$, then Theorem~\\ref{thm:cp_Adjacency_matrices_are_inner_products} implies\n \\[\n \\_B = \\delta_{\\substack{a=b\\\\i=r}} \\<\\xi, e_{js}^{(a)}\\cdot \\xi\\>_B = \\delta_{\\substack{a=b\\\\i=r}} \\frac{1}{\\delta^2}A(e_{js}^{(a)}) = \\< e_{ij}^{(a)}\\cdot\\epsilon, e_{rs}^{(b)}\\cdot \\epsilon\\>_B\n \\]\nfor all $1\\leq a, b\\leq d$, $1\\leq i,j\\leq N_a$, and $1\\leq r,s\\leq N_b$. It follows that for all $x,y,z,w\\in B$ that\n \\[\n \\_B = y^* \\< x\\cdot \\xi, z\\cdot \\xi\\>_B w = y^*\\< x\\cdot \\epsilon, z\\cdot \\epsilon\\>_B w = \\< x\\cdot \\epsilon\\cdot y, z\\cdot \\epsilon\\cdot w\\>_B.\n \\]\nTherefore $\\sum_i x_i\\cdot \\xi\\cdot y_i\\mapsto \\sum_i x_i\\cdot\\epsilon\\cdot y_i$ is a well-defined, inner product preserving (and hence injective), $B$-bilinear map from $B\\cdot\\xi\\cdot B=X$ onto $B\\cdot \\epsilon\\cdot B =E_\\cG$. In other words, it is a C*-correspondence isomorphism.\n\\end{proof}\n\n\nIt is straightforward to check that any homomorphism $A:B\\to B$ is a quantum adjacency matrix for a quantum space $(B,\\psi)$ (independent, in fact, of $\\psi$). Furthermore, every homomorphism of a C*-algebra that is completely positive must necessarily be $*$-preserving, so it is natural to ask if the one-to-one correspondence in Proposition~\\ref{prop:recognition} restricts to a one-to-one correspondence between $*$-homomorphisms and a subset of $\\{\\xi\\in B\\otimes_\\psi B\\colon \\xi\\#\\xi=\\xi,\\ (\\sigma_{i\/2}^\\psi\\otimes 1)(\\xi)^*=(\\sigma_{i\/2}^\\psi\\otimes 1)(\\xi)\\}.$ \n\n\\begin{proposition}\\label{prop:when_A_is_a_homomorphism}\nLet $(B,\\psi)$ be a finite quantum space, and suppose $A:B\\to B$ is a completely positive quantum adjacency matrix for $(B,\\psi).$ The following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $A$ is a homomorphism.\n\\item For all $x,y\\in B$, we have $(xy)\\cdot \\epsilon_\\cG=x\\cdot \\epsilon_\\cG\\cdot A(y)$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nSuppose for all $x,y\\in B$, we have $(xy)\\cdot \\epsilon_\\cG=x\\cdot \\epsilon_\\cG\\cdot A(y).$ Proposition~\\ref{prop:recognition} implies\n$$\n A(xy)\n =\\langle \\epsilon_\\cG, (xy)\\cdot \\epsilon_\\cG\\rangle_B\n =\\langle \\epsilon_\\cG, x\\cdot \\epsilon_\\cG\\cdot A(y)\\rangle_B\n =\\langle \\epsilon_\\cG, x\\cdot \\epsilon_\\cG\\rangle_B A(y)\n =A(x)A(y)\n$$\nfor all $x,y\\in B.$ Conversely, suppose $A$ is a homomorphism, and fix $x,y\\in B$. For any $a,b\\in B$,\n$$\n\\langle a\\cdot \\epsilon_\\cG \\cdot b, (xy)\\cdot \\epsilon_\\cG\\rangle_B\n=\nb^* A(a^*xy)%\n=\nb^* A(a^*x)A(y)\n=\n\\langle a\\cdot\\epsilon_\\cG\\cdot b, x\\cdot \\epsilon_\\cG\\cdot A(y)\\rangle_B\n$$\nAs elements of the form $a\\cdot \\epsilon_\\cG\\cdot b$ span $E_\\cG$, for any $\\xi\\in E_\\cG$, \n$$\\langle \\xi, (xy)\\cdot \\epsilon_\\cG - x\\cdot \\epsilon_\\cG \\cdot A(y)\\rangle_B=0\\quad \\text{for all}\\;x,y\\in B.$$ In particular, $\\xi_o:=(xy)\\cdot \\epsilon_\\cG - x\\cdot \\epsilon_\\cG \\cdot A(y)$ is an element of $E_\\cG$, so $\\langle \\xi_o,\\xi_o\\rangle_B=0$. By positive-definiteness of $\\langle \\cdot, \\cdot \\rangle_B,$ we may conclude $\\xi_o=0.$ Thus, $xy\\cdot \\epsilon_\\cG=x\\cdot \\epsilon_\\cG\\cdot A(y)$ for all $x,y\\in B.$\n\\end{proof}\n\nWhen $A$ is a $*$-automorphism, the quantum edge correspondence $E_\\cG$ arising from a quantum graph $\\cG:=(B,\\psi,A)$ is the span of $B\\cdot \\epsilon_\\cG$ or the span of $\\epsilon_\\cG\\cdot B$. Indeed, given $x\\cdot \\epsilon_\\cG\\cdot y\\in E_\\cG$ for some $x,y\\in B$, Proposition~\\ref{prop:when_A_is_a_homomorphism} implies that the maps $E_\\cG\\to E_\\cG$ given by $x\\cdot \\epsilon_G\\cdot y\\mapsto xA^{-1}(y)\\cdot \\epsilon_\\cG$ and $x\\cdot \\epsilon_\\cG\\cdot y\\mapsto \\epsilon_\\cG\\cdot A(x)y$ are both the identity map on $E_\\cG.$\n\n\\subsection{Faithfulness and fullness of the quantum edge correspondence}\n\n\nRecall the definitions of faithful and full for C*-correspondences from Section~\\ref{subsec:correspondences_and_CP_algebras}. The next theorem shows that these features for a quantum edge correspondence $E_\\cG$ are determined by the quantum adjacency matrix $A$.\n\n\\begin{theorem}\\label{thm:faithful_full_correspondence}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive, and let $E_\\cG$ be the quantum edge correspondence. If $1_a:=\\sum_{i=1}^{N_a} e_{ii}^{(a)}$ for each $1\\leq a\\leq d$, then\n \\[\n \\{x\\in B\\colon x\\cdot \\xi=0\\ \\forall \\xi\\in E_\\cG\\}= \\left(BA^{\\ast}(B)B\\right)^{\\perp}\n \n \\]\nand\n \\[\n \\overline{\\text{span}}\\_B = \n \n B\\cdot A(B) \\cdot B,\n \\]\n the two-sided ideal of $B$ generated by the range of $A$. \nIn particular, $E_\\cG$ is faithful if and only if $\\ker(A)$ does not contain a central summand of $B$, and $E_\\cG$ is full if and only if $A(B)$ is not orthogonal to a central summand of $B$.\n\\end{theorem}\n\\begin{proof}\nSuppose $x\\cdot \\xi=0$ for all $\\xi\\in E_\\cG$. This means precisely that for all $a,b \\in B$ we have $xa\\epsilon_{\\cG} b=0$, which, on the other hand, is equivalent to $\\langle c \\epsilon_{\\cG} d, xa \\epsilon_{\\cG} b\\rangle=0$ for all $c,d \\in B$. This expression is equal to $\\delta^{-2}d^* A(c^* xa) b=0$, so it is equal to zero if and only if $A(c^* x a)=0$. This time using the inner product on $B$, we see that this is equivalent to $0=\\langle y, A(c^* x a)\\rangle = \\langle c A^{\\ast}(x) a^*, x \\rangle $ for all $y \\in B$. Therefore $x \\in \\left(B A^{\\ast}(B) B\\right)^{\\perp}$.\n\n\n \n \n \n \n\n \n \n\n \n \n \n\nNext, observe that\n\\begin{align*}\n \\overline{\\text{span}}\\langle E_\\cG, E_\\cG\\rangle_B &= \\overline{\\text{span}}\\big\\{\\langle a \\cdot \\epsilon_\\cG \\cdot b, c \\cdot \\epsilon_\\cG \\cdot d\\rangle_B: a,b,c,d \\in B\\big\\} \\\\\n &= \\overline{\\text{span}}\\big\\{\\delta^{-2}b^*A(a^*c)d: a,b,c,d \\in B\\big\\} \\\\\n &=B \\cdot A(B) \\cdot B \n\\end{align*}\n\n\\end{proof}\n\n\\begin{remark}\nIn the case of a classical directed graph $\\cG=( C(V), \\psi, A)$, the central summands of $C(V)$ are indexed by $V$. A central summand belongs to $\\ker(A)$ when the corresponding vertex is a source (i.e. has no edges into it), and it is orthogonal to $A(C(V))$ when the corresponding vertex is a sink (i.e. has no edges out of it). Hence $E_\\cG$ is faithful when $A$ has no zero columns and is full when $A$ has no zero rows.\n\\end{remark}\n\nThe previous remark motivates the following definitions.\n\n\\begin{definition}\\label{def:quantum-sink-and-source}\n Let $\\cG=(B,\\psi, A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive. A \\textit{quantum sink} in $\\cG$ is a central summand of $B$ that is orthogonal to the range of $A$. A \\textit{quantum source} in $\\cG$ is a central summand of $B$ that lies in the kernel of $A$\n\\end{definition}\n\nIn the next section we will examine the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$ associated to the quantum edge correspondence. Consequently, it is important to understand how to express the left $B$-action in terms of compact operators $\\cK(E_\\cG)$.\n\n\\begin{theorem}\\label{thm:compact_operators}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive, and let $E_\\cG=B\\cdot \\epsilon_\\cG\\cdot B$ be the quantum edge correspondence. If $\\{ f_{ij}^{(a)}\\colon 1\\leq a\\leq d,\\ 1\\leq i,j\\leq N_a\\}$ are the adapted matrix units for $(B,\\psi)$, then\n \\[\n f_{ij}^{(a)}\\cdot \\xi = \\sum_{k=1}^{N_a} \\theta_{f_{ik}^{(a)}\\cdot \\epsilon_\\cG,f_{jk}^{(a)}\\cdot \\epsilon_\\cG}(\\xi)\n \\quad\n \\text{for all }\n \\xi\\in E_\\cG.\n \\]\n\n\\end{theorem}\n\\begin{proof}\nSince both sides are right $B$-linear in $\\xi$, it suffices to prove the equality for $\\xi=e_{rs}^{(b)}\\cdot \\epsilon_\\cG$, $1\\leq b\\leq d$ and $1\\leq r,s\\leq N_b$. Write $\\epsilon_\\cG=\\sum_\\alpha x_\\alpha\\otimes y_\\alpha$. Then using Proposition~\\ref{prop:properties_of_edge_generator}.(2) we have\n \\begin{align*}\n f_{ij}^{(a)}\\cdot (e_{rs}^{(b)}\\cdot \\epsilon_\\cG) &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} e_{is}^{(b)}\\cdot \\epsilon_\\cG\\\\\n &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} e_{is}^{(b)}\\cdot (\\epsilon_\\cG\\# \\epsilon_\\cG)\\\\\n &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} \\sum_{k,\\alpha,\\beta} (e_{is}^{(b)}x_\\alpha e_{kk}^{(a)} x_\\beta)\\otimes (y_\\beta y_\\alpha)\\\\\n &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} \\sum_{k,\\alpha,\\beta} (x_\\alpha)_{sk}^{(a)} (e_{ik}^{(a)}x_\\beta)\\otimes (y_\\beta y_\\alpha)\\\\\n &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} \\sum_{k,\\alpha,\\beta} \\frac{1}{\\psi(e_{kk}^{(a)})} \\psi( e_{ks}^{(a)}x_\\alpha)(e_{ik}^{(a)}x_\\beta)\\otimes (y_\\beta y_\\alpha)\\\\\n &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} \\sum_{k} \\frac{1}{\\psi(e_{kk}^{(a)})} e_{ik}^{(a)}\\cdot \\epsilon_\\cG \\cdot (\\psi\\otimes 1)(e_{ks}^{(a)}\\cdot \\epsilon_\\cG).\n \\end{align*}\nNow, using Proposition~\\ref{prop:properties_of_edge_generator}.(1) and Theorem~\\ref{thm:cp_Adjacency_matrices_are_inner_products} we see that\n \\[\n (\\psi\\otimes 1)(e_{ks}^{(a)}\\cdot \\epsilon_\\cG) = \\frac{1}{\\delta^2} A(e_{ks}^{(a)}) = \\< \\epsilon_\\cG, e_{ks}^{(a)}\\cdot \\epsilon_\\cG\\>_B = \\< e_{jk}^{(a)}\\cdot \\epsilon_\\cG , e_{js}^{(a)}\\cdot \\epsilon_\\cG\\>_B.\n \\]\nSo continuing our computation above, we have\n \\begin{align*}\n f_{ij}^{(a)}\\cdot (e_{rs}^{(b)}\\cdot \\epsilon_\\cG) &= \\delta_{\\substack{a=b\\\\ j=r}} [\\psi(e_{ii}^{(a)})\\psi(e_{jj}^{(a)})]^{-1\/2} \\sum_{k} \\frac{1}{\\psi(e_{kk}^{(a)})} e_{ik}^{(a)}\\cdot \\epsilon_\\cG \\cdot \\< e_{jk}^{(a)}\\cdot \\epsilon_\\cG , e_{js}^{(a)}\\cdot \\epsilon_\\cG\\>_B\\\\\n &= \\sum_{k} f_{ik}^{(a)}\\cdot \\epsilon_\\cG \\cdot \\< f_{jk}^{(a)}\\cdot \\epsilon_\\cG, e_{rs}^{(b)}\\cdot \\epsilon_\\cG\\>_B\\\\\n &= \\sum_k \\theta_{f_{ik}^{(a)}\\cdot \\epsilon_\\cG, f_{jk}^{(a)}\\cdot \\epsilon_\\cG}( e_{rs}^{(b)}\\cdot \\epsilon_\\cG),\n \\end{align*}\nas claimed.\n\\end{proof}\n\nThe following corollary is a rephrasing of Theorems~\\ref{thm:cp_Adjacency_matrices_are_inner_products} and \\ref{thm:compact_operators} in terms of linear maps, which will be useful in the next section. We leave the proof to the reader.\n\n\\begin{corollary}\\label{cor:abstract_Toeplitz}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive, and let $E_\\cG=B\\cdot \\epsilon_\\cG\\cdot B$ be the quantum edge correspondence. If $(\\pi,t)$ is a covariant representation of $E_\\cG$ on C*-algebra $D$, then the linear map $T\\colon B\\to D$ defined by $T(x):= t(x\\cdot \\epsilon_\\cG)$ satisfies\n \\begin{align*}\n \\mu_D(T^*\\otimes T) &= \\frac{1}{\\delta^2}\\pi A m,\\\\\n \\mu_D(T\\otimes T^*) m^* &=\\psi_t,\n \\end{align*}\nwhere $\\mu_D\\colon D\\otimes D\\to D$ is the multiplication map, $T^*(x):=T(x^*)^*$, and $\\psi_t\\colon \\cK(E_\\cG)\\to D$ is the $*$-homomorphism induced by $t$.\n\\end{corollary}\n\n\n\n\n\n\n\n\n\\section{Quantum Cuntz--Krieger algebras and local relations}\\label{sec:QCK_and_local_relations}\n\nIn this section we recall the quantum Cuntz--Krieger relations and define local quantum Cuntz--Krieger relations. The former differ slightly from those appearing in \\cite[Section 3.2]{BEVW20} (see Remark~\\ref{rem:unital_relation}). We will see in Theorem~\\ref{thm:universal_local_quantum_Cuntz--Krieger_algebra} below that the Cuntz--Pimsner algebra for a faithful quantum edge correspondence plays the role of the universal C*-algebra generated by local quantum Cuntz--Krieger relations. This in turn allows us to deduce that such Cuntz--Pimsner algebras are quotients of quantum Cuntz--Krieger algebras (see Corollary~\\ref{cor:quantum_Cuntz--Krieger_quotients}).\n\n\\begin{definition}\\label{def:QCK}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph. We define a \\textit{quantum Cuntz--Krieger $\\cG$-family} in a unital C*-algebra $D$ to be a linear map $s\\colon B\\to D$ such that:\n \\begin{enumerate}[label=(\\roman*)]\n \\item $\\mu_D(\\mu_D\\otimes 1)(s\\otimes s^* \\otimes s)(m^* \\otimes 1)m^*=s$ $\\hfill (\\textbf{QCK1})$\n \n \\item $\\mu_D(s^*\\otimes s)m^* = \\mu_D(s\\otimes s^*)m^*A$ $\\hfill (\\textbf{QCK2})$\n \n \\item $\\mu_D (s \\otimes s^*)m^*(1_B) = \\frac{1}{\\delta^2} 1_D$ $\\hfill (\\textbf{QCK3})$\n \\end{enumerate}\nwhere $\\mu_D\\colon D\\otimes D\\to D$ is the multiplication map for $D$ and $s^*(b)=s(b^*)^*$ for $b\\in B$. Then the \\textit{quantum Cuntz--Krieger algebra} associated to $\\cG$ is the universal unital C*-algebra $\\mathcal{O}(\\cG)$ generated by the image of a quantum Cuntz--Krieger $\\cG$-family $S\\colon B\\to \\cO(\\cG)$.\n\\end{definition} \n\nWe show that Definition~\\ref{def:QCK} gives the classical Cuntz--Krieger algebra when $\\cG$ is a classical graph.\n\n\\begin{example}\nLet $\\cG=(B,\\psi, A)$ be a classical graph, i.e., $B=C(V)$ is the finite-dimensional commutative C*-algebra arising from a simple finite directed graph $G=(V, E)$ on $N = |V|$ vertices, $\\psi$ is the normalized trace on $B$, and $A$ is defined by the adjacency matrix $A_G$ on $G$ in the usual way. If $\\left\\{e_1, \\ldots, e_N\\right\\}$ is the canonical basis of minimal projections in $B$, it is easy to see that $\\psi(e_i)=1\/N$, $m(e_i\\otimes e_j)=\\delta_{i=j}e_j$, and $m^*(e_i)=Ne_i\\otimes e_i$ for all $i, j$. Moreover, $\\psi$ is a $\\delta$-form with $\\delta^2=N$.\n\nLet $S:B\\to \\cO(\\cG)$ be a universal quantum Cuntz--Krieger $\\cG$-family and define $S_i=NS(e_i) \\in \\cO(\\cG)$. It was shown in \\cite[Proposition 4.1]{BEVW20} that the $S_i$ are partial isometries satisfying $S_i^*S_i=\\sum_{j=1}^NA_G(i,j)S_jS_j^*$ since $S:B\\to \\cO(\\cG)$ satisfies (\\textbf{QCK1}) and (\\textbf{QCK2}). One can see that the $S_i$ have mutually orthogonal range projections by observing that they sum to the identity in $\\cO({\\cG})$ since $S$ satisfies (\\textbf{QCK3}). Explicitly, observe that\n\\begin{align*}\n \\sum_{i=1}^N S_iS_i^*&=N^2\\left[\\sum_{i=1}^N S(e_i)S^*(e_i)\\right]\\\\\n &=N\\left[\\sum_{i=1}^n \\mu_{\\cO(\\cG)}(S\\otimes S^*)m^*(e_i)\\right]\\\\\n &= N\\left[\\frac{1}{\\delta^2}1_{\\cO(\\cG)}\\right]\n =1_{\\cO(\\cG)}.\n\\end{align*}\n\nThus, the $S_i$ form a Cuntz--Krieger $A_G$-family, which induces a $*$-homomorphism of $\\cO_{A_E}$ onto $\\cO(\\cG)$. \n\nConversely, given a universal Cuntz--Krieger $A_G$ family $\\left\\{S_i\\right\\}$, we can define $s:B \\to \\cO_{A_G}$ via $s(e_i)=\\frac{1}{N}S_i\\in \\cO_{A_G}$. As mentioned in \\cite[Proposition 4.1]{BEVW20}, one can check that $s$ satisfies (\\textbf{QCK1}) and (\\textbf{QCK2}). To see that $s$ satisfies (\\textbf{QCK3}), consider\n\\begin{align*}\n \\mu_{\\cO_{A_G}}(s\\otimes s^*)m^*(1_B) &= \\sum_{i=1}^N\\mu_{\\cO_{A_G}}(s(Ne_i)\\otimes s^*(e_i))\\\\\n &=\\frac{1}{N}\\sum_{i=1}^NS_iS_i^*\\\\\n &=\\frac{1}{N}1_{\\cO_{A_G}}.\n\\end{align*}\nHence, $s$ is a quantum Cuntz--Krieger $\\cG$-family, which induces a $*$-homomorphism of $\\cO(\\cG)$ onto $\\cO_{A_G}$. Checking that this map is the inverse of the previously induced $*$-homomorphism of $\\cO_{A_G}$ onto $\\cO(\\cG)$ yields $\\cO(\\cG)$ is isomorphic to $\\cO_{A_G}.$\n\\end{example}\n\n\n\\begin{remark}\\label{rem:unital_relation}\nIn \\cite{BEVW20}, a notion of quantum Cuntz--Krieger algebras was introduced without the relation $(\\textbf{QCK3})$ (see \\cite[Definition 3.7]{BEVW20}), which gives potentially non-unital, non-nuclear C*-algebras denoted $\\mathbb{F}\\cO(\\cG)$. As discussed in \\cite[Section 4.1]{BEVW20}, when $\\cG$ is a classical graph $\\mathbb{F}\\cO(\\cG)$ is a \\emph{free} Cuntz--Krieger algebra (see \\cite[Definition 2.5]{BEVW20}), whereas by the above example $\\cO(\\cG)$ is a Cuntz--Krieger algebra. Thus we will generally refer to $\\mathbb{F}\\cO(\\cG)$ as the \\emph{free} quantum Cuntz--Krieger algebra, and reserve the terminology ``quantum Cuntz--Krieger algebra'' for $\\cO(\\cG)$.\n\\end{remark}\n\nUsing adapted matrix units, one can produce a more explicit presentation of the quantum Cuntz--Krieger relations. This is essentially the content of \\cite[Proposition 3.9]{BEVW20}. More precisely, if $s_{ij}^{(a)}:=s(f_{ij}^{(a)})$, where $\\{f_{ij}^{(a)}\\colon 1\\leq a \\leq d,\\ 1\\leq i,j\\leq N_a\\}$ are the adapted matrix units for $(B,\\psi)$, then $s:B \\to D$ is a quantum Cuntz--Krieger $\\cG$-family if and only if the following relations hold:\n\n \\begin{align}\n \\sum_{k,\\ell=1}^{N_a} s_{ik}^{(a)}(s_{\\ell k}^{(a)})^*s_{\\ell s}^{(a)} &=s_{is}^{(a)} \\label{eqn:QCK1}\\\\\n \\sum_{\\ell =1}^{N_a}(s_{\\ell i}^{(a)})^*s_{\\ell j}^{(a)} &= \\sum_{c=1}^d\\sum_{\\ell,m,n =1}^{N_c} A_{ija}^{\\ell m c} s_{\\ell n}^{(c)} (s_{m n}^{(c)})^* \\label{eqn:QCK2}\\\\\n \\sum_{c=1}^d \\sum_{\\ell,m=1}^{N_c} \\psi(e_{\\ell,\\ell}^{(c)}) s_{\\ell m}^{(a)} (s_{\\ell m}^{(c)})^* &=\\frac{1}{\\delta^2} 1_D \\label{eqn:QCK3}\n \\end{align}\n\n\nWe now introduce localized versions of the above quantum Cuntz--Krieger relations.\n\n\\begin{definition}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form. We define a \\textit{local quantum Cuntz--Krieger $\\cG$-family} in a unital C*-algebra $D$ to be a linear map $s\\colon B\\to D$ such that\n \\begin{enumerate}[label=(\\roman*)]\n \\item $\\mu_D(\\mu_D \\otimes 1)(s \\otimes s^* \\otimes s)(m^* \\otimes 1) = \\frac{1}{\\delta^2}s m$ $\\hfill(\\textbf{LQCK1})$\n \n \\item $\\mu_D (s^* \\otimes s) = \\frac{1}{\\delta^2} \\mu_D(s \\otimes s^*)m^*Am$ $\\hfill(\\textbf{LQCK2})$\n \n \\item $\\mu_D (s \\otimes s^*)m^*(1_B) = \\frac{1}{\\delta^2} 1_D$ $\\hfill (\\textbf{LQCK3})$\n \n \\end{enumerate}\nwhere $\\mu_D\\colon D\\otimes D\\to D$ is the multiplication map for $D$, $s^*(b)=s(b^*)^*$ for $b\\in B$, and $m^*$ is the adjoint of $m$ with respect to the inner product given by $\\psi$.\n\\end{definition}\nOnce again, using adapted matrix units, one can produce a more explicit presentation of the local quantum Cuntz--Krieger relations. This is the content of the following proposition whose proof is omitted as it is very similar to the proof of \\cite[Proposition 3.9]{BEVW20}.\n\n\\begin{proposition}\\label{prop:explicit_local_relations}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form, and let $s\\colon B\\to D$ be a linear map into a unital C*-algebra. Denote $s_{ij}^{(a)}:=s(f_{ij}^{(a)})$, where $\\{f_{ij}^{(a)}\\colon 1\\leq a \\leq d,\\ 1\\leq i,j\\leq N_a\\}$ are the adapted matrix units for $(B,\\psi)$. Then $s$ is a local quantum Cuntz--Krieger $\\cG$-family if and only if the following relations hold:\n \\begin{align}\n \\sum_{k=1}^{N_a} s_{ik}^{(a)}(s_{jk}^{(a)})^*s_{rs}^{(b)} &=\\delta_{\\substack{a=b \\\\ j=r}} \\frac{1}{\\delta^2\\psi(e_{jj}^{(a)})}s_{is}^{(a)} \\label{eqn:QCP1}\\\\\n (s_{ij}^{(a)})^*s_{rs}^{(b)} &= \\delta_{\\substack{a=b\\\\ i=r}}\\frac{1}{\\delta^2\\psi(e_{ii}^{(a)})}\\sum_{c=1}^d\\sum_{\\ell,m,n =1}^{N_c} A_{jsa}^{\\ell m c} s_{\\ell n}^{(c)} (s_{m n}^{(c)})^* \\label{eqn:QCP2}\\\\\n \\sum_{c=1}^d \\sum_{\\ell,m=1}^{N_c} \\psi(e_{\\ell,\\ell}^{(c)}) s_{\\ell m}^{(a)} (s_{\\ell m}^{(c)})^* &=\\frac{1}{\\delta^2} 1_D \\label{eqn:QCP3}\n \\end{align}\nfor all $1\\leq a,b\\leq d$, $1\\leq i,j\\leq N_a$, and $1\\leq r,s\\leq N_b$.\n\\end{proposition}\n\nJust as in the case of the quantum Cuntz--Krieger algebras above, one can also define, for any quantum graph $\\cG = (B,\\psi, A)$, a corresponding {\\it local quantum Cuntz--Krieger algebra}. The local quantum Cuntz--Krieger algebra is the C$^\\ast$-algebra generated by a universal local quantum Cuntz--Krieger $\\cG$-family.\nThe following theorem is the main result of this section. It shows that local quantum Cuntz--Krieger algebras are in fact familiar objects -- under rather mild assumptions, they are precisely the Cuntz--Pimsner algebras of quantum edge correspondences. \n\n\n\\begin{theorem}\\label{thm:universal_local_quantum_Cuntz--Krieger_algebra}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive, and let $E_\\cG=B\\cdot \\epsilon_\\cG\\cdot B$ be its quantum edge correspondence. Let $(\\pi_{E_\\cG},t_{E_\\cG})$ denote the universal covariant representation of $E_\\cG$ on the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$.\nAssume $\\cG$ has no quantum sources.\nThen $S\\colon B\\to \\mathcal{O}_{E_\\cG}$ defined by $S(x):=\\frac{1}{\\delta} t_{E_{\\cG}}(x\\cdot \\epsilon_\\cG)$ is a local quantum Cuntz--Krieger $\\cG$-family whose image generates $\\cO_{E_\\cG}$. Moreover, given any local quantum Cuntz--Krieger $\\cG$-family $s\\colon B\\to D$ in a unital C*-algebra $D$ there exists a $*$-homomorphism $\\rho\\colon \\mathcal{O}_{E_\\cG}\\to D$ such that $s=\\rho\\circ S$.\n\\end{theorem}\n\\begin{proof}\nWe begin by showing that $S$ is a local quantum Cuntz--Krieger $\\cG$-family. Let $\\psi_{t_{E_\\cG}} \\colon \\cK(E_\\cG)\\to \\cO_{E_\\cG}$ be the $*$-homomorphism induced by $t_{E_\\cG}$, and let $\\mu\\colon \\cO_{E_\\cG}\\otimes \\cO_{E_\\cG}\\to \\cO_{E_\\cG}$ denote the multiplication map. Define a linear map $T\\colon B\\to \\cO_{E_\\cG}$ by $T(x):=t_{E_\\cG}(x\\cdot \\epsilon_\\cG)$, so that $S=\\frac{1}{\\delta} T$. Note that the quantum edge correspondence $E_\\cG$ is faithful by Theorem~\\ref{thm:faithful_full_correspondence} and finite dimensional so that the Katsura ideal $J_{E_\\cG}=B$. Moreover, faithfulness allows us to identify $B\\subset \\cL(E_\\cG)=\\cK(E_\\cG)$, and under this identification we have $\\psi_{t_{E_\\cG}}|_B = \\pi$. With this observation in hand, we can readily verify the local quantum Cuntz--Krieger relations.\n\n\n\n \\begin{itemize}\n \\item[]\\textbf{(LQCK1):} The second equation in Corollary~\\ref{cor:abstract_Toeplitz} implies\n \\begin{align*}\n \\mu(\\mu\\otimes 1)(S\\otimes S^*\\otimes S)(m^*\\otimes 1) &= \\frac{1}{\\delta^3} \\mu(\\mu\\otimes 1)(T\\otimes T^*\\otimes T)(m^*\\otimes 1)\\\\\n &= \\frac{1}{\\delta^3} \\mu(\\psi_{t_{E_\\cG}}\\otimes T)\\\\\n &=\\frac{1}{\\delta^3}\\mu(\\pi\\otimes T)\\\\\n &= \\frac{1}{\\delta^3} Tm= \\frac{1}{\\delta^2} Sm,\n \\end{align*}\n where the second-to-last equality follows from the relation $\\pi_{E_\\cG}(x)t_{E_\\cG}(\\xi) = t_{E_\\cG}(x\\cdot \\xi)$ for $x\\in B$ and $\\xi\\in E_\\cG$.\n \n \\item[]\\textbf{(LQCK2):} Using both equations in Corollary~\\ref{cor:abstract_Toeplitz} gives\n \\begin{align*}\n \\mu(S^*\\otimes S) &= \\frac{1}{\\delta^2} \\mu(T^*\\otimes T) = \\frac{1}{\\delta^4} \\pi A m = \\frac{1}{\\delta^2} \\psi_{t_{E_\\cG}} A m\\\\\n &= \\frac{1}{\\delta^4} \\mu(T\\otimes T^*)m^* Am = \\frac{1}{\\delta^2} \\mu(S\\otimes S^*)m^*Am.\n \\end{align*}\n \n \\item[]\\textbf{(LQCK3):} Using the second equation in Corollary~\\ref{cor:abstract_Toeplitz} we have\n \\[\n \\mu(S\\otimes S^*)m^*(1_B) = \\frac{1}{\\delta^2} \\mu(T\\otimes T^*)m^*(1_B) = \\frac{1}{\\delta^2} \\pi_{E_{\\cG}}(1_B) = \\frac{1}{\\delta^2} 1.\n \\]\n Note that $\\pi_{E_{\\cG}}$ is necessarily unital since $1\\cdot \\xi=\\xi$ for all $\\xi\\in E_{\\cG}$.\n \\end{itemize}\nThus $S$ is a local quantum Cuntz--Krieger $\\cG$-family. We also have $C^*(S(B))=\\cO_{E_\\cG}$. Indeed, $\\pi_{E_\\cG}(B)\\subset C^*(S(B))$ by Corollary~\\ref{cor:abstract_Toeplitz}, and $t_{E_\\cG}(\\xi)\\pi_{E_\\cG}(b)=t_{E_\\cG}(\\xi\\cdot b)$ implies $t_{E_\\cG}(E_\\cG)\\subset C^*(S(B))$.\n\nNow, suppose $s\\colon B\\to D$ is a local quantum Cuntz--Krieger $\\cG$-family in a unital C*-algebra $D$. To obtain the desired homomorphism, we will construct a covariant representation of $E_\\cG$ on $D$ and invoke the universal property of the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$. Define a linear map $\\pi\\colon B\\to D$ by\n \\[\n \\pi := \\delta^2 \\mu_D(s\\otimes s^*)m^*,\n \\]\nwhere $\\mu_D\\colon D\\otimes D\\to D$ is the multiplication map. Then $\\pi$ is unital by (\\textbf{LQCK3}), $\\pi^*=\\pi$ in light of how the multiplication maps interact with the adjoint, and (\\textbf{LQCK1}) implies\n \\begin{align*}\n \\mu_D(\\pi\\otimes \\pi) &= \\delta^4 \\mu_D(\\mu_D\\otimes \\mu_D)(s\\otimes s^*\\otimes s\\otimes s^*)(m^*\\otimes m^*)\\\\\n &= \\delta^4 \\mu_D( \\left[\\mu_D(\\mu_D\\otimes 1)(s\\otimes s^*\\otimes s)(m^*\\otimes 1)\\right]\\otimes s^*)(1\\otimes m^*)\\\\\n &= \\delta^2 \\mu_D( (sm)\\otimes s^*)(1\\otimes m^*)\\\\\n &= \\delta^2 \\mu_D( s\\otimes s^*)m^*m = \\pi m.\n \\end{align*}\nHence $\\pi$ is a $*$-homomorphism. \n\nNext define a linear map $t\\colon E_\\cG\\to D$ by\n \\[\n t(x\\cdot \\epsilon_\\cG\\cdot y):=\\delta s(x)\\pi(y) \\qquad \\qquad x,y\\in B.\n \\]\nWe will see below that $t(\\xi)^*t(\\eta)=\\pi(\\<\\xi,\\eta\\>_B)$ for $\\xi,\\eta\\in B$, which in particular will show this map is well-defined. By (\\textbf{LQCK1}) we have\n \\begin{align*}\n \\mu_D(\\mu_D\\otimes 1)(\\pi\\otimes s\\otimes \\pi) &= \\delta^2 \\mu_D(\\mu_D\\otimes 1)(\\mu_D\\otimes 1\\otimes 1)(s\\otimes s^*\\otimes s\\otimes \\pi)(m^*\\otimes 1\\otimes 1)\\\\\n &= \\mu_D( s\\otimes \\pi)(m\\otimes 1).\n \\end{align*}\nThus for $x,y,z\\in B$ we have\n \\begin{align*}\n \\pi(x)t(y\\cdot \\epsilon_\\cG\\cdot z) &= \\delta \\mu_D(\\mu_D\\otimes 1)(\\pi\\otimes s\\otimes \\pi)(x\\otimes y\\otimes z)\\\\\n &= \\delta \\mu_D(s\\otimes \\pi)(m\\otimes 1)(x\\otimes y \\otimes z) = s(xy)\\pi(z) = t(xy\\cdot \\epsilon_\\cG\\cdot z).\n \\end{align*}\nand so $\\pi(x)t(\\xi)=t(x\\cdot \\xi)$ for all $x\\in B$ and $\\xi\\in E_\\cG$. Using (\\textbf{LQCK2}) and the definition of $\\pi$, we have\n \\[\n \\mu_D(s^*\\otimes s) = \\frac{1}{\\delta^2} \\mu_D(s\\otimes s^*)m^* Am = \\frac{1}{\\delta^4} \\pi A m.\n \\]\nThus for $x,x',y,y'\\in B$ we have\n \\begin{align*}\n t(x\\cdot \\epsilon_\\cG \\cdot y)^* t(x'\\cdot \\epsilon_\\cG \\cdot y') &= \\delta^2 (\\mu_D\\otimes \\mu_D)(1\\otimes \\mu_D\\otimes 1)(\\pi^*\\otimes s^*\\otimes s\\otimes \\pi)(y^*\\otimes x^*\\otimes x'\\otimes y')\\\\\n &=\\frac{1}{\\delta^2} (\\mu_D\\otimes \\mu_D)(\\pi \\otimes (\\pi A)\\otimes \\pi)(1\\otimes m\\otimes 1)(y^*\\otimes x^*\\otimes x'\\otimes y')\\\\\n &= \\frac{1}{\\delta^2} \\pi( y^* A(x^*x') y')\\\\\n &= \\pi\\left(y^* \\_B y'\\right) = \\pi\\left(\\< x\\cdot \\epsilon_\\cG\\cdot y, x'\\cdot \\epsilon_\\cG\\cdot y'\\>_B \\right),\n \\end{align*}\nwhere the second-to-last equality follows from Theorem~\\ref{thm:cp_Adjacency_matrices_are_inner_products}. Thus $(\\pi,t)$ is a representation of $E_\\cG$ on $D$, and Theorem~\\ref{thm:compact_operators} implies it is covariant:\n \\begin{align*}\n \\psi_t(f_{ij}^{(a)}) &= \\sum_{k=1}^{N_a} \\psi_t\\left( \\theta_{f_{ik}^{(a)}\\cdot \\epsilon_\\cG, f_{jk}^{(a)}\\cdot \\epsilon_\\cG} \\right)\\\\\n &=\\sum_{k=1}^{N_a} t(f_{ik}^{(a)}\\cdot \\epsilon_\\cG) t( f_{jk}^{(a)}\\cdot \\epsilon_\\cG)^*\\\\\n &= \\delta^2 \\mu_D(s\\otimes s^*)m^*(f_{ij}^{(a)}) = \\pi(f_{ij}^{(a)}).\n \\end{align*}\nThus the universal property for the Cuntz--Pimsner algebra $\\cO_{E_\\cG}$ implies there is a $*$-homomorphism $\\rho\\colon \\cO_{E_\\cG}\\to D$ satisfying $\\pi=\\rho\\circ \\pi_{E_\\cG}$ and $t=\\rho\\circ t_{E_\\cG}$. In particular, we have\n \\[\n s(x) = \\frac{1}{\\delta} t(x\\cdot \\epsilon_\\cG) = \\frac{1}{\\delta} \\rho\\circ t_{E_\\cG}(x\\cdot \\epsilon_\\cG) = \\rho(S(x)).\n \\]\nHence $s=\\rho\\circ S$.\n\\end{proof}\n\n\n\n\n\n\nObserve that any local quantum Cuntz--Krieger $\\cG$-family is a (non-local) quantum Cuntz--Krieger $\\cG$-family. Indeed, since $\\psi$ is a $\\delta$ form one has $mm^*=\\delta^2 \\text{id}$. Thus if $s\\colon B\\to D$ satisfies (\\textbf{LQCK1}) and (\\textbf{LQCK2}), then applying $m^*$ to the right-hand sides of these relations yields (\\textbf{QCK1}) and (\\textbf{QCK2}), respectively. Also (\\textbf{LQCK3}) and (\\textbf{QCK3}) are identical. Hence the universal property for $\\mathcal{O}(\\cG)$ yields a unique $*$-homomorphism onto $C^*(s(D))$. In particular, if $\\ker(A)$ does not contain a central summand of $B$, then the previous theorem yields the following:\n\n\\begin{corollary}\\label{cor:quantum_Cuntz--Krieger_quotients}\nLet $\\cG=(B,\\psi,A)$ be a directed quantum graph such that $\\psi$ is a $\\delta$-form and $A$ is completely positive, and let $E_\\cG$ be its quantum edge correspondence. Assume that $\\ker(A)$ does not contain a central summand of $B$. Then $\\mathcal{O}_{E_\\cG}\\cong \\mathcal{O}(\\cG)\/\\mathcal{I}$ where $\\mathcal{I}\\triangleleft \\mathcal{O}(\\cG)$ is the closed two-sided ideal generated by the relations (\\textbf{LQCK1}), (\\textbf{LQCK2}), and (\\textbf{LQCK3}).\n\\end{corollary}\n\n\\subsection{Behavior of $\\cO_{E_\\cG}$ under quantum graph isomorphisms}\nIn this section we briefly examine the relationship between the Cuntz--Pimsner algebras of quantum edge correspondences associated to quantum isomorphic quantum graphs. We begin by recalling the notion of quantum isomorphism from \\cite{BCEHPSWCMP19, BEVW20}. Let $\\cG_i=(B_i,\\psi_i,A_i)$, $i=1,2$, be directed quantum graphs. We say that $\\cG_1$ and $\\cG_2$ are quantum isomorphic if there exists a Hilbert space $\\cH$ and a unital $\\ast$-homomorphism \\[\\theta_1:B_1 \\to B_2 \\otimes B(\\cH)\\]\nwhich satisfies the following $\\psi_i$ and $A_i$ covariance conditions\n \\begin{align*}\n (\\psi_2 \\otimes \\text{id})\\theta_1 &=\\psi_1(\\cdot)1_{B(\\cH)}, \\\\\n (A_2 \\otimes \\text{id})\\theta_1 &= \\theta_1 \\circ A_1.\n \\end{align*}\nNote that the general theory guarantees that whenever a morphism $\\theta_1:B_1 \\to B_2 \\otimes B(\\cH)$ exists, there automatically exists a corresponding morphism $\\theta_2:B_2 \\to B_1 \\otimes B(\\cH)$ with analogous covariance conditions to those for $\\theta_1$. In particular, the notion of quantum isomorphism is symmetric in $\\cG_1$ and $\\cG_2$. \n\nQuantum isomorphisms between quantum graphs can be seen as relaxations (or generalizations) of the notion of an isomorphism between quantum graphs. Indeed, in the special case where $\\cH = \\bC$, a quantum isomorphism between $\\cG_1$ and $\\cG_2$ defines an ordinary isomorphism of quantum graphs. \n\nThe following theorem should be compared with \\cite[Theorem 6.13]{BEVW20}, which is a similar result in the context of (free) quantum Cuntz--Krieger algebas. Note, however, that the conclusion of the following theorem is slightly stronger in that one gets {\\it injective} morphisms $\\Theta_i$ between Cuntz--Pimsner algebras below (compare with \\cite[Remark 6.14]{BEVW20}). \n\n\\begin{theorem}\nLet $\\cG_i=(B_i,\\psi_i,A_i)$, $i=1,2$, be directed quantum graphs. If $\\cG_1$ is quantum isomorphic to $\\cG_2$ with $*$-homomorphisms\n \\[\n \\theta_1\\colon B_1\\to B_2\\otimes B(\\cH)\\qquad \\text{and} \\qquad \\theta_2\\colon B_2\\to B_1\\otimes B(\\cH),\n \\]\n as above, then there exist injective $*$-homomorphisms\n \\[\n \\Theta_1\\colon \\cO_{E_{\\cG_1}}\\to \\cO_{E_{\\cG_2}}\\otimes B(\\cH) \\qquad \\text{and} \\qquad \\Theta_2\\colon \\cO_{E_{\\cG_2}}\\to \\cO_{E_{\\cG_1}}\\otimes B(\\cH)\n \\]\nsatisfying\n \\[\n \\Theta_1\\pi_{E_{\\cG_1}} = (\\pi_{E_{\\cG_2}}\\otimes 1)\\theta_1 \\qquad \\text{and}\\qquad \\Theta_2\\pi_{E_{\\cG_2}} = (\\pi_{E_{\\cG_1}}\\otimes 1)\\theta_2.\n \\]\n\\end{theorem}\n\\begin{proof}\nUsing Corollary~\\ref{cor:quantum_graph_correspondence_as_cp_correspondence}, we can work with $B_i\\otimes_{A_i} B_i$ rather than $E_{\\cG_i}$, $i=1,2$. Denote $\\tilde{A}_2:=A_2\\otimes \\text{id}_{B(\\cH)}$, which is a completely positive map, and $X:=(B_2\\otimes B(\\cH))\\otimes_{\\tilde{A}_2} (B_2\\otimes B(\\cH))$. Note that $X\\cong (B_2\\otimes_{A_2} B_2)\\otimes B(\\cH)$, and so $\\cO_X \\cong \\cO_{B_2\\otimes_{A_2} B_2}\\otimes B(\\cH)$ by \\cite[Example 6.4]{Mor17}. \n\nNow, for $x,y\\in B_1$ and $\\xi\\in B_1\\otimes B_1$ we have\n \\[\n (\\tilde{A}_2\\otimes 1)( (\\theta_1\\otimes \\theta_1)(\\xi)^* (\\theta_1\\otimes \\theta_1)(\\xi) ) = (\\theta_1\\otimes \\theta_1)(A_1\\otimes 1)(\\xi^*\\xi).\n \\]\nIt follows that $\\theta_1\\otimes \\theta_1$ induces a linear map $T\\colon B_1\\otimes_{A_1} B_1\\to X$ satisfying\n \\[\n T(x\\cdot \\xi \\cdot y) = \\theta_1(x)\\cdot T(\\xi)\\cdot \\theta_1(y) \\qquad x,y\\in B_1,\\ \\xi\\in B_1\\otimes_{A_1} B_1,\n \\]\nand\n \\[\n \\_{B_2\\otimes B(\\cH)} = \\theta_1(\\<\\xi, \\eta\\>_{B_1}) \\qquad \\xi,\\eta\\in B_1\\otimes_{A_1} B_1.\n \\]\nThus if $(\\pi_X,t_X)$ is the universal covariant representation of $X$ (note that we can take $\\pi_X:= \\pi_{E_{\\cG_2}}\\otimes 1$ and $t_X:=t_{E_{\\cG_2}}\\otimes 1$), then $(\\pi_X\\circ \\theta_1, t_X\\circ T)$ is a covariant representation of $B_1\\otimes_{A_1} B_1$ on $\\cO_X$. One easily checks that this representation is injective and admits a gauge action, and so \\cite[Theorem 6.4]{Kat04} implies there is an injective $*$-homomomorphism $\\Theta_1\\colon \\cO_{B_1\\otimes_{A_1} B_1}\\to \\cO_X$. Reversing the roles of $B_1$ and $B_2$ yields $\\Theta_2$.\n\\end{proof}\n\n\\section{Examples}\\label{sec:simpler_examples}\n\nIn this section we consider three common types of quantum graphs and determine the isomorphism classes of the Cuntz--Pimsner algebras associated to their quantum edge correspondences. Notably, several of these examples can be realized as Exel crossed products associated to natural Exel systems. We refer the reader to \\cite{Exel03-2}, \\cite{Exel03}, and \\cite{Exel04} for details but provide a brief summary of the construction of Exel crossed products below.\n\nLet $\\cC$ be a C*-algebra and $\\alpha:\\cC \\to \\cC$ be $\\ast$-endomorphism. A {\\em transfer operator} for $(\\cC,\\alpha)$ is a positive linear map on $\\cC$ such that $\\mathcal{L}(\\alpha(f)g)=f\\mathcal{L}(g)$ for all $f,g\\in \\cC$. The Exel crossed product is obtained by first constructing a Toeplitz algebra $\\cT(\\cC,\\alpha,\\mathcal{L})$ which is the universl C$^\\ast$-algebra generated by a copy of $\\cC$ along with an element $S$ such that $Sf=\\alpha(f)S$ and $\\mathcal{L}(f)=S^*fS$ for all $ f\\in \\cC.$ Two elements $f,k\\in \\cT(\\cC,\\alpha,\\mathcal{L})$ form a {\\em redundancy} if $f\\in \\cC, k\\in \\overline{\\cC SS^*\\cC},$ and $fgS=kgS$ for all $g\\in \\cC.$ The {\\em Exel crossed product} $\\cC\\rtimes_{\\alpha,\\mathcal{L}} \\mathbb{N}$ is the quotient of $\\mathcal{T}(\\cC,\\alpha,\\mathcal{L})$ by the ideal generated by differences $f-k$ of redundancies $(f,k)$. \n\nGiven an $n\\times n$ $\\{0,1\\}$-matrix $A$, one can construct the Markov subshift space of infinite paths in the graph associated to $A$, denoted by $X_A$. $X_A$ is a compact space, and admits a natural shift action $\\sigma:X_A \\to X_A$ given by $\\sigma (x_1,x_2,x_3, \\ldots ) = (x_2,x_3, \\ldots)$. A nice class of commutative Exel systems $(C(X_A),\\alpha,\\cL)$ arises from this construction, where $\\alpha(f) = f \\circ \\sigma$ for $f \\in C(X_A)$ and \\[\\mathcal L(f) (x) = |\\{y \\in X_A \\ | \\ \\sigma(y) = x\\}|^{-1} \\Big(\n\\sum_{\\{y \\in X_A \\ | \\ \\sigma(y) = x\\}}f(y)\\Big).\\] It is known that the Cuntz--Krieger algebra $\\cO_A$ is isomorphic to the Exel crossed product $ C(X_A)\\rtimes_{\\alpha,\\cL} \\bN$. Below, we explore the analogous relationship between noncommutative Exel systems and local quantum Cuntz--Krieger algebras.\n\n\\subsection{Complete quantum graphs}\\label{subsection:completequantumgraphs}\nGiven a finite quantum space $(B,\\psi)$ with $\\delta$-form $\\psi$, we denote by $K(B,\\psi)$ the {\\em complete quantum graph} $(B,\\psi, A)$, whose quantum adjacency matrix $A$ is given by the rank-one map $A=\\delta^2\\psi(\\cdot)1_B$ \\cite{BCEHPSWCMP19, BEVW20}. The quantum edge correspondence for $K(B,\\psi)$ is $E_K=B\\otimes B$ since an adapted matrix unit computation yields that the quantum edge indicator is $\\epsilon_K=1_B\\otimes 1_B.$ The quantum Cuntz--Krieger algebra $ \\cO(K(B,\\psi))$ is generated by $\\{\nS_{ij}^{(a)}: 1 \\le a \\le d, \\ 1 \\le i,j \\le N_a\\}$\naccording to the relations\n\\begin{align} \n\\label{QK1}\n&(S^{(a)})^*S^{(a)} \n= \n1 \\quad \\text{in } M_{N_a}\\otimes \\mathcal{O}(K(B,\\psi)) \\qquad (1 \\le a \\le d)\\\\\n\\label{QK2}\n& \\sum\\limits_{ars} \\delta^2 \\psi(e_{rr}^{(a)})S_{rs}^{(a)}S_{rs}^{(a)*}\n=\n1 \\quad \\text{in } \\mathcal{O}(K(B,\\psi))\n\\end{align}\n\nIn \\cite[Theorem 4.5]{BEVW20}, the authors establish that their definition of a quantum Cuntz--Krieger algebra associated to $K(B,\\psi)$ is isomorphic to $\\cO_{\\dim B}$ whenever $\\delta^2 \\in \\mathbb N$. It is unknown if this also holds for $\\delta^2\\not\\in \\mathbb{N}$.\nThe following proposition resolves this for local quantum Cuntz--Krieger algebras.\n\n\\begin{proposition} \\label{prop:Cuntz--Pimsner-is-a-Cuntz-algebra}\n$\\cO_{E_K}$ is isomorphic to the Cuntz algebra $\\cO_{n}$ where $n = \\dim B$. \n\\begin{proof}\nBy Theorem \\ref{thm:universal_local_quantum_Cuntz--Krieger_algebra} and Proposition \\ref{prop:explicit_local_relations}, the Cuntz--Pimsner algebra $\\mathcal{O}_{E_{K}}$ is a quotient of the quantum Cuntz--Krieger algebra $\\mathcal{O}(K(B,\\psi))$ subject to relations (\\ref{eqn:QCP1}), which remains the same, and (\\ref{eqn:QCP2}), which in this case becomes: \n\n\\begin{align}\n \n \n \n \n (S_{ij}^{(a)})^*(S_{rs}^{(b)})\n &=\\delta_{\\substack{a=b\\\\ i=r}}\n \\frac{1}{\\delta^2 \\psi(e_{ii}^{(a)})}\n \\sum\\limits_{c\\ell m}\n \\left[\n \\delta^2 \\delta_{\\substack{\\ell=m\\\\ j=s}}\\psi(e_{\\ell \\ell}^{(c)})\n \\sum\\limits_n S_{\\ell n}^{(c)}(S_{mn}^{(c)})^*\n \\right].\\label{eq:2}\n\\end{align}\nSimplifying equation \\eqref{eq:2} using the unit relation (\\ref{eqn:QCP3}), we get\n\n\\[\n(S_{ij}^{(a)})^*(S_{rs}^{(b)})=\\frac{\\delta_{\\substack{a=b\\\\ i=r\\\\ j=s}}}{\\delta^2 \\psi(e_{ii}^{(a)})}\\left(\\sum\\limits_{c\\ell n}\\delta^2\\psi(e_{\\ell \\ell}^{(c)}) S_{\\ell n}^{(c)}(S_{\\ell n}^{(c)})^*\\right) = \\frac{\\delta_{\\substack{a=b\\\\ i=r\\\\ j=s}}}{\\delta^2 \\psi(e_{ii}^{(a)})}(1_{\\mathcal{O}(E_{K})}).\n\\]\nThus, $\\left\\{\\delta\\psi(e_{ii}^{(a)})^{1\/2}S_{ij}^{(a)}:1\\leq a\\leq d, 1\\leq i,j\\leq N_a\\right\\}$ are generating isometries for $\\mathcal{O}_{E_{K}}$ whose range projections sum to the identity, and it follows that $\\mathcal{O}_{E_{K}}$ is isomorphic to the Cuntz algebra $\\mathcal{O}_{\\dim B}.$\n\\end{proof}\n\\end{proposition}\n\nHere, we define a particular noncommutative Exel system $(B^{\\otimes \\bN}, \\alpha, \\mathcal{L})$ and show that the Exel crossed product $B^{\\otimes \\bN}\\rtimes _{\\alpha,\\mathcal{L}}\\bN$ is the Cuntz algebra on $\\dim B$ generators. Define $\\alpha:B^{\\otimes \\bN}\\to B^{\\otimes \\bN}$ by \\[\\alpha(f):=1_B\\otimes f \\qquad (f\\in B^{\\otimes \\bN}).\\] Then $\\mathcal{L}:B^{\\otimes \\bN} \\to B^{\\otimes \\bN}$ given by \\[\\mathcal{L}(f_1\\otimes g):=\\psi(f_1)g, \\qquad (f_1 \\in B, g \\in B^{\\otimes \\bN}),\\] is a transfer operator for the dynamical system $(B^{\\otimes \\bN},\\alpha)$. In the rest of this section, we let $1$ denote the identity in $B^{\\otimes \\bN}$ and continue to denote the identity in $B$ by $1_B$. For each $1\\leq a \\leq d$ and $1\\leq i,j \\leq N_a$, define $E_{ij}^{(a)}:=e_{ij}^{(a)}\\otimes 1$, and let $S$ denote the operator which arises in the construction of $\\cT(B^{\\otimes \\bN},\\alpha,\\mathcal{L})$. We will show that this particular Exel crossed product is $\\mathcal{O}_{\\dim B}$, generated by $\\{v_{ij}^{(a)}:1\\leq a\\leq d, 1\\leq i,j\\leq N_a\\}$, where $$v_{ij}^{(a)}:=\\frac{1}{\\psi(e_{jj}^{(a)})^{-1\/2}}E_{ij}^{(a)}S.$$ \nThe $v_{ij}^{(a)}$ should be regarded as an analogue of the characteristic function of words in $X_A$ that ``begin with the quantum letter $e_{ij}^{(a)} \\in B$.''\n\n\\begin{lemma}\\label{lem:Complete_graph_Exel_redundancies}\nFor all $1\\leq a\\leq d$, $1\\leq i,j \\leq N_a$, the pair $\\left(E_{ij}^{(a)}, \\sum_{p} v_{ip}^{(a)}{v_{jp}^{(a)}}^*\\right)$ is a redundancy.\n\\end{lemma}\n\\begin{proof}\nFix $a\\in \\{1,...,d\\}$ and $i,j\\in \\{1,...,N_a\\}$. For $e_{k\\ell}^{(b)}\\otimes f\\in B^{\\otimes \\bN},$ note $v_{jp}^{(a)*}(e_{k\\ell}^{(b)}\\otimes f)=0$ unless $a=b$ and $j=k.$ Thus, it suffices to consider elements of the form $e_{j\\ell}^{(a)}\\otimes f$ in $B^{\\otimes \\bN}$. Observe:\n\\begin{align*}\n \\psi(e_{pp}^{(a)})(v_{ip}^{(a)}{v_{jp}^{(a)*}})(e_{j\\ell}^{(a)}\\otimes f)S\n \n &=E_{ip}^{(a)}SS^*[E_{pj}^{(a)}(e_{j\\ell}^{(a)}\\otimes f)]S\\\\\n &=\n E_{ip}^{(a)}S[S^*(e_{p\\ell}^{(a)}\\otimes f)S]\\\\\n &=\n E_{ip}^{(a)}S\\mathcal{L}(e_{p\\ell}^{(a)}\\otimes f)\\\\\n &=\n \\psi(e_{p\\ell}^{(a)})E_{ip}^{(a)}Sf\\\\\n &=\n \\psi(e_{p\\ell}^{(a)})E_{ip}^{(a)}\\alpha(f)S\\\\\n &=\n \\psi(e_{p\\ell}^{(a)})E_{ip}^{(a)}(1_B\\otimes f)S\\\\\n &=\n \\psi(e_{pp}^{(a)})\\delta_{p=\\ell}(e_{i\\ell}^{(a)}\\otimes f)S\\\\\n &=\n \\psi(e_{pp}^{(a)})\\delta_{p=\\ell}(e_{ij}^{(a)}e_{j\\ell}^{(a)}\\otimes f)S\\\\\n &= \n \\psi(e_{pp}^{(a)})\\delta_{p=\\ell}E_{ij}^{(a)}(e_{j\\ell}^{(a)}\\otimes f)S\n\\end{align*}\nAs elements of the form $e_{k\\ell}^{(b)}\\otimes f$ span $B^{\\otimes \\bN}$, for all $g\\in B^{\\otimes \\bN},$ we have \n$$\n\\left(\\sum_{p} v_{ip}^{(a)}{v_{jp}^{(a)}}^*\\right)gS\n=\\sum_{p} \\left(v_{ip}^{(a)}{v_{jp}^{(a)}}^*gS\\right)\n=\\sum_p \\left(\\delta_{p\\ell} E_{ij}^{(a)}gS\\right)\n=E_{ij}^{(a)}gS.\n$$\nBy definition, $\\left(E_{ij}^{(a)},\\sum_{p} v_{ip}^{(a)}{v_{jp}^{(a)}}^*\\right)$ is a redundancy.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:crossed-product-is-a-Cuntz-algebra}\nThe Exel crossed product $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\mathcal{L}}\\mathbb{N}$ is isomorphic to the Cuntz algebra $\\cO_n$ where $n=\\dim B$.\n\\end{theorem}\n\\begin{proof}\nWe first show that the C*-algebra generated by $\\{v_{ij}^{(a)}:1\\leq a\\leq d, 1\\leq i,j\\leq N_a\\}$ is $\\mathcal{O}_n,$ where $n=\\dim B.$ Observe that each $v_{ij}^{(a)}$ is an isometry in $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\mathcal{L}}\\mathbb{N}$:\n$$\n {v_{ij}^{(a)}}^*v_{ij}^{(a)}\n \n =\\frac{1}{\\psi(e_{jj}^{(a)})}S^*E_{ji}^{(a)}E_{ij}^{(a)}S\n \n =\\frac{1}{\\psi(e_{jj}^{(a)})}S^*E_{jj}^{(a)}S\n \n =\\frac{1}{\\psi(e_{jj}^{(a)})}\\mathcal{L}\\left(E_{jj}^{(a)}\\right\n \n \n =1.\n$$\nBy Lemma~\\ref{lem:Complete_graph_Exel_redundancies}, $\\sum_p v_{ip}^{(a)}{v_{ip}^{(a)}}^*=E_{ii}^{(a)}$, and thus, summing over $1\\leq a\\leq d$ and $1\\leq i\\leq N_a$ yields the identity. Therefore, each $v_{ij}^{(a)}$ is a Cuntz isometry, and the collection $\\{v_{ij}^{(a)}:1\\leq a\\leq d, 1\\leq i,j\\leq N_a\\}$ generates the Cuntz algebra $\\cO_n$ where $n=\\dim B$.\n\nTo verify that this copy of $\\cO_{\\dim B}$ inside $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\cL}\\bN$ is actually all of $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\cL}\\bN$, note that each $E_{ij}^{(a)}$ belongs to $\\cO_{\\dim B}$ by Lemma~\\ref{lem:Complete_graph_Exel_redundancies}. Further, since $S=1_BS=(\\sum_{a,i} E_{ii}^{(a)})S$, we also know that $S$ belongs to this copy of $\\cO_{\\dim B}$. We claim that elements of the form $E_{ij}^{(a)}$ along with $S$ generate all of $B^{\\otimes \\bN}$. Recall from \\cite{Exel03} that $(\\alpha(f),SfS^*)$ is a redundancy for all $f\\in B^{\\otimes \\bN}$, so for any $r\\in \\bN$, we have\n$$\n\\underbrace{1_B\\otimes ...\\otimes 1_B}_{r} \\otimes \\, e_{ij}^{(a)} \\otimes 1 \n= \\alpha^r(E_{ij}^{(a)})\n=S^rE_{ij}^{(a)}(S^*)^r\n\\in \\cO_{\\dim B}.\n$$\nAs elements of the above form generate the C*-algebra $B^{\\otimes \\bN}$ and are contained in $\\cO_{\\dim B}$, we may conclude $B^{\\otimes \\bN}$ is a subset of $\\mathcal{O}_{\\dim{B}}.$ Therefore, $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\cL}\\bN \\cong \\cO_n$ where $n=\\dim B.$\n\\end{proof}\n\nAs a result of Theorem~\\ref{thm:crossed-product-is-a-Cuntz-algebra}, the Cuntz--Pimsner algebra $\\cO_{E_K}$ for the quantum edge correspondence of the complete quantum graph can be realized as the Exel crossed product $B^{\\otimes \\bN}\\rtimes_{\\alpha,\\cL}\\bN$, whose underlying Exel system was a natural choice for the complete quantum graph. We may also conclude that the Cuntz algebra on $\\dim B$ generators is always a quotient of the quantum Cuntz-Krieger algebra $\\cO(K(B,\\psi))$ for the complete quantum graph.\n\n\n\\subsection{Trivial quantum graphs}\nDenote by $T(B,\\psi)$ the {\\it trivial quantum graph} $(B,\\psi, \\text{id}) $; in the following discussion $B$ and $\\psi$ will be fixed, so we will simply denote it by $T$. Using the adapted matrix units, we will now identify the edge correspondence and prove that it is isomorphic to the trivial $B$-correspondence, which will allow us to explicitly compute its Cuntz--Pimsner algebra. Recall the definition of the edge correspondence from \\Cref{def:edgecorrespondence}. It is generated, as a $B$-bimodule by the idempotent $\\epsilon_{T}:= \\frac{1}{\\delta^2} (\\text{id} \\otimes \\text{id})m^{\\ast}(1)=\\frac{1}{\\delta^2} m^{\\ast}(1)$. As $m^{\\ast}$ is a $B$-bimodule map, we get $x\\cdot \\epsilon_{T}\\cdot y = \\frac{1}{\\delta^2} m^{\\ast}(xy)$, hence the edge correspondence is equal as a vector space to the image of $m^{\\ast}$. We will actually show that the map $\\frac{1}{\\delta}m^{\\ast}$ identifies $B$ and the edge correspondence $E_{T}$ as $C^{\\ast}$-correspondences.\n\\begin{proposition}\nLet $B$ be viewed as a $C^{\\ast}$-correspondence, where $\\langle a, b\\rangle_B =a^{\\ast}b$ and the left and right actions are the usual ones. Then $\\frac{1}{\\delta} m^{\\ast}: B \\to E_{T}$ is an isomorphism of $C^{\\ast}$-correspondences.\n\\end{proposition}\n\\begin{proof}\nWe will use \\Cref{prop:recognition}. The vector $\\xi:=\\frac{1}{\\delta}1$ is cyclic for $B$ and $\\delta^2 \\langle \\xi, x\\cdot \\xi\\rangle = x$ is the quantum adjacency matrix of the trivial quantum graph. Therefore by \\Cref{prop:recognition} the assignment $\\xi \\mapsto \\epsilon_{T}$ extends to an isomorphism of correspondences $B$ and $E_{T}$. This map is precisely equal to $\\frac{1}{\\delta}m^{\\ast}$.\n\\end{proof}\n\\begin{corollary}\\label{cor:triv}\nThe Cuntz--Pimsner algebra $\\cO_{E_{T}}$ is isomorphic to $B\\otimes C(\\bT)$.\n\\end{corollary}\n\\begin{proof}\nFrom the previous proposition, $E_T$ is isomorphic to $B$. By \\cite[Example 3 p. 193]{Pimsner}, the Cuntz--Pimsner algebra of $B$ is isomorphic to the crossed of $B$ by the trivial action of $\\mathbb{Z}$, i.e. $B\\otimes C(\\bT)$.\n\\end{proof}\n\nConsider the trivial Exel system $(B,\\alpha,\\mathcal{L})$ where $\\alpha=\\mathcal{L}=\\text{id}$. Recall that $\\mathcal{T}(B,\\alpha,\\mathcal{L})$ is the universal C*-algebra generated by $B$ and an element $U$ subject to the relations $Ux=\\alpha(x)U$ and $U^*xU=\\mathcal{L}(x)$ for all $x\\in B.$ In this case, $Ux=xU$, which implies that $U$ is an isometry and commutes with every element of $B$.\n\n\\begin{proposition}\nThe Exel crossed product $B\\rtimes_{\\text{id},\\text{id}} \\bN$ is isomorphic to $B\\otimes C(\\bT)$\n\\end{proposition}\n\\begin{proof}\nRecall that $B\\rtimes_{\\text{id},\\text{id}} \\bN$ is the quotient of $\\mathcal{T}(B,\\text{id},\\text{id})$ by the ideal of generated by the set of redundancies, $\\{a-k:a\\in B,\\,k\\in \\overline{BUU^*B},\\,agU=kgU\\,\\,\\forall g\\in B\\}.$ Consider $k=UU^*$ and $a=1$. Then because $U$ is isometric and commutes with all elements of $B$, we have $UU^*gU=Ug=gU$ for all $g\\in B$. Therefore $UU^*$ and $1$ form a redundancy, so the quotient consists of a copy of $B$ and a unitary that commutes with $B$. Therefore, $B\\rtimes_{\\text{id}, \\text{id}}\\bN \\cong B\\otimes C(\\bT)$.\n\\end{proof}\n\n\\begin{remark}\nNote that the above proposition provides a second example of the Cuntz--Pimsner algebra for the quantum edge correspondence being implemented by a natural choice of Exel crossed product.\n\\end{remark}\n\n\n\n\\subsection{Rank-one quantum graphs}\nIn the special case when $B=M_n(\\mathbb{C})$ and $\\psi$ is a properly normalized trace, there is a nice correspondence between quantum adjacency matrices, projections in $M_n(\\mathbb{C}) \\otimes M_n(\\mathbb{C})^{{\\text{op}}}$, and subspaces of $M_n(\\mathbb{C})$. Just as before, a projection in $M_n(\\mathbb{C}) \\otimes M_n(\\mathbb{C})^{{\\text{op}}}$ can be viewed as the Choi matrix of a quantum adjacency matrix, but also as an orthogonal projection onto a subspace of $M_n(\\mathbb{C})$, when we identify the algebra of bounded operators on $M_n(\\mathbb{C})$ (viewed as a Hilbert space) with $M_n(\\mathbb{C}) \\otimes M_n(\\mathbb{C})^{{\\text{op}}}$ via the left-right action. \n\nFor the trivial quantum graph, the corresponding subspace of $M_n(\\mathbb{C})$ is the span of the identity matrix. It is therefore tempting to investigate other rank one subspaces. If such a subspace is spanned by a (suitably normalized) operator $T$, then the corresponding quantum adjacency matrix is equal to $A(x):= TxT^{\\ast}$. This is the inspiration for the following definition. Recall that the density matrices $\\{\\rho_a:1\\leq a\\leq d\\}$ associated to a $\\delta$-form $\\psi$ on $B$ are invertible and satisfy $\\text{Tr}(\\rho_{a}^{-1})=\\delta^2$.\n\n\\begin{definition}\n Let $T \\in B$ satisfy $\\text{Tr} (\\rho_a^{-1} T^{\\ast}T) = \\delta^2$ for all $1\\leq a\\leq d$. We call $A(x):= TxT^{\\ast}$ a quantum adjacency matrix of \\emph{rank one}.\n\\end{definition}\n\nBefore we proceed any further, we should first check that rank one quantum adjacency matrices are in fact quantum adjacency matrices. Because one can check the conditions separately on each matrix direct summand, we may assume that $B$ is a matrix algebra, so we can forget about the index $a$.\n\\begin{lemma}\\label{lem:rankone}\nIf $T \\in M_n(\\mathbb{C})$ satisfies $\\text{Tr} (\\rho^{-1} T^{\\ast}T)=\\delta^2$, where $\\rho$ is the density matrix of a $\\delta$-form $\\psi$ then $A(x):=TxT^{\\ast}$ is a quantum adjacency matrix.\n\\end{lemma}\n\\begin{proof}\nWe have to check that $m(A\\otimes A)m^{\\ast}(f_{ij}) = \\delta^2 A(f_{ij})$ for all adapted matrix units $f_{ij}$. The left-hand side is equal to $\\sum_{k} T f_{ik}T^{\\ast}T f_{kj} T^{\\ast}$. We will check that $\\sum_{k} f_{ik} S f_{kj} = \\text{Tr} (\\rho^{-1} S) f_{ij}$ and this will end the proof. First, we can write the matrix $S$ as $S = \\sum_{pt} S_{pt} e_{pt} = \\sum_{pt} S_{pt}\\sqrt{\\psi(e_{pp})\\psi(e_{tt})} f_{pt}$. We have $f_{ik} f_{pt} f_{kj} = \\delta_{pk}\\delta_{tk} \\frac{1}{\\psi(e_{kk})^2} f_{ij}$, so $f_{ik} S f_{kj} = \\frac{S_{kk}}{\\psi(e_{kk})} f_{ij}$. We obtain\n \\begin{align*\n \\sum_{k} f_{ik}S f_{kj} = \\left(\\sum_{k} \\frac{S_{kk}}{\\psi(e_{kk})}\\right) f_{ij} = \\text{Tr}(\\rho^{-1}S) f_{ij}. \n \\end{align*}\n\\end{proof}\n\\begin{proposition}\\label{prop:triv}\nLet $A$ be a rank one quantum adjacency matrix, given by $A(x)=TxT^{\\ast}$. Recall that $T$ is of the form $\\bigoplus_{a} T^{(a)}$ and we assume that each $T^{(a)}\\neq 0$. Then the Cuntz--Pimsner algebra of the edge correspondence is isomorphic to $B\\otimes C(\\bT)$.\n\\end{proposition}\n\\begin{proof}\nWe will once again resort to \\Cref{prop:recognition} to show that the edge correspondence is isomorphic to the trivial correspondence $B$. By our assumption on $T$, the element $\\xi:=\\frac{1}{\\delta}T^{\\ast}$ is cyclic for the trivial correspondence and we have $\\delta^2 \\langle \\xi, x\\xi\\rangle = TxT^{\\ast} = A(x)$. Therefore by \\Cref{prop:recognition} the edge correspondence is isomorphic to the trivial correspondence by a bimodular extension of the assignment $\\xi \\mapsto \\epsilon$. We conclude as in the proof of \\Cref{cor:triv} that the corresponding Cuntz--Pimsner algebra is isomorphic to $B\\otimes C(\\bT)$.\n\\end{proof}\n\\begin{remark}\nIf it happens that $T^{(a)}=0$ for some $a$'s, then the same proof shows that the Cuntz--Pimsner algebra is isomorphic to $B' \\otimes C(\\bT)$, where $B'$ is the direct sum of matrix algebras, but only over the $a$'s, for which $T^{(a)}\\neq 0$.\n\\end{remark}\n\n\\subsection{Quantum adjacency matrices arising from $*$-automorphisms} \nAnother way to generalize the trivial quantum graph is to let $\\alpha$ be a $*$-homomorphism of a finite dimensional C*-algebra $B$. As $\\alpha$ is multiplicative, we have \n$\n\\alpha\\circ m \n= m\\circ (\\alpha\\otimes \\alpha)\n$. \nIt follows that \n$\nm(\\alpha \\otimes \\alpha)m^{\\ast} \n= \n\\alpha mm^{\\ast} \n= \\delta^2 \\alpha\n$, i.e. $\\alpha$ is a quantum adjacency matrix. Then $\\cG_\\alpha:=(B,\\psi,\\alpha)$ is a quantum graph, and the associated quantum edge indicator for $\\cG_\\alpha$ is \n\\begin{align*}\n \\epsilon_{\\alpha}\n:=\\frac{1}{\\delta^2}(\\text{id}\\otimes \\alpha)m^*(1)\n =\\frac{1}{\\delta^2}\\sum_{aij}\\psi(e_{ii}^{(a)})f_{ij}^{(a)}\\otimes \\alpha(f_{ji}^{(a)}).\n\\end{align*}\n\nLet $E_{\\alpha} = B \\cdot \\epsilon_\\alpha \\cdot B$ denote the quantum edge correspondence for $\\cG_\\alpha$, and define $B_{\\alpha}$ to be $B$ as a C*-correspondence over itself with $\\langle a,b\\rangle=a^*b$, left action given by $x\\cdot \\eta:= \\alpha(x) \\eta$, and right action given by right multiplication.\n\\begin{lemma}\n$E_\\alpha\\cong B_{\\alpha}$ as C$^\\ast$-correspondences.\n\\end{lemma}\n\\begin{proof}\nThe element $\\xi:= \\frac{1}{\\delta} 1$ is cyclic for $B_{\\alpha}$ and $\\delta^2 \\langle \\xi, x\\cdot \\xi\\rangle = \\alpha(x)$. By \\Cref{prop:recognition} the edge correspondence is isomorphic to $B_{\\alpha}$.\n\\end{proof}\n\\begin{corollary}\\label{cor:triv2}\nWhen $\\alpha$ is a $*$-automorphism, the Cuntz--Pimsner algebra of $E_\\alpha$ is isomorphic to $B \\rtimes_\\alpha \\mathbb{Z}$.\n\\end{corollary}\n\\begin{proof}\nThis follows from Pimsner's work \\cite[Example 3, p. 193]{Pimsner}.\n\\end{proof}\n\nWe can make this crossed product a bit more explicit. First, recall that if we have an automorphism $\\alpha$ on a direct sum $B_{1}\\oplus B_2$, preserving the summands, then the corresponding crossed product splits as a direct sum as well, i.e. $(B_1\\oplus B_2)\\rtimes \\mathbb{Z} \\simeq (B_1\\rtimes \\mathbb{Z})\\oplus (B_2\\rtimes \\mathbb{Z})$. Second,\nautomorphisms of direct sums of matrix algebras are of a very special form. We can first collect all the matrix algebras of the same size and such a subalgebra has to be preserved by any automorphism. In this case we are dealing with an algebra of the form $M_n(\\mathbb{C}) \\otimes \\mathbb{C}^{k}$ and then any automorphism comes from a permutation of the set $\\{1,\\dots,k\\}$ followed by \\emph{inner} automorphisms on individual matrix algebras. Because crossed products are insensitive to inner perturbations (\\cite[II.10.3.17]{MR2188261}), we may assume that we are just dealing with a permutation automorphism. Moreover any permutation decomposes as a disjoint sum of cycles so we can treat those separately. To sum up, any crossed product $B\\rtimes \\mathbb{Z}$ will decompose as a direct sum of crossed products of the form $(M_n(\\mathbb{C})\\otimes \\mathbb{C}^k) \\rtimes \\mathbb{Z}$, where $\\mathbb{Z}$ acts by a cycle on $\\{1,\\dots,k\\}$ and as identity on $M_n(\\mathbb{C})$. Since it acts as identity on the matrix algebra, the resulting crossed product is isomorphic to $M_n(\\mathbb{C}) \\otimes (\\mathbb{C}^k \\rtimes \\mathbb{Z}) \\cong M_n(\\mathbb{C}) \\otimes M_k(\\mathbb{C}) \\otimes C(\\mathbb T)$ (\\cite[Section VIII.3, p. 230]{MR1402012}. \n\nAgain, there is a natural Exel system we may associate to the quantum graph $\\cG_\\alpha$. Consider $(B,\\alpha,\\alpha^{-1})$, where $\\alpha^{-1}$ plays the role of the transfer operator. As in the example of the complete quantum graph, the Exel crossed product for this system, $B\\rtimes_{\\alpha,\\alpha^{-1}} \\mathbb{N}$, is isomorphic to the Cuntz--Pimsner algebra $B\\rtimes_\\alpha \\mathbb{Z}$ for the quantum edge correspondence $E_\\alpha.$\n\n\n\n\n\n\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nWe consider an inverse boundary value problem for the wave equation \n$\n(\\frac {\\partial^2}{\\partial t^2} - c(x)^2 \\frac {\\partial^2}{\\partial x^2}) u(t,x) = 0,$$\nand introduce a discrete regularization strategy to recover the sound speed $c(x)$ by using the knowledge of perturbed and discetized Neumann-to-Dirichlet map $\\widetilde \\Lambda_{N_1}$.\nOur approach is based on the Boundary Control method \\cite{Belishev1987, Belishev1992, Tataru1995}.\n\nA variant of the Boundary Control method, called the iterative time-reversal control method, was introduced in \\cite{Bingham2008}.\nThe method was later modified in \\cite{Dahl2009}\nto focus the energy of a wave at a fixed time\nand in \\cite{Lauri2013} \nto solve an inverse obstacle problem for a wave equation. \nIn \\cite{jusa1} we introduced a modification to the iterative time-reversal control method\nthat is tailored for the 1+1 dimensional wave equation.\n\n \nThe novelty in this paper is that we analyze the effect of the discretization in the regularized solution of the inverse problem. We give a direct discrete regularization method for the\nnon-linear inverse problem for the wave equation. The result contains an\nexplicit (but not necessarily optimal) convergence rate.\n\nBy referring to direct methods for non-linear problems we mean the explicit construction of non-linear map to solve the problem\nwithout resorting to a local optimization method. In our case the map is given by (\\ref{Rstategia}), shown below.\nThe advantage of direct approaches\nis that they do not suffer from the possibility that the algorithm converges to a local minimum.\nIn particular, they do not require a priori knowledge\nthat the solution is in a small neighbourhood of a given function.\n\nClassical abstract regularization theory is explained in \\cite{Engl1996}. The iterative regularization\nof both linear and non-linear inverse problems and convergence rates are discussed\nin a Hilbert space setting in \\cite{Bissantz2004, Hanke1997, Hohage2008, LuPeRa2007, Mathe2008} and in a Banach space setting in \\cite{Hofmann2007, Kaltenbacher2006, Kaltenbacher2008, Kirsch1996, Ramlau2008, Ramlau2006, Resmerita2005}. In section \\ref{pamelapulkkinen} we compare our regularization strategy to Morozov's discrepancy principle (MDP). In the context of abstract regularization theory, this principle has been discussed, e.g., in \\cite{Scherzer1993}. \n\nThere are currently only a few regularized direct methods for non-linear inverse problems.\nAn example is a regularisation algorithm for\nthe inverse problem for the conductivity equation in \\cite{KLMS2009}.\n Also, a direct regularized inversion for blind\ndeconvolution is presented in \\cite{Justen2006}.\n\n\\section{Regularization Strategy}\n\\subsection{Continuity of forward map}\nWe define \n\\begin{align}\n\\label{normipoika}\n&\\norm{c}_{C^k(M)}=\n\\sum_{p=0}^{k}\\sup_{x\\in (0,\\infty)}|\\frac{\\partial^p c}{\\partial x^p}(x)|,\n\\end{align}\nwhere $M$ denotes the half axis $M=[0,\\infty)\\subset \\R$. We denote the set of bounded $C^k(M)$ -functions by \n\\begin{align*}\nC^k_b(M)=\\{c\\in C^k(M); \\norm{c}_{C^k(M)}< \\infty\n\\}.\n\\end{align*}\n Let $C_0,C_1,{L_0,L_1},m>0$ and define the space of {k times differentiable} velocity functions\n\\begin{align}\n\\label{nopeudet}\n {\\mathcal V^k}=&\\,\\{c\\in {C^k} (M);\nC_0\\le c(x)\\le C_1 ,\\\\\\nonumber &\\,\\norm{c}_{{C^k}(M)}\\le m,\\, c-1 \\in {C_0^k([L_0,L_1])} \\}.\n\\end{align}\nHere $C_0^k([L_0,L_1])$ is the subspace of functions in $C^k_b(M)$ that are supported on $[L_0,L_1]$.\nLet \n\\begin{align}\n\\label{aikapoika}\nT> \\frac{{L_1}}{C_0}.\n\\end{align}\nFor $c\\in{\\mathcal V^2}$ and \n$f\\in L^2(0,2T)$, the boundary value problem \n\\begin{align}\n\\label{dartwader}\n&(\\frac {\\partial^2}{\\partial t^2} - c(x)^2 \\frac {\\partial^2}{\\partial x^2}) u({x,t}) = 0 \\quad \\text{in ${M\\times (0,2T})$},\n\\\\\\nonumber\n&\\partial_x u({0,t}) = f(t),\n\\\\\\nonumber\n&u|_{t = 0} =0,\\quad \\partial_t u|_{t=0} = 0,\n\\end{align}\nhas a unique solution $u=u^f\\in H^1({M\\times (0,2T)})$. Using this solution we define the Neumann-to-Dirichlet operator $ \\Lambda= \\Lambda_c$,\n\\begin{align}\n\\label{lam}\n \\Lambda : L^{2}(0,2T) \\to L^{2}(0,2T),\\quad\n \\Lambda f=u^f|_{x=0}.\n\\end{align}\nFor a Banach space $E$ we define\n\\begin{align*}\n\\mathcal{L} (E)=\\{A:E \\to E;\nA\\text{ is linear and continuous}\\}.\n\\end{align*}\nLet ${Z=C^2_b(M)}$ and $Y=\\mathcal{L} (L^2(0,2T))$. The operator $\\mathcal A$ is defined in the domain $\\mathcal D(\\mathcal A)=\\mathcal V^3$ by setting\n\\begin{align}\n\\label{apina}\n\\mathcal A:{\\mathcal D(\\mathcal A)}\\subset Z\\to\\mathcal R (\\mathcal{A})\\subset Y,\\quad \n \\mathcal A(c)= \\Lambda_c. \n\\end{align}\nThe notation in (\\ref{apina}) means that the range {$\\mathcal R (\\mathcal{A})=\\mathcal A(\\mathcal V^3)$} and the domain {$\\mathcal D(\\mathcal A)$} are equipped with the topologies of $Y$ and $ Z$, respectively. Note that the maps (\\ref{lam}) and (\\ref{apina}) are continuous (see \\cite{jusa1}).\n\n\\subsection{Regularization strategies with discretizatized measurements}\n\nLet $T$ be as in (\\ref{aikapoika}) and $N\\in \\mathbb{Z}_+$. For $n\\in\\{1,2,3,...,2N\\}$ we define the basis functions as\n\\begin{align}\n\\label{kantafunktio}\n \\phi_{n,N}(t) = \\Big(\\frac{N}{T}\\Big)^{\\frac{1}{2}}1_{[\\frac{(n-1)T}{N},\\frac{nT}{N} )}(t), \\quad t \\in [0,2T).\n\\end{align}\nNote that the functions ${\\phi_{n,N}}$ are orthonormal in $L^2(0,2T)$. \nHaving (\\ref{kantafunktio}) we define the space of piecewise constant functions as \n\\begin{align}\n\\label{pcsfjoukko}\n\\mathcal P^N=span\\big\\{\\phi_{1,N},...,\\phi_{2N,N}\\big\\}\\subset L^2(0,2T) \n\\end{align}\nand \nan orthogonal projection as\n\\begin{align}\n\\label{pcsfjoukko2}\nP^N:L^2(0,2T)\\to\\mathcal P^N,\\quad P^N(f)=\\sum_{j=1}^{2N} \\langle f,\\phi_{j,N}\\rangle\n_{L^2(0,2T)}\\phi_{j,N}(t).\n\\end{align}\nLet $\\Lambda$ be as in (\\ref{lam}). Using (\\ref{pcsfjoukko2}) we define \n\\begin{align}\n\\label{pallokala}\n\\Lambda_N=P^N\\Lambda P^N.\n \\end{align}\\\\\nLet $E$ be a Banach space and $\\mathcal H\\in E$. We denote \n\\begin{align}\n\\label{pallo}\n \\mathcal{B}_E(\\mathcal H,\\epsilon)=\\{\\widetilde {\\mathcal H}\\in E : \\norm{\\widetilde{\\mathcal H}-\\mathcal H}_E < \\epsilon \\}.\n\\end{align}\n\n\\subsubsection{A model for a single discrete and noisy measurement}\n\\label{huippumalli1}\nLet $\\epsilon_0>0$ and we define\n\\begin{align}\n\\label{porraspossu}\nl_0(\\epsilon_0)=\\big\\lfloor\\frac{4}{7}\\log_2 \\epsilon_0^{-1}\\big\\rfloor\n\\end{align}\nand \n\\begin{align}\n\\label{porraspossu2}\nN_0(\\epsilon_0)=2^{l_0}.\n\\end{align}\nLet $N=2^l\\ge N_0$, where $l\\in \\mathbb{Z}_+$. Let $P^N$ be as in (\\ref{pcsfjoukko2}) and $\\Lambda $ be as in (\\ref{lam}). \nLet us define\n\\begin{align}\n\\label{porras}\n H(t) = \n\\begin{cases}\n1, & t\\ge 0, \\\\\n0, & t<0.\n\\end{cases}\n\\end{align}\nLet us define \n \\begin{align}\n\\label{mittauspoika}\n \\widetilde m_{N,\\epsilon_0}=P^N\\Lambda H + n_{N,\\epsilon_0} ,\n\\end{align}\nin which $n_{N,\\epsilon_0} \\in \\mathcal P^N$ represents the error and $\\norm{n_{N,\\epsilon_0} }_{L^2(0,2T)}\\le \\epsilon_0$. We consider the quantity $(\\epsilon_0,N_0,\\widetilde m_{N,\\epsilon_0})$ that we call a measurement. \nLet $\\mathcal A$ be as in (\\ref{apina}) and $H$ be as in (\\ref{porras}). We define \n\\begin{align}\n\\label{apina4}\n\\mathcal A_0:\\mathcal D(\\mathcal A_0)\\subset Z\\to\\mathcal R (\\mathcal A_0)\\subset L^2(0,2T),\\quad \n \\mathcal A_0(c) = \\mathcal A(c) H= \\Lambda H, \n\\end{align}\nwhere $\\mathcal D(\\mathcal A_0)=\\mathcal V^3$ . \nOur main result on the reconstruction of $c(x)$ from the measurement $(\\epsilon_0,N_0,\\widetilde m_{N,\\epsilon_0})$ is given by the following theorem. \n \n\\begin{theorem}\n\\label{kaiken_teoria2b}\nFor the operator $\\mathcal A_0:\\mathcal \\mathcal V^3\\subset Z\\to L^2(0,2T)$, there exists an admissible regularization strategy $\\mathcal R^{(0)}_{N_0,\\alpha_0}$ with the choice of parameter \n\\begin{align*}\n\\alpha_0(\\epsilon_0)=a_0\\epsilon_0^{\\frac{4}{45}},\n\\end{align*}\nsatisfying the following: For every $c\\in{\\mathcal V^3}$\nthere are $\\widetilde\\epsilon_0>$, $a_0>0$, and $C>0$ such that\n\\begin{align*} \n&\\sup\\Big\\{\\norm{\\mathcal R^{(0)}_{N_0,\\alpha_0}\\widetilde m_{N,\\epsilon_0}-c}_{ Z}:\n\\widetilde m_{N,\\epsilon_0}\\in\\mathcal P^{N},N=2^l\\ge N_0(\\epsilon_0),\n\\\\\\nonumber&\\qquad \\norm{\\widetilde m_{N,\\epsilon_0} -P^N\\Lambda H}_{L^2(0,2T)} \\le \\epsilon_0\\Big\\}\\le\nC\\epsilon_0 ^{\\gamma_0},\n\\end{align*}\nfor all $\\epsilon_0 \\in (0,\\widetilde\\epsilon_0)$. Here $\\gamma_0 =\\frac{1}{270}$ and $ N_0(\\epsilon_0)$ is as in (\\ref{porraspossu2}).\n\\end{theorem}\nAn explicit bound $\\widetilde\\epsilon_0$ and the value for constant $a_0$ are given in the proof.\nThe proof of Theorem \\ref{kaiken_teoria2b} is given in Section \\ref{convergense}.\n\\subsubsection{A model for several discrete and noisy measurements}\n\\label{virhesikapossu}\nLet \n\\begin{align}\n\\label{virheoperator}\n\\mathcal{E}_{N_1}:\\mathcal P^{N_1}\\to\\mathcal P^{N_1},\n\\end{align}\nwhere $N_1\\in\\mathbb{Z}_+$.\nHaving $\\Lambda_{N_1}$ as in (\\ref{pallokala}) we define a discrete and noisy measurement operator\n\\begin{align}\n\\label{akuaku}\n& \\widetilde\\Lambda_{N_1}: L^2(0,2T) \\to L^2(0,2T), \n\\quad \n\\\\\\nonumber&\\widetilde\\Lambda_{N_1}f = \n\\begin{cases}\n\\Lambda_{N_1}f+\\mathcal{E}_{N_1}f, & f \\in \\mathcal P^{N_1}, \\\\\n\\quad\\qquad 0, & f \\in (\\mathcal P^{N_1})^\\perp.\n\\end{cases}\n\\end{align}\nNote that $\\mathcal P^{N_1}\\cup (\\mathcal P^{N_1})^\\perp=L^2(0,2T)$. \nWith data corresponding to several boundary measurements $(\\epsilon_1, N_1, \\widetilde\\Lambda_{N_1})$ we get\nthe following results with improved error estimates.\n\\begin{theorem}\n\\label{kaiken_teoria2}\nFor the operator $\\mathcal A:\\mathcal \\mathcal V^3\\subset Z\\to Y$, there exists an admissible regularization strategy $\\mathcal R^{(1)}_{N_1,\\alpha_1}$ with the choice of parameter \n\\begin{align*}\n\\alpha_1(\\epsilon_1)=a_1\\epsilon_1^{\\frac{4}{9}}\n\\end{align*}\nthat satisfies the following: For every $c\\in{\\mathcal V^3}$\nthere are $\\widetilde\\epsilon_1>0$, $a_1>0$, and $C>0$ such that\n\\begin{align*} \n&\\sup\\Big\\{\\norm{\\mathcal R^{(1)}_{N_1,\\alpha_1}\\widetilde\\Lambda_{N_1}-c}_{ Z}:\nN_1\\ge\\epsilon_1^{-4},\\quad \\widetilde\\Lambda_{N_1}\\in \\mathcal L(\\mathcal P^{N_1}),\n\\\\\\nonumber&\\qquad \\norm{\\widetilde\\Lambda_{N_1}-\\Lambda_{N_1} }_{{Y}}\\le \\epsilon_1\\Big\\}\\le\nC\\epsilon_1 ^{\\gamma_1},\n\\end{align*}\nfor all $\\epsilon_1 \\in (0,\\widetilde\\epsilon_1)$. Here $\\gamma_1 =\\frac{1}{54}$.\n\\end{theorem}\nAn explicit bound $\\widetilde\\epsilon_1$ and the value for constant $a_1$ are given in the proof.\nThe proof of Theorem \\ref{kaiken_teoria2} is given in Section \\ref{convergense}.\nWe will give explicit choices for $\\mathcal R^{(0)}_{N_0,\\alpha_0}$ in formula (\\ref{Rstategia2}) and for $\\mathcal R^{(1)}_{N_1,\\alpha_1}$ in formula (\\ref{Rstategia2b}) below. For the convenience of the reader we give a short summary of the regularization strategy. \nAssume that we are given $\\widetilde \\Lambda_{N}\\in\\mathcal L(\\mathcal P^{N_1})\\subset Y$, that is,\n the discrete Neumann-to-Dirichlet map for the unknown wave speed $c(x)$ with measurements errors. Then\n the regularization strategy is obtained by the following steps:\n\n\\begin{enumerate}\n\\item Using operator $\\widetilde\\Lambda_{N_1}$ in (\\ref{sope3}) we constructed a source that produces a wave such that\n$u^{f_{\\alpha,r}}(t,x)|_{t=T}$ is close to the indicator function $1_{\\mathcal M(r)}(x)$ of the \ndomain of influence ${\\mathcal M(r)}$---see (\\ref{siiri}). \n \\item Using sources $\\widetilde f^N_{\\alpha,r}$ we approximately compute the \nvolumes $V(r)$ of the domains of influences---see (\\ref{operaattorit4}), (\\ref{sope2}), and Proposition \\ref{PSN}. \n \\item Using finite differences we compute approximate values of the \n derivatives of the volumes of the domain influences $\\partial_r V(r)$---see (\\ref{timojutila2c}). \n \\item We interpolate the obtained values of $\\partial_r V(r)$. This determines\n the approximate values of the wave speed $v(r)$ in the travel time coordinates---see (\\ref{operaattorit5}) and (\\ref{sebastianahoc}). \n \\item Finally, we change the coordinates from travel time coordinates to Euclidean\n coordinates to obtain the approximate values of the wave speed $c(x)$ for $x\\in M$---see (\\ref{kenguru}).\n \n\\end{enumerate}\n\\subsection{Previous literature}\nFrom the point of view of uniqueness questions, the inverse problem\nfor the 1+1 dimensional wave equation is equivalent to the one-dimensional inverse boundary spectral problem. The latter problem was\nthoroughly studied in the 1950s \\cite{GeLe1951, Krein1951, Marchenko1950} and we refer to \\cite[pp. 65--67]{Kabanikhin2005} for a historical overview. In the\n1960s Blagove{\\v{s}}{\\v{c}}enski{\\u\\i} \\cite{Blagovevsvcenskiui1969, Blagovevsvcenskiui1971} developed an\napproach to solving the inverse problem for the 1+1 dimensional wave\nequation without reducing the problem to the inverse boundary spectral\nproblem. This and later dynamical methods have the advantage over\nspectral methods that they only require data on a finite time\ninterval. Applications of one-dimensional inverse probems have been discussed widely in\n\\cite{Blagovevsvcenskiui1966, Kabanikhin2005,KKL2001}. \n\nThe method in the present paper is a variant of the Boundary Control\n method that was pioneered by M. Belishev \\cite{Belishev1987}\nand developed by M. Belishev and Y. Kurylev \\cite{BeKu1987,Belishev1992} in the late 1980s and early 1990s.\nOf crucial importance for the method is the result by D.\\ Tataru\n\\cite{Tataru1995} concerning\na Holmgren-type uniqueness theorem for non-analytic coefficients.\nThe Boundary Control method for multidimensional inverse problems has been summarized in \\cite{Belishev1997, KKL2001},\nand considered for 1+1 dimensional scalar problems in \\cite{BeKa1989, BeRFi1994, jusa1} and for multidimensional scalar problems\nin \\cite{Katchalov1998, KKLM2004, Kury1995, LaOk2014, LaOk2014b}.\nFor systems it has been considered in \\cite{Kurylev2009, Kurylev2006,Kurylev2015}.\nStability results for the method have been considered in \\cite{Anderson2004}\nand \\cite{Katsuda2007}, and\ncomputational implementations in\n\\cite{Belishev1999,Belishev2016,Hoop2017,Ivanov2016,Pestov2010}.\nAn application of the method to blockage detection in water pipes is in preparation \\cite{eemeli}.\n\nThe inverse problem for the wave equation can also be solved by using complex geometrical optics solutions. These solutions were developed in the context of elliptic\ninverse boundary value problems \\cite{Sylvester1987}, and in \\cite{Nachman1988} they were employed to solve an inverse\nboundary spectral problem. Local stability results can be proven using\n(real) geometrical optics solutions \\cite{Bellassoued2011, StUh1998, uulmanni2013}, and in \\cite{lauri2012} a stability result was\nproved by using ideas from the Boundary Control method together with complex\ngeometrical optics solutions. \n\n There is an important method\nbased on Carleman estimates \\cite{Bukhgeuim1981}, often called the Bukhgeim-Klibanov method after its founders, that can be used to show stability\nresults requiring only a\nsingle instance of boundary values, when the initial data for the wave equation is non-vanishing. \nWe mention the interesting recent computational work \\cite{Baudouin} that is based on this method, and also\nanother reconstruction method that uses a single measurement \\cite{beilina2012approximate, Beilina2008}. \nThis method is based\non a reduction to a non-linear integro-differential equation, and\nthere are several papers on\nhow to solve this equation (or an approximate version of it)---see \\cite{Klibanov2017, Klibanov2015} for recent results\nincluding computational implementations. \n \n\\subsection{Notations}\nWe will define $\\mathcal R^{(1)}_{N_1,\\alpha_1}$ as a discrete version of the regulation strategy given in \\cite{jusa1}. For that we recall some notation from \\cite{jusa1}. \n\nWe denote the indicator function of a set $E$ by \n$$\n1_E(x) = \n\\begin{cases}\n1, & x \\in E,\n\\\\\n0, & \\text{otherwise}.\n\\end{cases}\n$$\nWe define \n\\begin{align}\n\\label{operaattorit}\nJ : L^2(0,2T) \\to L^2(0,2T), \\quad\nJ f(t) = \\frac{1}{2} \\int_0^{2 T} 1_\\blacktriangle (t,s)f(s) ds,&\n\\end{align}\nwhere\n$\n\\blacktriangle = \\{(t,s) \\in (0,2T)^2;\\ \\text{$t + s \\le 2 T$ and $s > t > 0$}\\}.\n$\nWe define the time reversal operator as\n\\begin{align}\n\\label{operaattorit2}\nR : L^2(0,2T) \\to L^2(0,2T), \\quad\nR f(t) = f(2 T - t),\n\\end{align}\nand the projections as\n\\begin{align}\n\\label{operaattorit3}\n P_r f(t) = 1_{(T-r,T)}(t)f(t), \\quad r \\in [0,T].\n\\end{align}\nUsing (\\ref{operaattorit}), (\\ref{operaattorit2}), and (\\ref{operaattorit3}) we define that \n\\begin{align}\n\\label{Koo}\nK : Y \\to Y, \\quad\nKL = JL-RLRJ\n\\end{align}\nand\n\\begin{align}\n\\label{Hoo}\n\\boldsymbol{H} &:Y \\mapsto C([0,T],Y), \n\\quad\\boldsymbol{H}L(r) = \nP_r(K L)P_r.\n\\end{align}\n\nWe define a regularized inversion with cutoff as\n\\begin{align}\n\\label{Zalfa}\n{Z}_{\\alpha} : Y \\to Y,\n\\quad\nZ_\\alpha(L) = \\eta_Y(L,\\alpha) (L+\\alpha)^{-1},\n\\end{align}\nwhere $\\eta_Y : Y \\times (0,\\infty) \\to \\R$ is, for example, a continuous function that satisfies\n$\\eta_Y(L,\\alpha) = 1$ when $d(L,Y^+) \\le \\alpha \/ 4$ and \n$\\eta_Y(L,\\alpha) = 0$ when $d(L,Y^+) \\ge \\alpha \/ 2$. Here \n$$\nd(L,Y^+) = \\inf \\{\\norm{L-L^+}_Y;\\ \\text{$L^+ \\in Y$ is positive semidefinite}\\}.\n$$\nWe denote by $\\boldsymbol{Z}_{\\alpha}$ the lift of ${Z}_{\\alpha}$\nto $C([0,T],Y)$, that is,\n$\\boldsymbol{Z}_{\\alpha}(L)(r)$ is ${Z}_{\\alpha}(L(r))$.\nMoreover, we define $b(t) = 1_{(0,T)}(t) (T-t)$ and\n\\begin{align}\n\\label{innerproduct}\n& \\boldsymbol{S}: C([0,T],Y) \\to C([0,T]),\n\\quad \\boldsymbol SL(r) =\\langle L(r)P_r b,b\\rangle _{L^2(0,2T)}.\n\\end{align}\n\nWe define the travel time coordinates by\n\\begin{align}\n\\label{def_tau}\n\\tau:[0,\\infty)\\to[0,\\infty), \\quad \\tau(x) = \\int_0^x \\frac{1}{c(t)} dt, \\quad x \\in M,\n\\end{align}\nand the domain of influence \n\\begin{align}\n\\label{siiri}\n\\mathcal M(r) = \\{x \\in M; \\tau(x) \\le r\\}, r \\ge 0.\n\\end{align}\nThe function $\\tau$ \nis strictly increasing and we denote its inverse by $\\chi$.\nMoreover, $V(r)$ denotes the volume of $\\mathcal M(r)$ with respect to the measure $dV = c^{-2}dx$, \nwhere $c$ is the speed of sound in (\\ref{dartwader}).\nFrom \\cite[Eq. (21)]{jusa1} we see that \n\\begin{align}\n\\label{operaattorit4}\nV = \\lim_{\\alpha \\to 0} \n(\\boldsymbol S \\circ \\boldsymbol Z_\\alpha \\circ \\boldsymbol H)(\\Lambda).\n\\end{align}\nMoreover, according to \\cite[Eq. (19), (20)]{jusa1}, the speed of sound in travel time coordinates $v = c \\circ \\chi$ satisfies\n\\begin{align}\n\\label{operaattorit5}\nv(r) = \\frac 1 {\\partial_r V(r)}, \\quad \\chi(r) = \\int_0^r v(t) dt, \\quad r > 0. \n\\end{align}\nThus $c$ can be computed from $V$. We will next recall how the formula (\\ref{operaattorit5}) is regularized in \\cite{jusa1}. For small $h>0$ we consider the partition\n\\begin{align} \n\\label{jako}\n(0,T)=(0,h) \\cup[h,2h)\\cup[2h,3h)\\cup...\\cup\n[N_hh-h,N_hh)\\cup [N_hh,T),\n\\end{align}\nwhere $N_h\\in \\mathbb{N}$ satisfies $T-h\\le N_hh < T$.\nWe define a discretized and regularized approximation of the derivative operator $\\partial_r$ by\n\\begin{align} \n\\label{operaattori6}\nD_{h}:C([0,T])\\to L^\\infty(0,T), \n\\end{align}\n\\begin{displaymath}\nD_{h}(f)(t)= \\left\\{ \\begin{array}{cl}\n\\frac{f(h)}{h}, & \\textrm{if\\quad $t \\in (0,h)$},\\\\\n\\frac{f(jh+h\n)-f(jh)}{h}, & \\textrm{if\\quad $t \\in [jh,jh+h)$},\\\\\n \\frac{f(T)-f(N_hh)}{T-N_hh},& \\textrm{if\\quad $t \\in [N_hh,T)$.}\\\\ \n\\end{array} \\right.\n\\end{displaymath}\nLet us have \n\\begin{align}\n\\label{arskasika}\n\\vartheta (t;a,b) = a 1_{(-\\infty, a)}(t) + t 1_{[a, b]}(t)+\nb 1_{( b,\\infty)}(t).\n\\end{align}\nWe define an inversion with a cutoff that takes into account the a priori bounds in (\\ref{nopeudet}) by \n\\begin{align}\n\\label{rampo}\nz : \\R \\to \\R\n\\quad\nz(t) &= \\frac{1}{\\vartheta (t;C_0,C_1)}.\n\\end{align}\nWe denote by $\\boldsymbol z$ the lift of $z$\nto $L^\\infty(0,T)$, that is,\n$\\boldsymbol z(f)(t) = z(f(t))$.\nWe define the extension by one\n\\begin{align}\n\\label{arska}\nE : L^\\infty(0,T) \\to L^\\infty(0,\\infty), \n\\quad Ef(t) = \n\\begin{cases}\nf(t), & t \\in (0,T), \\\\\n1, & \\text{otherwise},\n\\end{cases}\n\\end{align}\nand set $W = E \\circ \\boldsymbol z$. We define \n\\begin{align}\n\\label{arskapiks}\n\\tilde \\chi : L^\\infty(0,\\infty) \\to C(0,\\infty), \\quad\n\\tilde \\chi(f)(r) = \\int_0^r f(t) dt.\n\\end{align}\nNote that having $f>0$ in (\\ref{arskapiks}) we have it that $\\tilde \\chi(f)$ is a strictly increasing function. \nHaving $L_0,L_1$, as in (\\ref{nopeudet}), we define \n\\begin{align} \n\\label{operaattori16}\n\\theta_\\R: L^\\infty(0,\\infty) \\to L^\\infty(\\mathbb{R}), \n\\end{align}\n\\begin{displaymath}\n\\theta_\\R(f)(t)= \\left\\{ \\begin{array}{cl}\n1, & \\textrm{if\\quad $t \\in (-\\infty,L_0]$},\\\\\nf(t), & \\textrm{if\\quad $t \\in (L_0,L_1)$},\\\\\n 1,& \\textrm{if\\quad $t \\in [L_1,\\infty)$.}\\\\ \n\\end{array} \\right.\n\\end{displaymath}\nHaving $f>0$ and using (\\ref{arskapiks}) and (\\ref{operaattori16}) we define\n\\begin{align}\n\\label{hiiri} \n\\Phi : L^\\infty(0,\\infty) \\to L^\\infty(\\mathbb{R}),\n\\quad\n\\Phi(f) = \\theta_\\R(f \\circ (\\tilde \\chi(f))^{-1}).\n\\end{align}\nLet us define $\\eta\\in C^\\infty (\\mathbb{R})$ by\n\\begin{align} \n\\label{konvooluutti}\n{ \\eta(x)= \\left\\{ \\begin{array}{cl}\n C\\exp\\big (\\frac{1}{x^2-1}\\big ), & \\textrm{if\\quad $x \\in (-1,1)$},\\\\\n0, & \\textrm{if\\quad $|x|\\ge 1$},\\\\\n\\end{array} \\right.}\n\\end{align}\n{where the constant $C>0$ is selected so that $\\int_\\mathbb{R} \\eta (x)=1$.} For $\\nu >0$ we define\n\\begin{align} \n\\label{molliolli}\n{ \\eta_\\nu (x)=\\frac{1}{\\nu }\\eta \\Big (\\frac{x}{\\nu } \\Big ).}\n\\end{align}\nBy using convolution we define a smooth approximation to a given function $f\\in L^\\infty(\\mathbb{R} )$ by setting\n\\begin{align} \n\\label{smootapp}\n\\Gamma_\\nu:L^\\infty(\\mathbb{R})\\to C^\\infty (\\mathbb{R} ),\\quad \\Gamma_\\nu ( f)= \\eta_\\nu \\ast f.\n\\end{align}\n Using (\\ref{Hoo}),\n(\\ref{Zalfa}), (\\ref{innerproduct}), (\\ref{operaattori6}), (\\ref{arska}), (\\ref{hiiri}), and (\\ref{smootapp}) we define the family of operators for the regularization strategy used in \\cite{jusa1} by\n\\begin{align}\n\\label{Rstategia}\n&\\mathcal R_{\\alpha}:Y\\to Z,\n\\\\\\nonumber&\\mathcal R_{\\alpha}=\\Gamma_\\nu\\circ\\Phi\\circ W\\circ D_{h}\\circ\\boldsymbol{S}\\circ \\boldsymbol{Z}_{\\alpha}\\circ\\boldsymbol{H},\n\\end{align}\nwhere $\\nu=C\\epsilon^{\\frac{1}{54}}$, $h=C\\epsilon^{\\frac{1}{18}}$ and $\\alpha(\\epsilon)=2^{\\frac{13}{9}}T^{\\frac{4}{9}}\\epsilon^{\\frac{4}{9}}$. Note that in \\cite{jusa1} we considered perturbations of the Neumann-to-Dirichlet operator of the form \n\\begin{align}\n\\label{virhemitta}\n&\\widetilde\\Lambda=\\Lambda+\\mathcal{E}, \n\\end{align}\nwhere $\\mathcal{E}\\in Y$\nmodels the measurement error and $\\norm{\\mathcal{E}}_Y \\le \\epsilon$. \nBelow we will introduce a discretized version of regularization strategy (\\ref{Rstategia}) that takes in discretized measurements. To this end we start with auxiliary lemmas.\n\\subsection{The proofs of the main results}\n\\label{convergense}\n\n\\begin{lemma}\n\\label{DLa} \nLet $N\\in \\mathbb{Z}_+$. Let $\\Lambda $ be as in (\\ref{lam}) and $\\Lambda_N$ as in (\\ref{pallokala}). Then we have\n\\begin{align*}\n \\norm{\\Lambda_{N}-\\Lambda }_{{Y}}\\le CN^{-\\frac{1}{4}}.\n\\end{align*}\n\\end{lemma}\nHere $C=C(T)>0$ depends on $T$.\n\\begin{proof}\nBy (\\ref{lam}) and the trace theorem we have\n\\begin{align}\n\\label{Lamestim}\n \\norm{\\Lambda}_{L^2(0,2T)\\to H^{\\frac{1}{2}}(0,2T)}\n\\le C_{Lam}.\n\\end{align}\nBy (\\ref{pcsfjoukko2}) we have\n\\begin{align}\n\\label{virheprojektiolle7}\n \\norm{I-P^N}_{L^2(0,2T)\\to L^2(0,2T)}\n\\le2.\n\\end{align}\nLet $f\\in H^2(0,2T)$. By (\\ref{pcsfjoukko2}) and $H^2(0,2T)\\hookrightarrow C^1([0,2T])$ we have\n\\begin{align}\n\\label{virheprojektiolle3}\n \\norm{f -P^N f}_{L^\\infty (0,2T)}\\le\n \\frac{T}{N}\\norm{f}_{C^1([0,2T])}.\n\\end{align}\nThus \n\\begin{align}\n\\label{virheprojektiolle4}\n \\norm{f -P^N f}_{L^2(0,2T)}\\le(2T)^{\\frac{1}{2}}\\norm{f -P^N f}_{L^\\infty (0,2T)}\n\\le(2T)^{\\frac{1}{2}} \\frac{T}{N}\\norm{f}_{C^1([0,2T])}.\n\\end{align}\nBy (\\ref{virheprojektiolle4}) and having $C_{sob}=\\norm{I}_{H^2(0,2T)\\to C^1([0,2T])}$ we get\n\\begin{align}\n\\label{virheprojektiolle5}\n \\norm{I-P^N}_{H^2(0,2T)\\to L^2(0,2T)}\n\\le(2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}.\n\\end{align}\nUsing interpolation theory with (\\ref{virheprojektiolle7}) and (\\ref{virheprojektiolle5}) we have\n\\begin{align}\n\\label{virheprojektiolle8}\n \\norm{I-P^N}_{H^{\\frac{1}{2}}(0,2T)\\to L^2(0,2T)}\n\\le 2^{\\frac{3}{4}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{4}}.\n\\end{align}\nFor self-adjoint operators $P^N$ and $\\Lambda$ we have\n\\begin{align}\n\\label{virheprojektiolle10}\n \\norm{\\Lambda -\\Lambda P^N }_{{Y}}=\n \\norm{(\\Lambda -\\Lambda P^N )^*}_{Y}=\n \\norm{(I - P^N)\\Lambda }_{{Y}}.\n\\end{align}\nBy (\\ref{Lamestim}) and (\\ref{virheprojektiolle8}) we have\n\\begin{align}\n\\label{virheprojektiolle12}\n \\norm{\\Lambda -\\Lambda P^N }_{{Y}}=\\norm{(I - P^N)\\Lambda }_{{Y}}\\le\n 2^{\\frac{3}{4}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{4}}C_{Lam}.\n\\end{align}\nThus\n\\begin{align}\n\\label{virheprojektiolle13}\n \\norm{P^N \\Lambda -P^N \\Lambda P^N }_{{Y}}\\le\n 2^{\\frac{3}{4}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{4}}C_{Lam}.\n\\end{align}\nBy (\\ref{virheprojektiolle10}), (\\ref{virheprojektiolle12}), and (\\ref{virheprojektiolle13}) \nwe have\n\\begin{align}\n\\label{virheprojektiolle14}\n \\norm{\\Lambda -P^N \\Lambda P^N }_{{Y}}\\le\n 2^{\\frac{7}{4}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{4}}C_{Lam}\\le\nCN^{-\\frac{1}{4}}.\n\\end{align}\n\\end{proof}\n\\begin{proposition}\n\\label{DLb} \nLet $\\epsilon>0$, $N\\in \\mathbb{Z}_+$ and $N\\ge\\epsilon^{-4}$ . Let $\\Lambda $ be as in (\\ref{lam}) and $\\Lambda_N$ be as in (\\ref{pallokala}). Assume that $\\widetilde\\Lambda_{N}\\in \\mathcal{B}_Y (\\Lambda_N,\\epsilon)$, then \n\\begin{align*}\n \\norm{\\widetilde\\Lambda_{N}-\\Lambda }_{{Y}}\\le C_2\\epsilon.\n\\end{align*}\n\\end{proposition}\nHere $C_2=C_2(T)>0$ depends on $T$.\n\\begin{proof}\n\n\nUsing Lemma \\ref{DLa} and having $N\\ge\\epsilon^{-4}$\nwe get\n\\begin{align}\n\\label{virheprojektiolle15}\n \\norm{\\widetilde\\Lambda_{N}-\\Lambda }_{{Y}}\\le \\epsilon +\n CN^{-\\frac{1}{4}}\\le (C+1)\\epsilon.\n\\end{align}\n\\end{proof}\n Let $J$ be as in (\\ref{operaattorit}) and using (\\ref{pcsfjoukko2}) we define\n\\begin{align}\n\\label{pridekulkue}\nJ_N=P^NJP^N.\n\\end{align}\n\\begin{lemma}\n\\label{JN}\nLet $N\\in\\mathbb{Z}_+$. Let $J$ be as in (\\ref{operaattorit}) and $J^N$ be as in (\\ref{pridekulkue}). Then we have\n\\begin{align*}\n \\norm{J-J_N}_{Y}\\le\nCN^{-\\frac{1}{2}}.\n\\end{align*}\n\\end{lemma}\nHere $C=C(T)>0$ depends on $T$.\n\\begin{proof}\nBy (\\ref{operaattorit}) we have\n\\begin{align}\n\\label{JnormiH}\n \\norm{J}_{L^2(0,2T)\\to H^{1}(0,2T)}\\le C_{J}.\n\\end{align}\nUsing interpolation with (\\ref{virheprojektiolle7}) and (\\ref{virheprojektiolle5}) we have\n\\begin{align}\n\\label{virheprojektiolle8b}\n \\norm{I-P^N}_{H^{1}(0,2T)\\to L^2(0,2T)}\n\\le 2^{\\frac{1}{2}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{2}}.\n\\end{align}\nBy (\\ref{JnormiH}) and (\\ref{virheprojektiolle8b}) we have\n\\begin{align}\n\\label{virheprojektiolle9b}\n \\norm{(I-P^N)J}_{L^{2}(0,2T)\\to L^2(0,2T)}\n\\le C_{J} 2^{\\frac{1}{2}}\\big((2T)^{\\frac{1}{2}} \\frac{T}{N}C_{sob}\\big)^{\\frac{1}{2}}.\n\\end{align}\nWe have\n\\begin{align}\n\\label{virheprojektiolle9c}\n \\norm{J(I-P^N)}_{L^{2}(0,2T)\\to L^2(0,2T)}=\n \\norm{(I-P^N)J^*}_{L^{2}(0,2T)\\to L^2(0,2T)}.\n\\end{align}\nBy (\\ref{JnormiH}), (\\ref{virheprojektiolle9b}), and (\\ref{virheprojektiolle9c})\nwe get\n\\begin{align}\n\\label{virheprojektiolle9d}\n \\norm{J-P^NJP^N}_{Y}\\le\n \\norm{(I-P^N)J}_{Y}+\\norm{P^N}_{Y}\\norm{J(I-P^N)}_{Y}\n\\le CN^{-\\frac{1}{2}}.\n\\end{align}\n\\end{proof}\n\nLet $J$ be as in (\\ref{operaattorit}), $J^N$ be as in (\\ref{pridekulkue}), and $R$ be as in (\\ref{operaattorit2}). We define \n\\begin{align}\n\\label{Koo2}\nK^N : Y \\to Y, \\quad\nK^N L = J_NL-RLRJ_N.\n\\end{align}\nBy (\\ref{operaattorit3}) and (\\ref{Koo2}) we define \n\\begin{align}\n\\label{Hn}\n\\boldsymbol{H}^N &:Y \\to C([0,T],Y), \n\\quad\\boldsymbol{H}^NL(r) = \nP_r(K^N L)P_r.\n\\end{align}\n\\begin{proposition}\n\\label{HN}\nLet $\\epsilon\\in (0,1)$, $\\kappa_3>0$, and $N\\ge\\kappa_3\\epsilon^{-4}$. Let $\\Lambda $ be as in (\\ref{lam}), $\\boldsymbol{H}$ be as in (\\ref{Hoo}), and $\\boldsymbol{H}^N$ be as in (\\ref{Hn}). Assume that $ \\widetilde\\Lambda_N\\in \\mathcal{B}_Y ( \\Lambda,\\epsilon)$, then we have \n\\begin{align*}\n \\norm{\\boldsymbol{H}^N\\widetilde\\Lambda_N -\\boldsymbol{H}\\Lambda }_{C([0,T],Y)}\\le C_3\\epsilon.\n\\end{align*}\n\\end{proposition}\nHere $C_3=C_3(T,\\kappa_3)>0$ depends on $T$ and $\\kappa_3$.\n\\begin{proof}\nBy (\\ref{Koo}) and (\\ref{Koo2}) we get\n\\begin{align}\n\\label{Kvirhe10b}\nK^N\\widetilde\\Lambda_N -K\\widetilde\\Lambda_N=\nR\\widetilde\\Lambda_NR(J-J_N)+(J_N-J)\\widetilde\\Lambda_N.\n\\end{align}\nUsing (\\ref{operaattorit2}), we have $\\norm{R}_Y\\le 1$. Using \\cite[Theorem 5]{jusa1}, we have $\\norm{\\Lambda}_Y \\le M_1<\\infty$. By (\\ref{Kvirhe10b}) and Lemma \\ref{JN} we have\n\\begin{align}\n\\label{Kvirhe10}\n \\norm{\\boldsymbol{H}^N\\widetilde\\Lambda_N(r) -\\boldsymbol{H}\\widetilde\\Lambda_N (r) }_Y\\le\\norm{K^N\\widetilde\\Lambda_N -K\\widetilde\\Lambda_N}_{Y}\\le\n2(M_1+1)CN^{-\\frac{1}{2}}.\n\\end{align}\nBy \\cite[Proposition 1]{jusa1} we have\n\\begin{align}\n\\label{Kvirhepapru}\n \\norm{\\boldsymbol{H}\\widetilde\\Lambda_N -\\boldsymbol{H}\\Lambda}_{C([0,T],Y)}\\le\nT\\epsilon.\n\\end{align}\nBy (\\ref{Kvirhe10}) and (\\ref{Kvirhepapru}), when $\\epsilon\\in (0,1)$ and $N\\ge\\kappa_3\\epsilon^{-4}$ we have \n\\begin{align}\n\\label{tohtorisaksala}\n \\norm{\\boldsymbol{H}^N\\widetilde\\Lambda_N -\\boldsymbol{H}\\Lambda }_{C([0,T],Y)}\\le 2(M_1+1)C\\Big(\\frac{1}{\\kappa_2}\\Big)^{\\frac{1}{2}}\\epsilon^{2}\n +T\\epsilon\\le C_3\\epsilon.\n\\end{align}\n\\end{proof}\n\nLet $P_r$ be as in (\\ref{operaattorit3}), $b$ be as in (\\ref{innerproduct}), and $P^N$ be as in (\\ref{pcsfjoukko2}). We define \n\\begin{align}\n\\label{innerproduct2}\n& \\boldsymbol S^N: C([0,T],Y) \\to C([0,T]),\n\\quad \\boldsymbol S^NL(r) =\\langle P^N L(r)P_r b,b\\rangle _{L^2(0,2T)}.\n\\end{align}\n\\begin{lemma}\n\\label{SN}\nLet $L\\in C([0,T],Y)$ and $\\boldsymbol{S}$ be as defined in (\\ref{innerproduct}). Then\n\\begin{align*}\n\\norm{\\boldsymbol S^NL-\\boldsymbol SL}_{C([0,T])}\n\\le \\frac{T^3}{6N}\\norm{L}_{C([0,T],Y)}.\n\\end{align*} \n\\end{lemma}\n\\begin{proof} By using (\\ref{innerproduct}), (\\ref{innerproduct2}) with the self-adjointness of operator $P^N$ we get \n\\begin{align*}\n|\\boldsymbol S^NL(r)-\\boldsymbol SL(r)|\\le \\norm{L(r)}_Y \\norm{P_rb}_{L^2(0,2T)}\\norm{P^Nb-b}_{L^2(0,2T)}.\n\\end{align*}\nWe have $\\norm{P_rb}_{L^2(0,2T)}\\le(\\frac{T^3}{3})^{\\frac{1}{2}}$ and $\\norm{P^Nb-b}_{L^2(0,2T)}\\le(\\frac{T^3}{12N^2})^{\\frac{1}{2}}.$\n\\end{proof}\nLet $K$ be as in (\\ref{Koo}) and $K^N$ be as in (\\ref{Koo2}). Let $\\boldsymbol{H}$ be as in (\\ref{Hoo}) and $\\boldsymbol{H}^N$ be as in (\\ref{Hn}). \nLet $\\boldsymbol{Z}_{\\alpha}$ be as in (\\ref{Zalfa}). For $r\\in[0,T]$ and $\\alpha>0$ we denote\n\\begin{align}\n\\label{elvisonkunkku}\n& Z_{\\alpha,r}=\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda)(r),\n \\\\\\nonumber &\\widetilde Z_{\\alpha,r}=\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N)(r).\n\\end{align} \nLet $\\boldsymbol{S}$ be as in (\\ref{innerproduct}). Using (\\ref{elvisonkunkku}) we denote\n\\begin{align}\n\\label{sope2}\n& s_\\alpha = \\boldsymbol S \\circ \\boldsymbol Z_\\alpha \\circ \\boldsymbol H\\Lambda,\n \\\\\\nonumber &\\widetilde s_\\alpha = \\boldsymbol S \\circ \\boldsymbol Z_\\alpha \\circ \\boldsymbol{H}^N\\widetilde\\Lambda_N.\n\\end{align} \nLet $\\boldsymbol S^N$ be as in (\\ref{innerproduct2}). Using (\\ref{elvisonkunkku}) we denote\n\\begin{align}\n\\label{sope3}\n\\widetilde s^N_\\alpha = \\boldsymbol S^N \\circ \\boldsymbol Z_\\alpha \\circ \\boldsymbol{H}^N\\widetilde\\Lambda_N,\\quad \\widetilde s^N_\\alpha (r)\n =\\langle P^N\\widetilde Z_{\\alpha,r} P_r b,b\\rangle _{L^2(0,2T)}.\n\\end{align} \n\\begin{proposition}\n\\label{PSN}\nLet $\\epsilon\\in (0,1)$ and $\\kappa_4>0$. Let $\\alpha=2\\epsilon^{\\frac{4}{9}}$ and $N\\ge\\kappa_4\\epsilon^{-4}$. Let $s_\\alpha$ be as defined in (\\ref{sope2}) and $\\widetilde s^N_\\alpha$ be as defined in (\\ref{sope3}). Let $\\boldsymbol{H} \\Lambda(r)\\in Y$ be bounded and positive semidefinite for all $r\\in[0,T]$. Assume that $\\boldsymbol{H}^N\\widetilde\\Lambda_N\\in \\mathcal{B}_{C([0,T],Y)} (\\boldsymbol{H} \\Lambda,\\epsilon)$, then we have\n\\begin{align*}\n\\norm{\\widetilde s^N_\\alpha-s_\\alpha}_{C([0,T])}\n\\le C_4\\epsilon^{\\frac{1}{9}}.\n\\end{align*} \nHere $C_4=C_4(T,\\kappa_3)>0$ depends on $T$ and $\\kappa_3$.\n\\end{proposition}\n\n\\begin{proof}\nWe have \n\\begin{align}\n\\label{soinintimppa}\n\\norm{\\widetilde s^N_\\alpha-s_\\alpha}_{C([0,T])}\n\\le\\norm{\\widetilde s^N_\\alpha-\\widetilde s_\\alpha}_{C([0,T])}+\\norm{\\widetilde s_\\alpha-s_\\alpha}_{C([0,T])} .\n\\end{align} \nLet $\\boldsymbol{S}$ be as defined in (\\ref{innerproduct}). We have $\\norm{\\boldsymbol{S}}_{C([0,T],Y)\\to C([0,T])}\\le \\frac{T^3}{3}$, see \\cite[Proposition 2]{jusa1}. Having $\\boldsymbol{H}^N\\widetilde\\Lambda_N\\in \\mathcal{B}_{C([0,T],Y)} (\\boldsymbol{H} \\Lambda,\\epsilon)$ we get $\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N) -\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda) }_{C([0,T],Y)}\\le\\frac{1}{2}\\epsilon^{\\frac{1}{9}}$, see \\cite[Proposition 2]{jusa1}. Using (\\ref{sope2}), for the second part of the sum in the right-hand side we get\n\\begin{align}\n\\label{rape}\n\\norm{\\widetilde s_\\alpha-s_\\alpha}_{C([0,T])}\n\\le\\norm{\\boldsymbol{S}}_\\Omega\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N) -\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda) }_{C([0,T],Y)}\\le\\frac{T^3}{3}\\frac{1}{2}\\epsilon^{\\frac{1}{9}} ,\n\\end{align} \nwhere we denote $\\Omega=C([0,T],Y)\\to C([0,T])$.\nUsing (\\ref{sope2}) and (\\ref{sope3}) with Lemma \\ref{SN}, for the first part of the sum in the right-hand side we get\n\\begin{align}\n\\label{rape2}\n\\norm{\\widetilde s^N_\\alpha-\\widetilde s_\\alpha}_{C([0,T])}\n\\le\\frac{T^3}{6N}\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N)}_{C([0,T],Y)}.\n\\end{align} \nWe have \n\\begin{align}\n\\label{rape3}\n\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N)}_\\Omega\n\\le\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H}^N\\widetilde\\Lambda_N)-\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda) }_\\Omega+\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda) }_\\Omega,\n\\end{align} \nwhere we denote $\\Omega=C([0,T],Y)$. Using \\cite[Eq. (32)]{jusa1} we have\n\\begin{align}\n\\label{rape4}\n\\norm{\\boldsymbol{Z}_{\\alpha}(\\boldsymbol{H} \\Lambda (r)) }_Y\\le\\alpha^{-1}.\n\\end{align} \nHaving $\\epsilon\\in (0,1)$, $N\\ge\\kappa_4\\epsilon^{-4}$ and $\\alpha=2\\epsilon^{\\frac{4}{9}}$ with use of (\\ref{rape2}), (\\ref{rape3}), and (\\ref{rape4}) we get\n\\begin{align}\n\\label{rape5}\n\\norm{\\widetilde s^N_\\alpha-\\widetilde s_\\alpha}_{C([0,T])}\n\\le\\frac{T^3}{6}\\frac{\\epsilon^4}{\\kappa_4}\\Big(\\frac{1}{2}\\epsilon^{\\frac{1}{9}}+\\frac{1}{2}\\epsilon^{-\\frac{4}{9}}\\Big)\\le \\frac{T^3}{3\\kappa_4}\\epsilon^{\\frac{32}{9}}.\n\\end{align} \nBy using (\\ref{soinintimppa}) (\\ref{rape}) and (\\ref{rape5}) we get the estimate.\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{kaiken_teoria2}]\n\\label{hauki}\n\\hfill \\break\nLet us consider the measurement ($\\epsilon_1, N_1, \\widetilde\\Lambda_{N_1}$).\nThere is $N_1\\ge\\epsilon_1^{-4}$ and $\\widetilde\\Lambda_{N_1}$, for which $\\norm{\\widetilde\\Lambda_{N_1}-\\Lambda_{N_1}}_{Y}\\le \\epsilon_1$, and Proposition \\ref{DLb} gives us\n\n\\begin{align*}\n \\norm{\\widetilde\\Lambda_{N_1}-\\Lambda }_{{Y}}\\le C_2\\epsilon_1.\n\\end{align*}\n\n Let $\\epsilon_1\\in (0,C_2^{-1})$ and $\\epsilon=C_2\\epsilon_1$. We choose $\\kappa_3=C_2^4$ and thus $ N_1\\ge\\epsilon_1^{-4}=C_2^4\\epsilon^{-4}$. Having $\\widetilde\\Lambda_{N_1}\\in \\mathcal{B}_{Y} (\\Lambda,\\epsilon)$, Proposition \\ref{HN} gives us \n\\begin{align}\n\\label{patelaine2c}\n \\norm{\\boldsymbol{H}^{N_1}\\widetilde\\Lambda_{N_1} -\\boldsymbol{H}\\Lambda}_{C([0,T],Y)}\\le C_3C_2\\epsilon_1.\n\\end{align}\nLet $\\epsilon_1\\in(0,C_2^{-1}C_3^{-1})$ and $\\epsilon=C_2C_3\\epsilon_1$. We choose $\\kappa_4=C_2^4C_3^4$ and thus $ N_1\\ge\\epsilon_1^{-4}=C_2^4C_3^4\\epsilon^{-4}$. Let $\\alpha_1=2C_2^{\\frac{4}{9}}C_3^{\\frac{4}{9}}\\epsilon_1^{\\frac{4}{9}}$. Having (\\ref{patelaine2c}), Proposition \\ref{PSN} gives us\n\\begin{align}\n\\label{timojutilac}\n\\norm{\\widetilde s^{N_1}_\\alpha-s_{\\alpha}}_{C([0,T])} \\le\nC_4C_3^{\\frac{1}{9}}C_2^{\\frac{1}{9}}\\epsilon_1^{\\frac{1}{9}}.\n\\end{align}\nLet $V$ be as defined as in (\\ref{operaattorit4}) and $D_{h}$ be as defined as in (\\ref{operaattori6}). Let $\\epsilon_1\\in(0,\\widehat\\epsilon)$, where $\\widehat\\epsilon$ as defined in \\cite[Proposition 4]{jusa1}. Let $\\kappa_5 >0$ and $h=\\kappa_5\\epsilon_1^{\\frac{1}{18}}$. Having (\\ref{timojutilac}) and using \\cite[Proposition 4]{jusa1} we get \n\\begin{align} \n\\label{timojutila2c}\n\\norm{D_{h}(\\widetilde s^{N_1}_\\alpha)-\\partial_r V}_{L^\\infty (0,T)}\n\\le C_5 C_4^{\\frac{1}{2}} C_3^{\\frac{1}{18}}C_2^{\\frac{1}{18}}\\epsilon_1^{\\frac{1}{18}},\n\\end{align} \nwhere $C_5=C_5(\\kappa_5)$. Note that we use the parameter $\\kappa_5$ to control the size of discretization in (\\ref{operaattori6}).\nLet $v$ be as defined in (\\ref{operaattorit5}) and $W$ be as defined in (\\ref{arska}). Let us denote $\\widetilde{w}^{N_1}_{\\alpha}=W(D_{h}(\\widetilde s^{N_1}_\\alpha))$. \nHaving (\\ref{timojutila2c}) and using \\cite[Proposition 5]{jusa1} we get \n\\begin{align}\n\\label{sebastianahoc}\n\\norm{ \\widetilde{w}^{N_1}_{\\alpha}-v}_{L^\\infty (M)} \n\\le C_6 C_5 C_4^{\\frac{1}{2}} C_3^{\\frac{1}{18}} C_2^{\\frac{1}{18}}\\epsilon_1^{\\frac{1}{18}}.\n\\end{align}\nLet $\\nu =\\kappa_6\\epsilon_1^\\frac{1}{54}$ and $c\\in\\mathcal V^3$. Let $\\Phi$ be as in (\\ref{hiiri}) and $\\eta_\\nu$ be as in (\\ref{molliolli}). Let us denote $\\widetilde c^{N_1}_{\\alpha}=\\eta_\\nu \\ast\\Phi(\\widetilde w^{N_1}_\\alpha)$. Having (\\ref{sebastianahoc}) and using \\cite[Proposition 6]{jusa1} we get \n\\begin{align}\n\\label{kenguru}\n\\norm{\\widetilde c^{N_1}_{\\alpha} -c}_{C^2(M)} \n\\le C_7C_6^{\\frac{1}{3}} C_5^{\\frac{1}{3}} C_4^{\\frac{1}{6}} C_3^{\\frac{1}{54}} C_2^{\\frac{1}{54}}\\epsilon_1^{\\frac{1}{54}},\n\\end{align}\nwhere $C_7=C_7(\\kappa_6)$. Note that we use the parameter $\\kappa_6$ to control the support of $\\eta_\\nu$ in (\\ref{molliolli}).\nWe define\n\\begin{align}\n\\label{epsilooninolla}\n \\widetilde\\epsilon_1=min\\Big\\{C_2^{-1},C_2^{-1}C_3^{-1},\\widehat\\epsilon\\Big\\}.\n\\end{align} \n\nBy using (\\ref{Zalfa}), (\\ref{operaattori6}), (\\ref{arska}), (\\ref{hiiri}), (\\ref{smootapp}), (\\ref{Hn}), and (\\ref{innerproduct2}) we define \n\\begin{align}\n\\label{Rstategia2}\n& \\mathcal R^{(1)}_{N_1,\\alpha_1}:Y\\to Z,\n\\\\\\nonumber&\\mathcal \\mathcal R^{(1)}_{N_1,\\alpha_1}=\\Gamma_\\nu\\circ\\Phi\\circ W\\circ D_{h}\\circ\\boldsymbol{S}^{N_1}\\circ \\boldsymbol{Z}_{\\alpha_1}\\circ\\boldsymbol{H}^{N_1},\n\\end{align}\nand have the estimate\n\\begin{align*}\n\\norm{\\mathcal R^{(1)}_{N_1,\\alpha_1}(\\widetilde\\Lambda_{N_1})-c}_{ Z} \\le\na_1\\epsilon_1^{\\frac{1}{54}},\n\\end{align*}\nwhen $\\epsilon_1 \\in (0,\\widetilde\\epsilon_1)$ and $a_1=C_7C_6^{\\frac{1}{3}} C_5^{\\frac{1}{3}} C_4^{\\frac{1}{6}} C_3^{\\frac{1}{54}} C_2^{\\frac{1}{54}}$. \n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{kaiken_teoria2b}]\n\\label{ahven}\n\\hfill \\break\nLet ${\\phi_{j,N_0}}$ be as in (\\ref{kantafunktio}), where $j\\in\\{1,2,3,...,2N_0\\}$. \nLet $r\\in [0,2T)$ and we define\n\\begin{align}\n\\label{translaatio}\n\\mathcal T_r: L^2(0,2T) \\to L^2(0,2T), \n\\quad T_rf(t) = \n\\begin{cases}\n\\,\\quad 0, & t \\in (0,r), \\\\\nf(t-r), & t\\in [r,2T).\n\\end{cases}\n\\end{align}\n\nUsing (\\ref{mittauspoika}) and (\\ref{translaatio}) we define\n\\begin{align}\n\\label{mittausaikapoika}\n \\widetilde\\Lambda_{\\mathcal P}\\phi_{j,N_0}=\\Big(\\frac{N_0}{T}\\Big)^{\\frac{1}{2}}\\Big(\\mathcal T_{\\frac{(j-1)T}{N_0}} -\\mathcal T_{\\frac{jT}{N_0}}\\Big)P^{N_0} \\widetilde m_{N,\\epsilon_0}.\n\\end{align}\nAs $\\big\\{\\phi_{1,N_0},...,\\phi_{2N_0,N_0}\\big\\}$ span $\\mathcal P^{N_0}$ this defines a linear map\n\\begin{align}\n\\label{mittahulg}\n \\widetilde\\Lambda_{\\mathcal P}:\\mathcal P^{N_0}\\to \\mathcal P^{N_0}.\n\\end{align}\nUsing (\\ref{mittausaikapoika}) and (\\ref{mittahulg}) we define a perturbed and discretizatized Neumann-to-Dirichlet operator\n\n\\begin{align}\n\\label{mittamulgi}\n&\\widetilde\\Lambda_{N_0}:L^2(0,2T)\\to L^2(0,2T),\n\\\\\\nonumber& \\widetilde\\Lambda_{N_0} |_{\\mathcal P^{N_0}}= \\widetilde\\Lambda_{\\mathcal P},\n\\\\\\nonumber&\\widetilde\\Lambda_{N_0} |_{(\\mathcal P^{N_0})^\\perp}=0,\n\\end{align}\nwhere $\\mathcal P^{N_0}\\oplus (\\mathcal P^{N_0})^\\perp = L^2(0,2T)$. Using (\\ref{mittauspoika}), (\\ref{mittausaikapoika}), (\\ref{mittahulg}), and (\\ref{mittamulgi}) we define\n\\begin{align}\n\\label{erkkipoika}\nE_{N_0} : \\mathcal P^N \\to Y,\\quad\n E_{N_0} \\widetilde m_{N,\\epsilon_0}=\\widetilde\\Lambda_{N_0}.\n\\end{align}\nLet $\\epsilon_0>0$. Let $l_0(\\epsilon_0)$ be as in (\\ref{porraspossu}) and $N_0(\\epsilon_0)$ be as in (\\ref{porraspossu2}). Let $N=2^l\\ge N_0$, where $l\\in\\mathbb{Z}_+$. Let $\\Lambda_{N_0} $ be as in (\\ref{pallokala}) and $\\widetilde\\Lambda_{N_0}$ be as in (\\ref{mittamulgi}). Let $P^N$ be as in (\\ref{pcsfjoukko2}) and $\\phi_{1,N}$ be as in (\\ref{kantafunktio}). Let $\\widetilde m_{N,\\epsilon_0}$ be as in (\\ref{mittauspoika}). Assume that $\\widetilde m_{N,\\epsilon_0}\\in \\mathcal{B}_{L^2(0,2T)} (P^N\\Lambda H,\\epsilon_0)$ and $\\norm{n_{N,\\epsilon_0} }_{L^2(0,2T)}\\le\\epsilon_0$. Then \n\\begin{align}\n\\label{kumiukko}\n \\norm{\\widetilde\\Lambda_{N_0}-\\Lambda_{N_0} }_{{Y}}\\le C_{P_1}\\epsilon_0^{\\frac{1}{5}}.\n\\end{align}\n\nHere $C_{P_1}=C_{P_1}(T)>0$ depends on $T$.\nHaving (\\ref{kantafunktio}), (\\ref{porras}), and (\\ref{translaatio}) we get \n \\begin{align}\n\\label{mittauspoikanolla}\n\\Lambda_{N_0}\\phi_{1,{N_0}}=\\Big(\\frac{N_0}{T}\\Big)^{\\frac{1}{2}}P^{N_0}\\Lambda P^{N_0}(I-\\mathcal T_{\\frac{T}{N_0}})H.\n\\end{align}\nAs $N=2^l\\ge N_0=2^{l_0}$ we have $P^{N_0}P^N=P^{N_0}$. Using $\\mathcal T_r \\Lambda=\\Lambda\\mathcal T_r$ and $P^{N_0}\\mathcal T_{\\frac{T}{N_0}}=\\mathcal T_{\\frac{T}{N_0}}P^{N_0}$ with (\\ref{mittauspoika}), (\\ref{mittausaikapoika}), and (\\ref{mittauspoikanolla}) we get \n \\begin{align}\n\\label{mittauspoikan}\n \\widetilde\\Lambda_{N_0}\\phi_{1,{N_0}}-\\Lambda_{N_0}\\phi_{1,{N_0}}=\\Big(\\frac{N_0}{T}\\Big)^{\\frac{1}{2}}(I-\\mathcal T_{\\frac{T}{N_0}})n_{N,\\epsilon_0}.\n\\end{align}\nUsing $N_0\\le\\epsilon_0^{-\\frac{4}{5}}$, $\\norm{\\mathcal T_{\\frac{T}{N_0}} }_{L^2(0,2T)}\\le 1$ and $\\norm{n_{N,\\epsilon_0} }_{L^2(0,2T)}\\le\\epsilon_0$ we get \n\\begin{align}\n\\label{sorsa}\n \\norm{\\widetilde\\Lambda_{N_0}\\phi_{1,{N_0}}-\\Lambda_{N_0}\\phi_{1,{N_0}} }_{L^2(0,2T)}\\le 2T^{-\\frac{1}{2}}\\epsilon_0^{\\frac{3}{5}}.\n\\end{align}\nLet $f\\in L^2(0,2T)$. By (\\ref{pcsfjoukko2}) we get\n\\begin{align}\n\\label{virheporrasfunktiolle2}\n \\norm{\\widetilde\\Lambda_{N_0} f-\\Lambda_{N_0} f}_{L^2(0,2T)}\\le \\sum_{j=1}^{2N_0} \\langle f,\\phi_{j,N_0}\\rangle\n_{L^2(0,2T)}\\norm{P^{N_0}\\widetilde\\Lambda\\phi_{j,{N_0}}-P^N\\Lambda\\phi_{j,{N_0}}}_{L^2(0,2T)}.\n\\end{align}\nUsing (\\ref{kantafunktio}), (\\ref{mittausaikapoika}), (\\ref{mittauspoikanolla}), and (\\ref{mittauspoikan}) we have\n\\begin{align}\n\\label{virheporrasfunktiolle2cc}\n \\norm{\\widetilde\\Lambda_{N_0} f-\\Lambda_{N_0} f}_{L^2(0,2T)}\\le\\norm{P^{N_0}\\widetilde\\Lambda\\phi_{1,{N_0}}-P^{N_0}\\Lambda\\phi_{1,{N_0}}}_{L^2(0,2T)} \\sum_{j=1}^{2{N_0}} \\langle f,\\phi_{j,{N_0}}\\rangle\n_{L^2(0,2T)}.\n\\end{align}\nThus (\\ref{sorsa}) and (\\ref{virheporrasfunktiolle2cc}) gives us\n\\begin{align}\n\\label{virheporrasfunktiolle2ccg}\n\\norm{\\widetilde\\Lambda_{N_0} f-\\Lambda_{N_0} f}_{L^2(0,2T)}\\le (2N_0)^{\\frac{1}{2}}\\norm{f}_{L^2(0,2T)}2T^{-\\frac{1}{2}}\\epsilon_0^{\\frac{3}{5}}.\n\\end{align}\nWe have $N_0\\le\\epsilon_0^{-\\frac{4}{5}}\\le 2N_0$ and this proves (\\ref{kumiukko}).\n\nWe define \n\\begin{align}\n\\label{anttiaffauros}\n \\epsilon_1=C_{P_1}\\epsilon_0^{\\frac{1}{5}}.\n\\end{align}\nUsing $N_0\\le\\epsilon_0^{-\\frac{4}{5}}\\le 2N_0$ and (\\ref{anttiaffauros}) we get\n\\begin{align}\n\\label{anttiaffauros2}\n N_0\\ge 2^{-1}C_{P_1}^{4}\\epsilon_1^{-4}.\n\\end{align}\nBy (\\ref{anttiaffauros2}), $\\widetilde\\Lambda_{N_0}\\in \\mathcal{B}_Y (\\Lambda_{N_0},\\epsilon_1)$, and Theorem \\ref{kaiken_teoria2} we get \n\\begin{align}\n\\label{katiska}\n\\norm{\\widetilde c^{N_0}_{\\alpha} -c}_{C^2(M)} \n\\le C_7C_6^{\\frac{1}{3}} C_5^{\\frac{1}{3}} \\widetilde C_4^{\\frac{1}{6}} \\widetilde C_3^{\\frac{1}{54}} \\widetilde C_2^{\\frac{1}{54}}\\epsilon_1^{\\frac{1}{54}}.\n\\end{align}\nNote that in the proof of Theorem \\ref{kaiken_teoria2} we have assumed $N_1\\ge\\epsilon_1^{-4}$. After replacing this by (\\ref{anttiaffauros2}), the proof is identical, only the constants $\\widetilde C_2, \\widetilde C_3, \\widetilde C_4$ change---see (\\ref{virheprojektiolle15}), (\\ref{tohtorisaksala}), (\\ref{rape4}), and (\\ref{kenguru}). Using (\\ref{anttiaffauros}) we get\n\\begin{align*}\n\\norm{\\widetilde c^{N_0}_{\\alpha} -c}_{C^2(M)} \n\\le C_7C_6^{\\frac{1}{3}} C_5^{\\frac{1}{3}} \\widetilde C_4^{\\frac{1}{6}} \\widetilde C_3^{\\frac{1}{54}}\\widetilde C_2^{\\frac{1}{54}}C_{P_1}^{\\frac{1}{54}}\\epsilon_0^{\\frac{1}{270}}.\n\\end{align*}\n\nWe define\n\\begin{align}\n\\label{epsilooninolla}\n \\widetilde\\epsilon_0=min\\Big\\{C_{P_1}^{4}\\widetilde C_2^{-5},C_{P_1}^{4}\\widetilde C_2^{-5}C_3^{-5},\\widehat\\epsilon\\Big\\},\n\\end{align} \nwhere $\\widehat\\epsilon$ can be specified by using \\cite[Proposition 4]{jusa1}. \nBy using (\\ref{Zalfa}), (\\ref{operaattori6}), (\\ref{arska}), (\\ref{hiiri}), (\\ref{smootapp}), (\\ref{erkkipoika}), (\\ref{Hn}), and (\\ref{innerproduct2}), we define \n\\begin{align}\n\\label{Rstategia2b}\n&\\mathcal R^{(0)}_{N_0,\\alpha_0}:\\mathcal P^{N}\\to Z,\n\\\\\\nonumber&\\mathcal R^{(0)}_{N_0,\\alpha_0}=\\Gamma_\\nu\\circ\\Phi\\circ W\\circ D_{h}\\circ\\boldsymbol{S}^{N_0}\\circ \\boldsymbol{Z}_{\\alpha}\\circ\\boldsymbol{H}^{N_0}\\circ E^{N_0},\n\\end{align}\nand have the estimate\n\\begin{align*}\n\\norm{\\mathcal R^{(0)}_{N_0,\\alpha_0}(\\widetilde m_{N,\\epsilon_0})-c}_{ Z} \\le\na_0\\epsilon_0^{\\frac{1}{270}},\n\\end{align*}\nwhen $\\epsilon_0 \\in (0,\\widetilde\\epsilon_0)$ and $a_0=C_7C_6^{\\frac{1}{3}} C_5^{\\frac{1}{3}} \\widetilde C_4^{\\frac{1}{6}} \\widetilde C_3^{\\frac{1}{54}}\\widetilde C_2^{\\frac{1}{54}}C_{P_1}^{\\frac{1}{54}}$. \n\\end{proof}\n\n\n\n\n\\section{Numerical examples}\n\\label{sec_computations}\n\nIn this section we describe a computational implementation of the regularization strategy in Theorem \\ref{kaiken_teoria2}. We will also compare this with a heuristic variant of MDP---see (\\ref{morozov}).\nWe will begin by describing how the data---that is, the noisy discretized Neumann-to-Dirichlet map $\\widetilde \\Lambda_{N_1}$---is simulated. \n\n\n\\subsection{The simulation of measurement data}\n\nWe choose $T = 0.6$ in all the simulations.\nWe use $k$-Wave \\cite{K-Wave} to solve the\nboundary value problem (\\ref{dartwader}) with \n$f =\\phi_{1,N_1} \\in \\mathcal P_{N_1}$, \nwhere $N_1 = 2^{10}$, and denote the solution by $u^{(sim)}$.\nRecall that \n\\begin{align}\n\\label{ekakantafunktio}\n \\phi_{1,N_1}(t) = h^{-\\frac{1}{2}}1_{[0,h)}(t), \\quad t \\in [0,2T),\n\\end{align} \nwhere $h = T \/ N_1$.\nIn order to simulate $u^{(sim)}$ for $2T$ time units, a fine discretization needs to be used, and we choose a regular mesh with $2^{13}$ spatial and $N_2 = 2^{15}$ temporal cells. \nThen we define the simulated Neumann-to-Dirichlet map, acting on the first basis function,\n\\begin{align}\n\\label{possuli}\n\\Lambda^{(sim)} \\phi_{1,N_1}(t) = \\sum_{j=1}^{2N_1} u^{(sim)}(t_j, 0) \\phi_{j,N_2}(t), \\quad t \\in [0,2T),\n\\end{align}\nwhere $t_j=\\frac{(j-1)2T}{N_2}$, $j\\in\\{1,2,...,N_2\\}$, are the temporal grid points.\nThe output of k-Wave is, of course, only an approximation of $u^{(sim)}$ but we do not analyze this simulation error and use the same notation for both $u^{(sim)}$ and its approximation. \n\nOur primary object of interest is the following discretized version of the Neumann-to-Dirichlet map \n\\begin{align}\n\\label{lamdad}\n \\Lambda_{N_1}^{(d)} : \\mathcal P^{N_1} \\to \\mathcal P^{N_1},\\quad\\Lambda_{N_1}^{(d)}f=\\sum_{j=1}^{2N_1}\\sum_{k=1}^{j}f_k\\Lambda_{j-k+1}\\phi_{j,N_1}, \n\\end{align}\nwhere $\\Lambda_j=\\langle \\Lambda^{(sim)} \\phi_{1,N_1},\\phi_{j,N_1}\\rangle_{L^2(0,2T)}$ and $f_k$, $k=1,2,\\dots,2N_1$,\nare the coefficients of $f$ on the basis of $\\mathcal P^{N_1}$.\nObserve that $\\Lambda_{N_1}^{(d)} \\phi_{1,N_1}$ is \nsimply the projection of $\\Lambda^{(sim)} \\phi_{1,N_1}$ on $\\mathcal P^{N_1}$, and that $\\Lambda_{N_1}^{(d)}f$ is then defined by using the fact that the wave equation (\\ref{dartwader}) is invariant with respect to translations in time.\n\nWe will now describe how the noise is simulated. \nConsider \n\\begin{equation}\n\\label{the_noise}\nn=(n_1,n_2,...,n_{2N_1})\\in\\mathbb{R}^{2N_1},\n\\end{equation}\nwhere\n$n_j\\in\\mathcal{N}(0,1)$, that is, $n_j$ is a normally distributed random variable with zero mean and unit variance.\nWe compute a realization of $n$ by using the randn function of MATLAB,\nand use the same notation for $n$ and its realization.\nLet $\\epsilon_0^{(d)}>0$ and define,\nanalogously to (\\ref{lamdad}),\na noisy, discretized version of the Neumann-to-Dirichlet map \n\\begin{align}\n\\label{lamdadvir}\n \\widetilde\\Lambda_{N_1}^{(d)} : \\mathcal P^{N_1} \\to \\mathcal P^{N_1},\\qquad\\widetilde\\Lambda_{N_1}^{(d)} f=\\sum_{j=1}^{2N_1}\\sum_{k=1}^{j}f_k \\widetilde\\Lambda_{j-k+1}\\phi_{j,N_1},\n\\end{align}\nwhere $\\widetilde\\Lambda_j=\\Lambda_j+\\widehat n_j$ and \n\\begin{align}\n\\label{mittavirhe_mod}\n\\widehat n_j= \\frac{\\epsilon_0^{(d)}}{\\norm{n}_{l^2}}n_j, \n\\quad j=1,2,\\dots,2N_1.\n\\end{align} \nFollowing the formulation of Theorem \\ref{kaiken_teoria2},\nrather than using $\\epsilon_0^{(d)}>0$, we prefer to \nparametrize the noise level in terms of \n\\begin{align}\n\\label{laskennalvirhe}\n\\epsilon^{(d)}_1:= \\norm{\\widetilde \\Lambda_{N_1}^{(d)}-\\Lambda_{N_1}^{(d)} }_{\\mathcal P^{N_1}}.\n\\end{align}\nIn what follows, we will consider \nthe quantity $(\\epsilon^{(d)}_1, \\widetilde \\Lambda_{N_1}^{(d)})$, \na simulated analogue of the noisy measurements in Section \\ref{virhesikapossu}. \n\n\n\\subsection{Implementation of the regularization strategy}\n\nFor the a priori bounds in (\\ref{nopeudet}) we use values $C_0=0.01$, $C_1=10$, $L_0=0.01$, and $L_1=0.6$.\nThe crux of the regularization strategy $\\mathcal R^{(1)}_{N_1,\\alpha_1}$ is the computation of the inverse in (\\ref{Zalfa}). \nWhen starting from the simulated measurement $(\\epsilon^{(d)}_1, \\widetilde \\Lambda_{N_1}^{(d)})$, the analogue of (\\ref{Zalfa}) is to solve $X_j$ in the equation\n\\begin{align}\n\\label{yhtalot}\n(P_{r_j}K^{N_1} \\widetilde \\Lambda_{N_1}^{(d)} P_{r_j}+\\alpha_1)X_j=P_{r_j}b.\n\\end{align}\nHere $P_r$ is the projection in (\\ref{operaattorit3}), and\nwe choose $r_j=jh$, $j = 1,2,\\dots,N_1$.\nThe choice of the regularization parameter $\\alpha_1$ is discussed in detail below. \n\nWe use the restarted generalized minimal residual (GMRES) method to solve the system of linear equations (\\ref{yhtalot}) and choose six as the maximum number of outer iterations and 10 as the number of inner iterations (restarts). We use the initial guess $f = 0$ and the tolerance of the method is set to 1e-12.\n\nAfter this we simply follow the regularization strategy \n(\\ref{Rstategia2}), that is, we get an approximation of $c$ \nby setting\n\\begin{align}\n\\label{the_rec}\n\\widetilde c^{N_1}_{\\alpha_{1}}=\\Gamma_\\nu(\\Phi(W(D_h(\\widetilde s^N_{\\alpha_1})))),\n\\end{align}\nwhere $\\widetilde s^{N_1}_{\\alpha_1} (r_j)\n =\\langle X_j,b\\rangle _{L^2(0,2T)}$, $j = 1,2,\\dots,N_1$.\nThe scaling of $\\nu$ is chosen as follows \n\\begin{align*}\n\\nu=0.01(\\epsilon_1^{(d)})^{\\frac{1}{54}}.\n\\end{align*}\nIn the numerical computations the parameter $h$ was fixed to be $h=\\frac{T}{N_1}$, that is, the discrete derivative $D_h$ was computed in the grid that is used in (\\ref{ekakantafunktio}) to represent the basis functions $\\phi_{1,N_1}$. Observe that this deviates from the theoretical choice $h=C\\epsilon_1^{\\frac{1}{18}}$ used in (\\ref{Rstategia}). We will describe next how the regularization parameter $\\alpha_1$ is chosen, and then we will study how the error $\\widetilde c^{N_1}_{\\alpha_{1}} - c$ behaves as function of $\\epsilon^{(d)}_1$.\n\n\n\\subsection{Calibration of the regularization strategy}\n\nRecall that in Theorem \\ref{kaiken_teoria2} \nthe choice of regularization parameter is of the form\n$\\alpha_1=C_{reg1}(\\epsilon_1^{(d)})^p$, where $p={\\frac{4}{9}}$.\nIn particular, the choice is explicit apart from the constant $C_{reg1}$. In this section we choose $C_{reg1}$ so that it gives a good reconstruction of a particular velocity function $c_c$---see Figure \\ref{testprofile}. Then the same constant is used in all the subsequent computational examples. \n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.18]\n{calibfun.jpg}\n\\caption{The velocity function $c_c$\nused in the calibration of the regularization strategy.\n} \n\\label{testprofile}\n\\end{figure}\n\nIn the regularization strategy we consider 10 values for measurement errors, as defined in (\\ref{laskennalvirhe})\n\\begin{align}\n\\label{epsihommeli}\n\\epsilon^{(d)}_{1,k}\\in\\{k\\cdot 10^{-2}|k=1,2,3,...,10\\},\n\\end{align} \nand nine values for the multiplicative constant $C_{reg1}=10^{-j}$, $j=1,3,...,9$.\nThen we consider the error in the reconstruction as a function of $j$,\n\\begin{align}\nerror(j)=\\norm{\\widetilde c^{N_1}_{\\alpha_{j,k}} -c_c}_{L^2 (M)},\n\\end{align}\nwhere for each error level, the reconstruction $\\widetilde c^{N_1}_{\\alpha_{j,k}}$ is computed by (\\ref{the_rec}).\nThese computations are summarized in Figure \\ref{choosing_C2}.\nWe see that the choice $j=4$, that is, \n\\begin{align}\n\\label{the_alpha}\n\\alpha_1 = 10^{-4} (\\epsilon_1^{(d)})^p, \\quad p={\\frac{4}{9}},\n\\end{align}\ngives a good reconstruction on all the error levels. \nIn what follows we will systematically use this choice. \n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.18]\n{cvalinta.jpg}\n\\caption{\nThe reconstruction error as a function of the multiplicative constant $C_j = C_{reg1}$. Each curve corresponds to a noise level in (\\ref{epsihommeli}).\nAs expected, the reconstruction error is monotonous as a function of the noise level: The highest line corresponds to $\\epsilon_1^{(d)}=0.1$\nand the lowest one to $\\epsilon_1^{(d)}=0.01$. \nWe also observe that the reconstruction error becomes more sensitive to the choice of $C_{reg1}$ as the noise level grows. \n} \n\\label{choosing_C2}\n\\end{figure}\n\n\\subsection{Reconstruction results based on the analysis}\nWe will now consider the reconstruction (\\ref{the_rec}), \nwith the choice of regularization parameter (\\ref{the_alpha}),\nin two test cases. \nWe begin with with a smooth velocity function $c_s$(see Figure \\ref{estimatkunkku}), where reconstructions of two different noise levels are shown. \n\n \\begin{figure}\n\\centering\n\\includegraphics[scale=0.14]\n{kuvauusisilee1.jpg}\n\\includegraphics[scale=0.14]\n{kuvauusisilee2.jpg}\n\\caption{\nTwo reconstructions (the solid blue lines) of a smooth velocity function $c_s$ (the dashed lines).\n{\\em Top:} Noise level $\\epsilon^{(d)}_1= \n 0.1$. {\\em Bottom:} Noise level $\\epsilon^{(d)}_1= \n 0.01$.\n }\n\\label{estimatkunkku}\n\\end{figure}\nTo study the order of convergence of our reconstruction method, \nwe consider 10 noise levels,\n\\begin{align}\n\\label{epsihommeli2}\n\\epsilon^{(d)}_{1}\\in\\{k\\cdot 10^{-2}|k=1,2,3,...,10\\},\n\\end{align} \nand simulate noisy measurements with five different realizations of the random vector\n$n$ in (\\ref{the_noise}) at each noise level.\nThe corresponding reconstruction errors \n$\\norm{\\widetilde c^{N_1}_{\\alpha_{1}} -c_s}_{L^2 (M)}$\nare summarized in Figure \\ref{fourestimates}.\nComputations suggest that the order of convergence is 0.40.\nThis is better than $\\frac{1}{54}$ in Theorem (\\ref{kaiken_teoria2}). \n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.18]\n{suppenema.jpg}\n\\caption{\nThe reconstruction error as a function of the noise level $\\epsilon^{(d)}_{1}$ (in log--log axes). We have used here 5 different realizations of the noise. The solid line is the average of these. \nLinear fitting (the dotted line) gives the estimated convergence the order 0.40.\n}\n\\label{fourestimates}\n\\end{figure}\n\n\nWe also tested the method with a non-smooth velocity function $c_p$(see Figure \\ref{estimatikka23}), where reconstructions of two different noise levels are shown. This case is not covered by the above analysis, but the reconstruction method is also robust in this case. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.14]\n{kuvauusiporras1.jpg}\n\\includegraphics[scale=0.14]\n{kuvauusiporras2.jpg}\n\\caption{\nTwo reconstructions (the solid blue lines) of a piecewise constant velocity function $c_p$ (the dashed lines).\n{\\em Top:} Noise level $\\epsilon^{(d)}_1= \n 0.1$. {\\em Bottom:} Noise level $\\epsilon^{(d)}_1= \n 0.01$.\n}\n\\label{estimatikka23}\n\\end{figure}\n\n\n\n\\subsection{Reconstruction results based on MDP}\n\\label{pamelapulkkinen}\nHere we use a heuristic version of MDP as a parameter choice\nrule for $\\alpha=\\alpha(\\epsilon)$.\nTypically MDP is applied to a Tikhonov regularization of the form\n\\begin{equation}\n\\label{classical_Tikhonov}\n\\min_x \\norm{F(x) - y}^2 + \\alpha \\norm{x - x^*}^2,\n\\end{equation}\nwhere $F$ is a model for the measurements, $y$ is the data, and $x^*$ plays the role of a selection criterion. \nIn our case, the model $F$ corresponds to $c \\mapsto \\Lambda_{N_1}^{c}\\phi_{1,N_1}$\nand $y = \\widetilde\\Lambda_{N_1}^{c}\\phi_{1,N_1}$ gives the measurement data, but we do not cast the inverse problem as a minimization problem and our regularization method is not of the Tikhonov type.\nIn particular, our method does not depend on the choice of the auxiliary parameter $x^*$, which can be viewed as an initial guess, and that chooses a local minimum of the non-linear optimization problem (\\ref{classical_Tikhonov}).\nDue to these differences, the existing results on MDP do not apply to our method, and (\\ref{morozov}) below is only a heuristic analogue of the classical MDP.\nWe refer to \\cite{Scherzer1993} for a study of MDP in an abstract context of the form (\\ref{classical_Tikhonov}), with non-linear $F$. \n\nThe heuristic principle that we use is as follows. We fix tuning parameters $h > 1$ and small $\\delta > 0$ and search for a regularization parameter $\\alpha$ in such a way that the following consistency condition holds:\n\\begin{align}\n\\label{morozov}\n(h-\\delta)\\epsilon\\le\\norm{\\Lambda_{N_1}^{\\widetilde c^{N_1}_{\\alpha}}\\phi_{1,N_1}-y}_{L^2(0,2T)}\n\\le(h+\\delta)\\epsilon.\n\\end{align} \nHere $\\epsilon > 0$ is the noise level, \n$y = \\widetilde\\Lambda_{N_1}^{c}\\phi_{1,N_1}$ is again the measurement data, and $\\Lambda_{N_1}^{\\widetilde c^{N_1}_{\\alpha}}\\phi_{1,N_1}$ is the corresponding data computed with the velocity function $\\widetilde c^{N_1}_{\\alpha}$, given by the reconstruction method. \nObserve that (\\ref{morozov}) is a relaxed version of (1.7) in \\cite{Scherzer1993}.\n\nWe choose $h=1.1$ and $\\delta = 0.01$ and use a bisection search to find $\\alpha$. Our implementation was unable to find $\\alpha$ satisfying the constraint (\\ref{morozov}) for noise levels $\\epsilon^{(d,k)}>0.02$.\nFor smaller noise levels, the regularization parameters found using the principle are summarized in Figure \\ref{estimatee}.\nWe see that, with the above choice of tuning parameters, the heuristic MDP always gives a larger regularization parameter than (\\ref{the_alpha}). The reconstructions are consistently worse than those produced by the choice (\\ref{the_alpha}).\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.15]\n{uusimorob.jpg}\n\\caption{\nRegularization parameter $\\alpha$, given by MDP as a function of the noise level (in log--log axes).\nThe straight line represents the relation (\\ref{the_alpha}).\n}\n\\label{estimatee}\n\\end{figure}\n\n\n\n\n\\noindent{\\bf Acknowledgements.} We thank Samuli Siltanen for inspiring discussions on the regularization on inverse problems. \n\nL.\\ Oksanen was partly supported by \nEPSRC, project EP\/P01593X\/1.\nJ.\\ Korpela and M.\\ Lassas were supported by the Academy of Finland, projects 263235, 273979, 284715, and 312119.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:introduction}\nMasked image modeling (MIM) \\cite{bao2021beit} is at the center of self-supervised representation learning, showing good potentials on various downstream tasks, including image classification, semantic segmentation, object detection, and instance segmentation. \nIt masks out some patches in the images with a pre-defined \\emph{mask ratio} and adds the reconstruction supervision on a set of patches at some specific positions, \\textit{i.e.}, \\emph{supervision position}. Specifically, in most MIM methods \\cite{bao2021beit, he2022masked, xie2022simmim, chen2022context}, the supervision positions are only associated with the masked patches, \\textit{i.e.}, only adding the supervisions on the masked patches. \n\n\n\n\\par\nThe reconstruction loss of MIM can be applied in different domains or targets, such as \nRGB~\\cite{he2022masked,xie2022simmim}, HOG~\\cite{wei2022masked}, discrete visual tokens~\\cite{bao2021beit,chen2022context,el2021large,peng2022beit,dong2021peco}, momentum encoders~\\cite{tao2022siamese,chen2022sdae,wu2022extreme}, and pretrained models~\\cite{wei2022mvp,wei2022masked}.\nRecently, MVP~\\cite{wei2022mvp} applies the reconstruction loss on the image representations of CLIP~\\cite{radford2021learning}, \\textit{i.e.}, minimizing the reconstruction error in the domain of CLIP representation. Benefiting from the rich multi-modality information and informative representation, the CLIP-targeted MVP performs very well. \n\n\n\n\n\\par\nDespite that, it is still under-explored how the detailed ways of applying CLIP in MIM affect performance.\nUnlike most MIM methods~\\cite{bao2021beit, he2022masked, xie2022simmim, chen2022context} applying the reconstruction supervision on the masked patches, MVP supervises both masked and unmasked patches.\nIt raises a question: how will the \\emph{supervision position} influence the CLIP-targeted MIM?\nOn the other hand, the \\emph{mask ratio} performs differently for different supervision targets~\\cite{he2022masked,bao2021beit}. \nWith CLIP as the target, it is unclear how the mask ratio behaves. \n\n\n\n\n\\par\nIn this paper, we study these two critical ingredients, \\textit{i.e.}, \\textit{supervision position} and \\textit{mask ratio}, in MIM with the CLIP representation as the supervision. \nTo conduct the study, we develop a simple \nMIM pipeline, \\textit{i.e.}, context autoencoder with CLIP target ({CAE v2}\\xspace).\nWe will first analyze how the supervision position and the mask ratio influence the performance of {CAE v2}\\xspace. \nThen, relying on the analyses, we implement \n{CAE v2}\\xspace as a concise yet effective MIM model, producing superior performance on \nvarious downstream tasks. \n\n\\par\n\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[trim =0mm 0mm 0mm 0mm, clip, width=0.85\\linewidth]{figure\/framework.pdf}\n\\caption{\nOverview of the proposed {CAE v2}\\xspace. \n{CAE v2}\\xspace first masks the input image $\\pmb{\\mathrm{x}}$ with the mask ratio $\\gamma$, which is positively correlated with the model size of encoder.\n$\\propto$ represents the positive correlation.\nThen, {CAE v2}\\xspace inputs the visible patches $\\pmb{\\mathrm{X}}_v$ into the encoder to obtain the latent representation $\\pmb{\\mathrm{Z}}_v$.\nThe decoder receives $\\pmb{\\mathrm{Z}}_v$ and the mask token $\\pmb{\\mathrm{E}}_m$ to recover the latent representations of the masked patches $\\pmb{\\mathrm{Z}}_m$.\nAfter a lightweight head, $\\pmb{\\mathrm{Z}}_v$ and $\\pmb{\\mathrm{Z}}_m$ are projected to $\\pmb{\\mathrm{Y}}_v$ and $\\pmb{\\mathrm{Y}}_m$.\n{CAE v2}\\xspace also inputs $\\pmb{\\mathrm{x}}$ into the CLIP model to generate the target supervisions, which are split to $\\pmb{\\mathrm{T}}_v$ and $\\pmb{\\mathrm{T}}_m$ according to the absolute positions of $\\pmb{\\mathrm{X}}_v$ and $\\pmb{\\mathrm{X}}_m$.\nThe optimization is applied on the prediction $\\pmb{\\mathrm{Y}}_v$ and the target supervision $\\pmb{\\mathrm{T}}_v$ of visible patches.\nMeanwhile, the loss on $\\pmb{\\mathrm{Y}}_m$ and $\\pmb{\\mathrm{T}}_m$ for masked patches is optional.\n}\n\\vspace{-1.0em}\n\\label{fig:framework}\n\\end{figure*}\n\nFirst, we study on the influence of the supervision position.\nExcept for applying the CLIP supervision on the predictions of masked patches,\nwe consider to directly supervise the latent representations of visible patches with CLIP features.\nSurprisingly, we find that the supervision \\textit{only on visible patches} achieves remarkable performance, even better than that on masked patches.\nIt reveals that the visible patches can effectively extract rich semantic information from CLIP, performing like the feature distillation. \nMoreover, combining the supervisions of masked and visible patches together like \\cite{wei2022mvp} brings slight performance improvement.\nTherefore, we advocate that the supervision on \\textit{visible patches is a good way for the supervision position} in the CLIP-targeted MIM.\n\n\n\n\nNext we explore the behavior of the mask ratio.\nRecall that\nMAE~\\cite{he2022masked} points out a high mask ratio (75\\%) is good for the balance of efficiency and effectiveness.\nOthers like \\cite{bao2021beit,chen2022context} empirically set the mask ratio as 40-50\\%, and MVP simply follows \\cite{bao2021beit}.\nHere we want to explore what is the optimal mask ratio in the CLIP-targeted MIM.\nWe conduct a series of experiments on a battery of models with different sizes and sweep the mask ratio from 15\\% to 95\\%.\nThe results are interesting, showing that the smaller models favor lower mask ratios, while larger models prefer higher ones.\nThis provides a new perspective that the optimal mask ratio is \\textit{positively correlated with the model size}.\n\n\n\n\n\n\n\n\nBased on the above analyses,\nour {CAE v2}\\xspace achieves superior performance on various scales of models.\nEspecially, with 300 epoch pre-training, {CAE v2}\\xspace can boost a vanilla ViT-Large to 81.7\\% and 86.7\\% top-1 accuracy on linear probing and fine-tuning on ImageNet-1K, and 55.9\\% mIoU on ADE20K.\nWe hope our analyses and findings can be useful guidelines for the future MIM studies, especially for the pre-training on lightweight models.\nIn summary, our contributions are:\n\\begin{itemize}\n\\itemsep -.051cm\n\\item We develop a simple CLIP-targeted MIM pipeline,\nnamely {CAE v2}\\xspace, to study the supervision position and the mask ratio;\n\\item For the supervision position, we advocate that applying the supervision on visible patches is a good way;\n\\item For the mask ratio, we present that the optimal mask ratio positively correlates with the model size;\n\\item Driven by these analyses, \nour {CAE v2}\\xspace achieves superior performance on different scales of models on various downstream tasks.\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{{CAE v2}\\xspace}\nWith CLIP as the supervision target, this paper aims to study the two important elements in MIM, \\textit{i.e.}, the supervision position and the mask ratio.\nWe achieve this by developing a simple framework {CAE v2}\\xspace, in which the supervision target is CLIP and \nthe model structure is based on \\cite{chen2022context,wei2022mvp} with some specific modifications.\nWe introduce the details as follows.\n\n\\subsection{Overview}\nThe overview of our {CAE v2}\\xspace is illustrated in Figure~\\ref{fig:framework}.\nLet $\\pmb{\\mathrm{x}}\\in\\mathcal{D}$ denote an input image.\nFollowing previous MIM methods~\\cite{bao2021beit,he2022masked,wei2022masked,chen2022context}, {CAE v2}\\xspace first embeds $\\pmb{\\mathrm{x}}$ into a total length of $N$ patches,\nwhich are then randomly masked by a specific proportion $\\gamma$.\nThese $N$ patches are naturally split into two non-overlapped sets, \\textit{i.e.}, visible patches $\\pmb{\\mathrm{X}}_v$ and masked patches $\\pmb{\\mathrm{X}}_m$, where $N=\\left | v \\right |+\\left | m \\right |$.\nThe mask ratio is thus denoted as $\\gamma=\\left | m \\right |\/N$.\nFollowing ~\\cite{he2022masked,chen2022context}, the encoder $\\mathcal{F}$ maps the visible patches $\\pmb{\\mathrm{X}}_v$ to the latent representations $\\pmb{\\mathrm{Z}}_v$.\nThe decoder $\\mathcal{G}$ predicts the latent representations $\\pmb{\\mathrm{Z}}_m$ for the masked patches from mask tokens $\\pmb{\\mathrm{E}}_m$.\nAfter that, the predictions of visible patches $\\pmb{\\mathrm{Y}}_v$ and masked patches $\\pmb{\\mathrm{Y}}_m$ are obtained \nvia a head $\\mathcal{H}$.\n\nFor the target supervision, we directly input the intact image $\\pmb{\\mathrm{x}}$ into the CLIP vision model $\\mathcal{T}$ to generate the target supervision $\\pmb{\\mathrm{T}}$.\n$\\pmb{\\mathrm{T}}$ is then split into $\\pmb{\\mathrm{T}}_v$ and $\\pmb{\\mathrm{T}}_m$ corresponding to the absolute positions of $\\pmb{\\mathrm{X}}_v$ and $\\pmb{\\mathrm{X}}_m$.\nThe optimization is applied on $\\pmb{\\mathrm{Y}}_v$ and $\\pmb{\\mathrm{T}}_v$, \nand we also study to add the supervision on $\\pmb{\\mathrm{Y}}_m$ and $\\pmb{\\mathrm{T}}_m$.\n\n\n\n\\subsection{Architecture}\n{CAE v2}\\xspace contains four modules, \\textit{i.e.}, one encoder, one decoder, one MIM head, and one CLIP model.\n\n\\myparagraph{Encoder.}\nThe encoder $\\mathcal{F}$ only receives the visible patches $\\pmb{\\mathrm{X}}_v$ following \\cite{he2022masked,chen2022context}.\n$\\mathcal{F}$ maps the visible patches $\\pmb{\\mathrm{X}}_v$ to the latent representations $\\pmb{\\mathrm{Z}}_v$ across a stack of transformer blocks.\nThe operation of $\\mathcal{F}$ is based on self-attention.\nIn this paper, we utilize a series of ViTs~\\cite{dosovitskiy2020image} to form the encoder, including ViT-Tiny, ViT-Small, ViT-Base, and ViT-Large.\n\n\\myparagraph{Decoder.} The decoder $\\mathcal{G}$ predicts the latent representation $\\pmb{\\mathrm{Z}}_m$ for the masked patches from the mask tokens $\\pmb{\\mathrm{E}}_m$, conditioned on the\nvisible latent representation $\\pmb{\\mathrm{Z}}_v$.\nIt is inspired by the latent contextual regressor in CAE~\\cite{chen2022context}.\n$\\mathcal{G}$ performs as the same as cross-attention.\nHere, we utilizes a lighweight decoder in {CAE v2}\\xspace, \\textit{i.e.}, one-layer transformer block\\footnote{We find that one-layer transformer block as the decoder ready can produce promising performance. For parameter-friendly, we use this structure as the default setting.\nWe do not analyze the influence of the depth of decoder, since it is not the main concern in this paper.\n}.\n\n\\myparagraph{Head.} The head $\\mathcal{H}$ maps the latent predictions $\\pmb{\\mathrm{Z}}_v$ and the latent representations $\\pmb{\\mathrm{Z}}_m$ to $\\pmb{\\mathrm{Y}}_v$ and $\\pmb{\\mathrm{Y}}_m$, respectively.\n$\\pmb{\\mathrm{Y}}_v$ and $\\pmb{\\mathrm{Y}}_m$ share the same form with the target supervisions $\\pmb{\\mathrm{T}}_v$ and $\\pmb{\\mathrm{T}}_m$.\nIn this work, we only use a FC (fully-connected) layer followed by a LN (layernorm) layer in $\\mathcal{H}$.\n\n\\myparagraph{CLIP model.} The vision branch of CLIP model $\\mathcal{T}$ generates the target supervisions $\\pmb{\\mathrm{T}}$.\nIn the whole paper, we only use the ViT-Base of the CLIP model as $\\mathcal{T}$.\n$\\pmb{\\mathrm{T}}$ is then split into the target supervisions for visible patches $\\pmb{\\mathrm{T}}_v$ and for masked patches $\\pmb{\\mathrm{T}}_m$ according to the absolute positions of $\\pmb{\\mathrm{X}}_v$ and $\\pmb{\\mathrm{X}}_m$.\n\n\n\\subsection{Study, Discovery and Analysis}\n\\label{sec:study}\nWe pay attention to two critical elements in MIM, \\textit{i.e.}, the supervision position and the mask ratio.\n\n\\myparagraph{Supervision position.}\nMost previous MIM methods~\\cite{he2022masked,chen2022context,bao2021beit} apply the reconstruction supervision on the predictions of masked patches.\nWith CLIP as the target, MVP~\\cite{wei2022mvp} supervises both visible and masked patches.\nHere, we do experiments to study how will the\nsupervision position influence the CLIP-targeted MIM.\n\n\nWe systematically analyze three kinds of supervision positions based on {CAE v2}\\xspace, \\textit{i.e.}, only on the predictions of visible patches $\\pmb{\\mathrm{Y}}_m$, only on the predictions of masked patches $\\pmb{\\mathrm{Y}}_v$, and on both $\\pmb{\\mathrm{Y}}_v$ and $\\pmb{\\mathrm{Y}}_m$.\nThe loss function can be formulated as follows:\n\\begin{equation}\n\\centering\n\\begin{aligned}\nL=\\frac{\\delta_{v}\\cdot \\ell(\\pmb{\\mathrm{Y}}_{v},\\pmb{\\mathrm{T}}_{v}) + \\delta_{m}\\cdot \\ell(\\pmb{\\mathrm{Y}}_{m},\\pmb{\\mathrm{T}}_{m})}\n{\\delta_{v}\\cdot \\left | v \\right | + \\delta_{m}\\cdot \\left | m \\right |},\n\\end{aligned}\n\\label{eq:loss}\n\\end{equation} \nwhere $\\ell$ is the loss function.\nBy default, we use the cosine distance as the loss function.\n$\\delta_{v}$\/$\\delta_{m}$ is the indicative function, controlling whether to use visible\/masked patches for optimization.\nIf $\\delta_{m}=1-\\delta_{v}=1$, Eq.~\\eqref{eq:loss} reduces to only use the reconstruction loss for masked patches as in \\cite{he2022masked,chen2022context,bao2021beit}.\nIf $\\lambda_{m}=1-\\delta_{v}=0$, the loss becomes the feature distillation loss for visible patches.\nWhen setting $\\lambda_{m}=\\delta_{v}=1$, the optimization is imposed on both visible and masked patches as in \\cite{wei2022mvp}.\n\nFrom the experiments (see Table~\\ref{tab:supervision_position}), we observe an interesting phenomenon: \\textit{only applying the supervision on visible patches can achieve remarkable performance}.\nWe conjecture this benefit mainly comes from the powerful CLIP model by knowledge distillation.\nIntriguingly, the distillation loss works well on a subset of image patches, in which some contextual information are missing.\nMoreover, despite the relatively inferior performance when only using the reconstruction supervision on masked patches,\nit bring slightly positive improvement when\ncombining the reconstruction supervision on the masked patches with the feature distillation loss on the visible patches.\nWe believe that in this way, the supervision on masked patches performs as a regularization for the representation learning, which is beneficial to alleviating the over-fitting when training only on visible patches. \n\n\nOur findings are different from the common sense in the current MIM methods~\\cite{bao2021beit,he2022masked,chen2022context} that only compute the loss on the masked patches, which\nis inherited from BERT~\\cite{bert} in the NLP areal and has been verified by most current works.\nTherefore, in CLIP-targeted MIM, we provide a new perspective that the feature distillation on the partial image patches (here referred to visible patches) is a good choice for the model optimization. \n\n\n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[trim =0mm 0mm 0mm 0mm, clip, width=1.0\\linewidth]{figure\/comparison_mvp.pdf}\n\\caption{\nMVP~\\cite{wei2022mvp} \\textit{vs.\\ } our {CAE v2}\\xspace. We mainly study the supervision position and the mask ratio in the CLIP-targeted MIM.\n}\n\\vspace{-0.5cm}\n\\label{fig:comparison_mvp}\n\\end{figure}\n\n\\myparagraph{Mask ratio.}\nThe MIM methods generally mask a specific percentage of patches on $\\pmb{\\mathrm{x}}$ as the input for the model training.\nFor example, \\cite{he2022masked} utilizes the mask ratio of 75\\%, while\n~\\cite{bao2021beit,wei2022mvp} and \\cite{chen2022context} empirically set the mask ratio as 40\\% and 50\\%.\nDriven by the above experience that the CLIP supervision on visible patches can achieve good results in our {CAE v2}\\xspace, we naturally consider to study what is the optimal option for the mask ratio.\n\nWe begin from a high mask ratio $\\gamma$, \\textit{i.e.}, 75\\% as in \\cite{he2022masked}.\nWe find that despite the reasonable performance on the large model (\\textit{e.g.}, ViT-Base), it performs less than satisfactory on smaller models like ViT-Tiny\/Small (see Figure~\\ref{fig:mask_ratio}).\nWe thus gradually decrease the mask ratio, leading to more visible patches.\nThe performance on all scales of models starts to improve at the beginning of reducing the mask ratio.\nThis trend is especially true for the downstream task like semantic segmentation, \nverifying that the high mask ratio is not necessary in the CLIP-targeted MIM.\n\nWith the continued decreas of the mask ratio, the models with different sizes perform differently.\nWe observe that the smaller models favor lower proportions of mask ratios, while the larger models prefer relatively higher ones.\nThe underlying reason may be that it is hard for the small-size model to optimize on a small subset of patches where most contextual information are missing.\nSo it is better to reduce the difficulty by using more visible patches, \\textit{i.e.}, a lower mask ratio.\nOn the contrary, large-scale models learn representations from plenty of patches easily.\nTherefore, a high mask ratio can make MIM harder to ease the over-fitting. \n\nBased on the above observation, we point out that \\textit{the optimal mask ratio is positively correlated with the model size}.\nThat is to say, the larger the model, the higher the mask ratio, and preferring more challenging work; otherwise, the smaller model favors a lower mask ratio.\nWe believe that this discovery can be a useful guideline in the MIM pre-training area, especially for the small models.\n\n\\begin{table}\n\\small\n\\begin{center}\n\\setlength{\\tabcolsep}{3.0mm}\n\\renewcommand{\\arraystretch}{1.1}\n\\scalebox{1.0}{\n\\begin{tabular}{l|cc|c|c|c}\n\\hline\n\\multirow{2}{*}{Model} & \\multicolumn{2}{c|}{Supervision} & \\multicolumn{2}{c|}{IN-1K} & \\multicolumn{1}{c}{ADE20K} \\\\\n\\cline{2-6}\n & \\multicolumn{1}{c}{$\\pmb{\\mathrm{Y}}_m$} & \\multicolumn{1}{c|}{$\\pmb{\\mathrm{Y}}_v$} & LIN & FT & mIoU \\\\ \n\\hline\n\\hline\n\\multirow{3}{*}{ViT-Tiny} & \\checkmark & - & 64.9 & 77.2 & 44.1 \\\\\n& - & \\checkmark & 68.8 & 77.4 & 44.2 \\\\\n & \\baseline{\\checkmark} & \\baseline{\\checkmark} & \\baseline{\\textbf{69.3}} & \\baseline{\\textbf{77.8}} & \\baseline{\\textbf{44.7}} \\\\\n\\hline\n\\multirow{3}{*}{ViT-Small} & \\checkmark & - & 73.9 & 82.4 & 49.6 \\\\\n& - & \\checkmark & 77.3 & \\textbf{82.8} & 49.1 \\\\\n & \\baseline{\\checkmark} & \\baseline{\\checkmark} & \\baseline{\\textbf{77.5}} & \\baseline{82.7} & \\baseline{\\textbf{49.7}} \\\\\n\\hline\n\\multirow{3}{*}{ViT-Base} & \\checkmark & - & 78.4 & 85.0 & 52.7 \\\\\n& - & \\checkmark & 80.5 & 85.2 & \\textbf{53.1} \\\\\n & \\baseline{\\checkmark} & \\baseline{\\checkmark} & \\baseline{\\textbf{80.6}} & \\baseline{\\textbf{85.3}} & \\baseline{52.9} \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\vspace{-0.5cm}\n\\caption{Influences of the supervision position in our {CAE v2}\\xspace. Default settings are marked in \\colorbox{baselinecolor}{gray}.}\n\\vspace{-0.5cm}\n\\label{tab:supervision_position}\n\\end{table}\n\n\\myparagraph{Discussion.}\nThe most relevant work for ours is MVP~\\cite{wei2022mvp}.\nIt is noted that our {CAE v2}\\xspace is not contradictory to MVP, even though we both use CLIP as the target. \nWe focus on the analyses under the CLIP-targeted MIM and propose new perspectives, while MVP highlights the rich information brought by the CLIP target. \nSpecifically,\nin this work, we go further one step to study two important ingredients in MIM, including the supervision target and the mask ratio.\nFirst, we find only applying the CLIP supervision on visible patches already achieves comparable or even superior performance compared to the optimization on all patches as in MVP.\nSecond, MVP inherits the mask ratio (40\\%) from \\cite{bao2021beit} and applies it on both base- and large-size models.\nDifferently, we study the effect of the mask ratios on a battery of models with different scales, and find the mask ratio is highly related to the model size.\nIn addition, our {CAE v2}\\xspace with these two findings demonstrates large performance gains. \nWe illustrate the detailed paradigms of our {CAE v2}\\xspace and MVP in Figure~\\ref{fig:comparison_mvp} for better comparisons.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\n\\subsection{Settings}\n\\myparagraph{Model structures.} We study a series of vision transformer backbones~\\cite{dosovitskiy2020image}, including ViT-Tiny (12 layers with $dim$$=$192), ViT-Small (12 layers with $dim$$=$384), ViT-Base (12 layers with $dim$$=$768), and ViT-Large (24 layers with $dim$$=$1024).\nNote that for ViT-Tiny, we follow \\cite{wang2022closer} to increase the number of heads from 3 to 12, which gives better results on ImageNet-1K~\\cite{deng2009imagenet}. For other models, we strictly follow the model configurations as in~\\cite{dosovitskiy2020image}.\n\nTo eliminate the influence of different sizes of CLIP models, we adopt the vision branch ViT-Base\/16 of CLIP\\footnote{The official pre-trained CLIP model is available at \\url{https:\/\/github.com\/openai\/CLIP\/blob\/main\/clip\/clip.py}.} as the target for all pre-training experiments with ViT-Tiny\/Small\/Base\/Large~\\cite{dosovitskiy2020image}.\n\n\n\n\\begin{table}\n\\small\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.1}\n\\setlength{\\tabcolsep}{2.7mm}{\n\\begin{tabular}{l|c|c|c|c}\n\\hline\n\\multirow{2}{*}{Model} & \\multirow{2}{*}{Loss type} & \\multicolumn{2}{c|}{IN-1K} & \\multicolumn{1}{c}{ADE20K} \\\\\n\\cline{3-5}\n & & LIN & FT & mIoU \\\\ \n\\hline\n\\hline\n\\multirow{3}{*}{ViT-Tiny} & MSE & 69.1 & 77.3 & \\textbf{44.8} \\\\\n & Smooth-$l$1 & \\textbf{69.4} & 77.6 & 43.7 \\\\\n & \\baseline{Cosine distance} & \\baseline{69.3} & \\baseline{\\textbf{77.8}} & \\baseline{44.7} \\\\\n\\hline\n\\multirow{3}{*}{ViT-Small} & MSE & 77.3 & 82.7 & \\textbf{49.8} \\\\\n & Smooth-$l$1 & 77.4 & \\textbf{82.8} & \\textbf{49.8} \\\\\n & \\baseline{Cosine distance} & \\baseline{\\textbf{77.5}} & \\baseline{82.7} & \\baseline{49.7} \\\\\n\\hline\n\\multirow{3}{*}{ViT-Base} & MSE & 80.4 & \\textbf{85.3} & \\textbf{52.9} \\\\\n & Smooth-$l$1 & 80.5 & 85.2 & 52.0\\\\\n & \\baseline{Cosine distance} & \\baseline{\\textbf{80.6}} & \\baseline{\\textbf{85.3}} & \\baseline{\\textbf{52.9}} \\\\\n\\hline\n\\end{tabular}}\n\\end{center}\n\\vspace{-0.5cm}\n\\caption{Ablation on the loss type in our {CAE v2}\\xspace. \nWe use the cosine distance as the default loss function (marked in \\colorbox{baselinecolor}{gray}).\n}\n\\vspace{-0.3cm}\n\\label{tab:loss_type}\n\\end{table}\n\n\n\\myparagraph{Pre-training.}\nFollowing most previous MIM methods~\\cite{bao2021beit,he2022masked,chen2022context,wei2022mvp,wang2022closer}, \nwe use ImageNet-1K (IN-1K) dataset~\\cite{deng2009imagenet} for all pre-training experiments.\nThe input images are with the size of $224\\times 224$ and partitioned into $14\\times 14$ patches with the patch size being $16\\times 16$.\nWe apply random resized cropping and horizontal flipping during pre-training.\n\nThe pre-training settings are almost the same as CAE~\\cite{chen2022context}, except for the mask ratios that are analyzed in Section.~\\ref{sec:ablation}.\nBy default, we use $15\\%$, $25\\%$, $50\\%$, and $50\\%$ mask ratios on ViT-Tiny, ViT-Small, ViT-Base, and ViT-Large, respectively.\nWe use AdamW~\\cite{Loshchilov2019} for optimization and train the {CAE v2}\\xspace for 300 epochs for all scales of ViTs~\\cite{dosovitskiy2020image}.\nThe detailed pre-training settings are shown in the supplementary material.\n\n\n\\begin{figure*}[t]\\centering\n\\includegraphics[width=.95\\linewidth]{figure\/masking_ratio.pdf}\n\\vspace{-.7em}\n\\caption{Influences of the mask ratio in our {CAE v2}\\xspace on different model sizes, including (top row) ViT-Tiny, (middle row) ViT-Small and (bottom row) ViT-Base. The optimal mask ratio is positively correlated to the model size.\nA higher mask ratio is more appropriate to a larger model, while the smaller model prefers a lower mask ratio.\nThe y-axes is the Top-1 accuracy (\\%) on (left column) linear probing and (middle column) fine-tuning on ImageNet-1K, and (right column) mIoU (\\%) on ADE20K.\n}\n\\label{fig:mask_ratio}\n\\vspace{-1em}\n\\end{figure*}\n\n\n\n\\myparagraph{Evaluation.}\nWe evaluate our {CAE v2}\\xspace on various downstream tasks.\nFor image classification, we conduct evaluations on ImageNet-1K~\\cite{deng2009imagenet} with both linear probing (LIN) and fine-tuning (FT) protocols.\nWithout specification, in all experiments, we conduct the linear probing for 90 epochs and the fine-tuning for 100 epochs.\nFor semantic segmentation, we follow BEiT\\cite{bao2021beit} to use UperNet~\\cite{xiao2018unified} and report the mIoU on ADE20K~\\cite{zhou2017scene} dataset.\nFor objection detection and instance segmentation, we use COCO~\\cite{lin2014microsoft} as the evaluation dataset.\nWe adopt both Mask R-CNN~\\cite{he2017mask} and Cascade Mask R-CNN~\\cite{cai2018cascade} frameworks and report $\\text{AP}^{b}$ and $\\text{AP}^{m}$ on the COCO val split.\nPlease refer to the supplementary material for more training details on various downstream tasks.\n\n\n\n\n\\subsection{Main Properties}\n\\label{sec:ablation}\nWe mainly explore two critical ingredients, the supervision position and the mask ratio, in {CAE v2}\\xspace when using CLIP as the supervision target. Compared with previous pre-training methods, we observe different properties and trends. In addition, we also give investigations on the loss types and the masking types. Details are given below.\n\n\\myparagraph{Supervision position.} Modern pre-training methods~\\cite{bao2021beit,he2022masked,chen2022context} only give supervision on masked patches, as they find that learning with visible patches is an easy task and may leak information, result in trivial solutions, and degenerate the representation learning. When using CLIP~\\cite{radford2021learning} as the pre-training target, the situation is different. \n\nTable~\\ref{tab:supervision_position} provides the detailed results of the influence of the supervision position on ImageNet-1K~\\cite{deng2009imagenet} and ADE20K~\\cite{zhou2017scene} based on {CAE v2}\\xspace. \nDifferent from existing MIM methods~\\cite{bao2021beit,he2022masked,chen2022context}, we observe that only adding supervision on visible patches already presents remarkable results on both linear probing, image classification fine-tuning, and semantic segmentation fine-tuning. While only adding supervision on masked patches~\\cite{bao2021beit,he2022masked,chen2022context} shows worse performance with linear probing (64.9\\% \\textit{vs.} 68.8\\% with ViT-Tiny, 73.9\\% \\textit{vs.} 77.3\\% with ViT-Small, and 78.4\\% \\textit{vs.} 80.5\\% with ViT-Base), but competitive results with image classification fine-tuning and semantic segmentation fine-tuning (Table~\\ref{tab:supervision_position}). We conjecture that the rich semantics in CLIP~\\cite{radford2021learning} make the supervision on visible patches\nwork\nas a knowledge distillation task. It is different from previous methods where the semantics in the target is not precise enough to provide strong supervision on visible patches. Moreover, by combining the supervision on visible patches and masked patches (the rows marked with \\colorbox{baselinecolor}{gray} in Table~\\ref{tab:supervision_position}), we obtain a slight improvement over the one only with supervision on visible patches. Based on the observation, \nthe supervision on masked patches\nin our {CAE v2}\\xspace can be considered as a regularization for representation learning with CLIP as the target.\n\n\n\n\n\n\n\n\n\n\\myparagraph{Mask ratio.}\nGiven that \nthe supervision on masked patches is a regularization for the learning of {CAE v2}\\xspace, we suppose that it may not be appropriate to adopt a high mask ratio (75\\% in MAE~\\cite{he2022masked}, 40\\%-50\\% in BEiT~\\cite{bao2021beit}, CAE~\\cite{chen2022context}, and MVP~\\cite{wei2022mvp}) for all scales of ViTs. \nWe study different mask ratios, ranging from 15\\% to 90\\% (see Figure~\\ref{fig:sampling_strategy}), for a series of ViTs (Tiny, Small, and Base) with our {CAE v2}\\xspace.\n\n\n\nFigure~\\ref{fig:mask_ratio} gives the performance of ViT-Tiny\/ViT-Small\/ViT-Base~\\cite{dosovitskiy2020image} under various mask ratios using three evaluation methods: linear probing, image classification fine-tuning, and semantic segmentation fine-tuning. The results give a clear trend on the choice of mask ratios, \\textit{i.e.}, {\\em the best mask ratio is positively related to the model size}. From the curves in Figure~\\ref{fig:mask_ratio}, we also find that with the selected mask ratio exceeding the best mask ratio, the performances on downstream tasks drop quickly, especially fine-tuning. The evidence indicates the significance of our study on the mask ratio and provides a guideline for choosing mask ratios for different scales of ViTs.\nAccording to the experiments in Figure~\\ref{fig:mask_ratio}, we use a mask ratio of $15\\%$ for ViT-Tiny, $25\\%$ for ViT-Small, and $50\\%$ for ViT-Base.\n\n\n\\myparagraph{Loss type.} As the target of {CAE v2}\\xspace is the CLIP features, we explore the influence of different loss functions in the feature space.\nTable~\\ref{tab:loss_type} shows the results of {CAE v2}\\xspace on different scales of models with different loss functions, including the mean square error (MSE), Smooth-$l$1 and the cosine distance. We only observe slight differences ($\\leq$$0.5\\%$) in performance on ImageNet1K~\\cite{deng2009imagenet}. On ADE20K~\\cite{zhou2017scene}, the variances are slightly bigger, indicating the loss function of using CLIP as the target gives more impact on challenging downstream tasks. We adopt the cosine distance loss as the default choice.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[trim =0mm 0mm 0mm 0mm, clip, width=1.0\\linewidth]{figure\/mask_ratio_image.pdf} \\\\\n\\setlength{\\belowcaptionskip}{0.3cm} %\n\\figcaption{\nIllustration of corrupted images with different mask ratios $\\gamma$ via (top row) block-wise sampling strategy (our default) and (bottom row) random sampling strategy.\n}\n\\vspace{-0.8cm}\n\\label{fig:sampling_strategy}\n\\end{figure}\n\n\n\n\n\\myparagraph{Mask sampling strategy.} We also compare different mask sampling strategies in our {CAE v2}\\xspace, \\textit{i.e.}, random sampling~\\cite{he2022masked} and block-wise sampling~\\cite{bao2021beit,chen2022context,wei2022mvp} (as shown in Figure~\\ref{fig:sampling_strategy}). There are only $\\sim$$0.1\\%$ gaps between these two sampling strategies on linear probing and image classification fine-tuning (see Table~\\ref{tab:mask_strategy}). When it comes to semantic segmentation, the block-wise sampling shows better performance than the random sampling. In {CAE v2}\\xspace, we use block-wise sampling strategy by default.\n\n\n\n\\subsection{Main Results}\nBased on the analyses of the supervision position in Section~\\ref{sec:study}, we find that only applying the feature distillation supervision on visible patches can achieve good performance.\nWe denote this format as {CAE v2}\\xspace.\nMeanwhile, adding the reconstruction supervision on masked patches can further improve the performance, which is denoted as {CAE v2}\\xspace$+$.\nWe report both {CAE v2}\\xspace and {CAE v2}\\xspace$+$ in this subsection.\n\n\\begin{table}[t]\n\\small\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.15}\n\\setlength{\\tabcolsep}{2.5mm}{\n\\begin{tabular}{l|c|c|c|c}\n\\hline\n\\multirow{2}{*}{Model} & \\multirow{2}{*}{Mask strategy} & \\multicolumn{2}{c|}{IN-1K} & \\multicolumn{1}{c}{ADE20K} \\\\\n\\cline{3-5}\n & & LIN & FT & mIoU \\\\ \n\\hline\n\\hline\n\\multirow{2}{*}{ViT-Tiny} & Random & 69.1 & 77.4 & 43.8 \\\\\n & \\baseline{Blockwise} & \\baseline{\\textbf{69.3}} & \\baseline{\\textbf{77.8}} & \\baseline{\\textbf{44.7}} \\\\\n\\hline\n\\multirow{2}{*}{ViT-Small} & Random & 77.4 & \\textbf{82.7} & 49.0 \\\\\n & \\baseline{Blockwise} & \\baseline{\\textbf{77.5}} & \\baseline{\\textbf{82.7}} & \\baseline{\\textbf{49.7}} \\\\\n\\hline\n\\multirow{2}{*}{ViT-Base} & Random & \\textbf{80.6} & \\textbf{85.4} & 52.4 \\\\\n & \\baseline{Blockwise} & \\baseline{\\textbf{80.6}} & \\baseline{85.3} & \\baseline{\\textbf{52.9}} \\\\\n\\hline\n\\end{tabular}}\n\\end{center}\n\\vspace{-0.5cm}\n\\tabcaption{Ablation on the mask sampling strategy in our {CAE v2}\\xspace. We use the block-wise sampling by default (marked in \\colorbox{baselinecolor}{gray}).}\n\\vspace{-0.7cm}\n\\label{tab:mask_strategy}\n\\end{table}\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\scalebox{1.0}{\n\\renewcommand{\\arraystretch}{1.05}\n\\setlength{\\tabcolsep}{6.0mm}{\n\\begin{tabular}{lccccc} %\n\\toprule\n\\multirow{2}{*}{Methods} & \\multirow{2}{*}{\\#Epochs} & \\multirow{2}{*}{Target} & \\multicolumn{2}{c}{IN-1K} \n& \\multirow{1}{*}{ADE20K} \\\\\n\\cline{4-6}\n& & & LIN & FT & mIoU \\\\\n\\midrule\n\\midrule\n\\multicolumn{5}{l}{\\emph{Methods using ViT-Tiny}:} \\\\\nMAE-Tiny~\\cite{wang2022closer} & 400 & RGB & 23.4 & 76.2 & - \\\\\nCAE~\\cite{chen2022context}$^\\ddagger$ & 300 & DALL-E & 28.1 & 75.9 & 38.3 \\\\\nDistilled MAE-lite~\\cite{wang2022closer} & 400 & RGB & - & 76.5 & - \\\\\n\\textbf{{CAE v2}\\xspace} & 300 & CLIP-B & 68.8 & 77.4 & 44.2 \\\\\n\\textbf{{CAE v2}\\xspace$+$} & 300 & CLIP-B & \\textbf{69.3} & \\textbf{77.8} & \\textbf{44.7} \\\\\n\\midrule\n\\multicolumn{5}{l}{\\emph{Methods using ViT-Small}:} \\\\\nMoCo v3~\\cite{chen2021empirical}$^\\S$ & 300 & Self-EMA & 73.1 & 81.7$^\\dagger$ & - \\\\\nBEiT~\\cite{bao2021beit}$^\\S$ & 300 & DALL-E & 15.7 & 81.7$^\\dagger$ & - \\\\\nSplitMask~\\cite{el2021large} & 300 & DALL-E & - & 81.5 & - \\\\\nCAE~\\cite{chen2022context} & 300 & DALL-E & 51.8 & 82.0$^\\dagger$ & - \\\\\niBOT~\\cite{zhou2021ibot} & 3200 & Self-EMA & \\textbf{77.9} & 82.3$^\\dagger$ & 45.4 \\\\\n\\textbf{{CAE v2}\\xspace} & 300 & CLIP-B & 77.3 & 82.8 & 49.1 \\\\\n\\textbf{{CAE v2}\\xspace$+$} & 300 & CLIP-B & 77.5 & \\textbf{83.1}$^\\dagger$ & \\textbf{49.7} \\\\\n\\midrule\n\\multicolumn{5}{l}{\\emph{Methods using ViT-Base}:} \\\\\nMoCo v3~\\cite{chen2021empirical} & 300 & Self-EMA & 76.5 & 83.2 & 47.2 \\\\\nDINO~\\cite{caron2021emerging}$^\\S$ & 400 & Self-EMA & 77.3 & 83.3 & 47.2 \\\\\niBOT~\\cite{zhou2021ibot} & 1600 & Self-EMA & 79.5 & 84.0 & 50.0 \\\\\nBEiT~\\cite{bao2021beit} & 800 & DALL-E & 56.7 & 83.2 & 45.6 \\\\\nSimMIM~\\cite{xie2022simmim} & 800 & RGB & 56.7 & 83.8 & - \\\\\nMAE~\\cite{he2022masked} & 1600 & RGB & 68.0 & 83.6 & 48.1 \\\\\nCAE~\\cite{chen2022context} & 1600 & DALL-E & 70.4 & 83.9 & 50.2 \\\\\nSdAE~\\cite{chen2022sdae} & 300 & Self-EMA & 64.9 & 84.1 & 48.6 \\\\\nSIM~\\cite{tao2022siamese} & 1600 & Self-EMA & 76.4 & 83.8 & - \\\\\nMaskFeat~\\cite{wei2022masked} & 1600 & HOG & - & 84.0 & - \\\\\nSplitMask~\\cite{el2021large} & 300 & DALL-E & - & 83.6 & 45.7 \\\\\nPeCo~\\cite{dong2021peco} & 800 & VQGAN & - & 84.5 & 48.5 \\\\\ndata2vec~\\cite{baevski2022data2vec} & 800 & Self-EMA & - & 84.2 & - \\\\\nCMAE~\\cite{huang2022contrastive} & 1600 & RGB & - & 84.7 & 50.1 \\\\\nExtreMA~\\cite{wu2022extreme} & 300 & Self-EMA & 73.3 & 83.7 & 47.9 \\\\\nCLIP~\\cite{radford2021learning} & - & Text & - & 84.9 & 51.1 \\\\\nMaskCLIP \\cite{Dong2022MaskCLIPMS} & 1600 & Text & 72.9 & 84.1 & 50.8 \\\\\nMVP~\\cite{wei2022mvp} & 300 & CLIP-B & 75.4 & 84.4 & 52.4 \\\\\nBEIT V2~\\cite{peng2022beit} & 300 & VQ-CLIP-B & 80.1 & 85.0 & 52.7 \\\\\n\\textbf{{CAE v2}\\xspace} & 300 & CLIP-B & 80.5 & 85.2 & \\textbf{53.1} \\\\\n\\textbf{{CAE v2}\\xspace$+$} & 300 & CLIP-B & \\textbf{80.6} & \\textbf{85.3} & 52.9 \\\\\n\\midrule\n\\multicolumn{5}{l}{\\emph{Methods using ViT-Large}:} \\\\\nMoCo v3~\\cite{chen2021empirical}$^\\S$ & 300 & Self-EMA & - & 84.1 & 49.1 \\\\\nBEiT~\\cite{bao2021beit}$^\\S$ & 1600 & DALL-E & - & 85.2 & 53.3 \\\\\niBOT~\\cite{zhou2021ibot} & 1200 & Self-EMA & 81.0 & 84.8 & - \\\\\nMAE~\\cite{he2022masked} & 1600 & RGB & 75.8 & 85.9 & 53.6 \\\\\nCAE~\\cite{chen2022context} & 1600 & DALL-E & 78.1 & 86.3 & 54.7 \\\\\ndata2vec~\\cite{baevski2022data2vec} & 1600 & Self-EMA & - & 86.6 & - \\\\\nMVP~\\cite{wei2022mvp} & 300 & CLIP-B & - & 86.3 & 54.3 \\\\\nBEIT V2~\\cite{peng2022beit} & 300 & VQ-CLIP-B & - & 86.6 & 55.0 \\\\\n\\textbf{{CAE v2}\\xspace$+$} & 300 & CLIP-B & \\textbf{81.7} & \\textbf{86.7} & \\textbf{55.9} \\\\\n\\bottomrule\n\\end{tabular}\n}\n}\n\\caption{\nPre-training evaluation on the top-1 accuracy (\\%) on linear probing (LIN) and fine-tuning (FT) on ImageNet-1K~\\cite{deng2009imagenet}, and mIoU (\\%) on ADE20K~\\cite{zhou2017scene}.\n$\\dagger$ denotes the fine-tuning epoch is 200 for ViT-Small.\n$\\ddagger$ means our implementation using the officially released code.\n$^\\S$ means the results from \\cite{chen2022context}.\nAll other results except for ours are from the original papers.\n}\n\\label{tab:imagenet_seg_sota}\n\\end{table*}\n\n\\myparagraph{Image classification on ImageNet-1K.}\nTable~\\ref{tab:imagenet_seg_sota} shows the comparisons of different models with two evaluation methods: linear probing and fine-tuning.\n\nWith linear probing, {CAE v2}\\xspace shows significant improvements over previous methods with other targets, {\\em e.g.}, BEiT~\\cite{bao2021beit}, MAE~\\cite{he2022masked}, CAE~\\cite{chen2022context}, and MaskFeat~\\cite{wei2022masked}. These gains are expected as CLIP features contain rich semantics than other targets. Compared with the methods (MVP~\\cite{wei2022mvp} and BEIT V2~\\cite{peng2022beit}) use CLIP as the target, {CAE v2}\\xspace can also give superior performance (on ViT-Base with 300 epoch pre-training, {CAE v2}\\xspace \\textit{vs.\\ } MVP: 80.6\\% \\textit{vs.\\ } 75.4\\% and {CAE v2}\\xspace \\textit{vs.\\ } BEIT V2: 80.6\\% \\textit{vs.\\ } 80.1\\%). When we fine-tune the pre-trained model on ImageNet-1K, {CAE v2}\\xspace achieves the best results among various methods across all scales of ViTs (Table~\\ref{tab:imagenet_seg_sota}). Specifically, {CAE v2}\\xspace achieves $\\textbf{85.3\\%}$ top-$1$ accuracy, surpassing previous methods by large margins. Moreover, with ViT-Large, {CAE v2}\\xspace improves the performance to $\\textbf{86.7\\%}$ top-$1$ accuracy.\n\n\n\\myparagraph{Semantic segmentation on ADE20K.} \nSemantic segmentation is a challenging task that needs to classify all pixels to various semantic labels given an image. CLIP~\\cite{radford2021learning} as the target shows clear advantages in this task. As shown in Table~\\ref{tab:imagenet_seg_sota}, {CAE v2}\\xspace significantly improves the results over the methods pre-trained with other targets, {\\em e.g.}, by 2.7\\% mIoU over CAE~\\cite{chen2022context} with ViT-Base. When comparing with CLIP~\\cite{radford2021learning}, MVP~\\cite{wei2022mvp}, and BEIT V2~\\cite{peng2022beit}, {CAE v2}\\xspace outperforms them with the same or less pre-training epochs. The superior performance hold when we move to ViT-Large, with which {CAE v2}\\xspace achieves $\\textbf{55.9\\%}$ mIoU on ADE20K~\\cite{zhou2017scene}.\n\n\\myparagraph{Object detection and instance segmentation on COCO.}\nWe evaluate the pre-trained models on COCO~\\cite{lin2014microsoft} with Mask R-CNN~\\cite{he2017mask} (Table~\\ref{tab:cocodetection_mask}) and Cascade Mask R-CNN~\\cite{cai2018cascade,he2017mask} (Table~\\ref{tab:cocodetection_cascade}). We report the results of $1\\times$ (12 epochs) training schedule.\nCompared with other pre-training methods, {CAE v2}\\xspace performs better \nunder both two configurations. With Mask R-CNN, {CAE v2}\\xspace gives 4.9\/3.0 points higher on ViT-Small and 2.0\/0.9 points higher on ViT-Base on AP$^b$\/AP$^m$ than the previous best method~\\cite{chen2022context}. The superior performance remain when adopting Cascade Mask R-CNN as the fine-tuned model (see Table~\\ref{tab:cocodetection_cascade}).\n\n\\begin{table}[!t]\n \\centering\n\\small\n\\renewcommand{\\arraystretch}{1.05}\n\\setlength{\\tabcolsep}{4.8mm}{\n\\begin{tabular}{lccc}\n \\toprule\n \\multirow{2}{*}{Method} & \n \\multirow{2}{*}{\\#Epochs} & \n \\multicolumn{1}{c}{DET}& \n \\multicolumn{1}{c}{INS}\\\\ \n \\cline{3-4}\n & & { $\\text{AP}^{b}$} & {$\\text{AP}^{m}$} \\\\ \n \\midrule\n \\midrule\n \\multicolumn{3}{l}{\\emph{Methods using ViT-Small}:} \\\\\n DeiT \\cite{touvron2021training} & 300 & 43.1 & 38.4 \\\\\n MoCo v3$^*$ \\cite{chen2021empirical} & 300 & 43.3 & 38.8 \\\\\n BEiT \\cite{bao2021beit} & 300 & 35.6 & 32.6 \\\\\n CAE \\cite{chen2022context} & 300 & 44.1 & 39.2 \\\\\n \\textbf{{CAE v2}\\xspace$+$} & 300 & \\textbf{49.0} & \\textbf{42.2} \\\\\n \\midrule\n \\multicolumn{3}{l}{\\emph{Methods using ViT-Base}:}\\\\\n DeiT \\cite{touvron2021training} & 300 & 46.9 & 41.5 \\\\\n MoCo v3$^*$ \\cite{chen2021empirical} & 300 & 45.5 & 40.5 \\\\\n DINO$^*$ \\cite{caron2021emerging} & 400 & 46.8 & 41.5 \\\\\n BEiT \\cite{bao2021beit} & 800 & 42.1 & 37.8 \\\\\n MAE \\cite{he2022masked} & 1600 & 48.4 & 42.6 \\\\\n data2vec~\\cite{baevski2022data2vec} & 800 & 41.1 & 37.0 \\\\\n CAE \\cite{chen2022context} & 1600 & 50.0 & 44.0 \\\\\n \\textbf{{CAE v2}\\xspace} & 300 & 51.8 & 44.7 \\\\\n \\textbf{{CAE v2}\\xspace$+$} & 300 & \\textbf{52.0} & \\textbf{44.9} \\\\\n\n \\bottomrule \n \\end{tabular} \n }\n \\caption{Pre-training evaluation on object detection (DET) and instance segmentation (INS) on COCO~\\cite{lin2014microsoft}. \n Mask R-CNN~\\cite{he2017mask} is adopted and trained with $1\\times$ schedule.\n All results except for {CAE v2}\\xspace are from \\cite{chen2022context}.\n \\#Epochs refers to the pre-training epochs on ImageNet-$1$K.\n $^*$ denotes multi-crop pre-training augmentation.\n }\n \\vspace{-0.5em}\n \\label{tab:cocodetection_mask}\n\\end{table}\n\n\n\n\\section{Related Work}\n\nMasked image modeling (MIM) aims to learn transferable vision representations. It is inspired by the successful large-scale pre-training for transformers~\\cite{vaswani2017attention} with masked language modeling (MLM)~\\cite{bert,chen2020generative,gpt3,unilm} in NLP and can serve as a pretext task in self-supervised vision pre-training~\\cite{deepcluster, doersch2015unsupervised, cpc, ermolov2020whitening, goyal2021self, li2021prototypical, zbontar2021barlow,moco,mocov2,simclr,byol, Ge2-AE, BootMAE, HiViT, GreenMIM, MimCo, MSN, ConMIM, Li2022mcBEiTMD}. MIM methods~\\cite{bao2021beit,he2022masked,xie2022simmim,wei2022masked,baevski2022data2vec, Singh2022RevisitingWS} follow a mask-then-predict pipeline of (i) corrupting an image by masking several image patches based on a pre-defined mask ratio and (ii) learning to predict the missing content under specific supervision. {CAE v2}\\xspace uses the CLIP model~\\cite{radford2021learning} as the supervision target and studies on the above two aspects. Next, we discuss related works with respect to these two aspects.\n\n\n\n\\myparagraph{Supervision target.} There are several ways to represent the missing content when supervising a model. Existing MIM methods explore different supervision targets on their frameworks, including RGB pixels~\\cite{he2022masked,gao2022convmae}, HOG descriptors~\\cite{wei2022masked}, discrete visual tokens~\\cite{bao2021beit,chen2022context,el2021large,peng2022beit,dong2021peco}, and feature representation from momentum models~\\cite{tao2022siamese,chen2022sdae,wu2022extreme}. Among these methods, supervision is added to the model's predictions for masked patches. They give no supervision to the model's predictions for unmasked patches (visible patches). Recently, MVP~\\cite{wei2022mvp} explored changing the supervision target from other modalities and validated the effectiveness of the additional knowledge. In detail, MVP adopts the vision branch of the CLIP mode\\cite{radford2021learning} as the supervision target in MIM. Then MVP gives supervision on all patches of the image, including masked patches and unmasked patches. {CAE v2}\\xspace follows MVP~\\cite{wei2022mvp} to use the CLIP model as the supervision target and go one step further to study the supervision position in this paper.\n\n\\begin{table}[!t]\n \\centering\n\n\\small\n\\renewcommand{\\arraystretch}{1.05}\n\\setlength{\\tabcolsep}{4.8mm}{\n\\begin{tabular}{lccc}\n \\toprule\n \\multirow{2}{*}{Method} & \n \\multirow{2}{*}{\\#Epochs} & \n \\multicolumn{1}{c}{DET}& \n \\multicolumn{1}{c}{INS}\\\\ \n \\cline{3-4}\n & & { $\\text{AP}^{b}$} & {$\\text{AP}^{m}$} \\\\ \n \\midrule\n \\midrule\n \n \\multicolumn{3}{l}{\\emph{Methods using ViT-Small}:} \\\\\n iBOT$^*$~\\cite{zhou2021ibot} & 3200 & 49.4 & 42.6 \\\\\n \\textbf{{CAE v2}\\xspace$+$} & 300 & \\textbf{51.5} & \\textbf{43.9} \\\\\n \\midrule\n \\multicolumn{3}{l}{\\emph{Methods using ViT-Base}:} \\\\\n MAE \\cite{he2022masked} & 1600 & 51.3 & 44.3 \\\\\n CAE \\cite{chen2022context} & 1600 & 52.9 & 45.5 \\\\\n iBOT$^*$~\\cite{zhou2021ibot} & 1600 & 51.2 & 44.2 \\\\\n \\textbf{{CAE v2}\\xspace$+$} & 300 & \\textbf{53.9} & \\textbf{45.9} \\\\\n \\bottomrule \n \\end{tabular} \n }\n \\caption{\n Pre-training evaluation on object detection (DET) and instance segmentation (INS) on COCO~\\cite{lin2014microsoft} with Cascade Mask R-CNN~\\cite{cai2018cascade}. All models are trained with the $1\\times$ schedule.\n All results except for {CAE v2}\\xspace are from \\cite{chen2022context},\n and the results of iBOT are from the original paper~\\cite{zhou2021ibot}.\n \\#Epochs refers to the effective pre-training epochs on ImageNet-$1$K.\n $^*$ denotes the multi-crop pre-training augmentation.\n }\n \\vspace{-1.0em}\n \\label{tab:cocodetection_cascade}\n\\end{table}\n\n\\myparagraph{Mask ratio.} The mask ratio is a hyper-parameter that needs hand-design in both MLM and MIM. In MLM, BERT~\\cite{bert} uses a relatively small mask ratio (15\\%) for pre-training. ~\\cite{wettig2022should} argues that masking up to 40\\% may give higher performance. In MIM, a lot of works use a high mask ratio for pertaining. For example, MAE~\\cite{he2022masked} utilizes a mask ratio of\n75\\%, BEiT~\\cite{bao2021beit}, CAE~\\cite{chen2022context}, and MVP~\\cite{wei2022mvp} empirically set the mask ratio as 40\\% and 50\\%. In this paper, we study this interesting problem and provide a guideline for choosing the right mask ratio for different scales of ViTs.\n\n\n\nConcurrent with our work, \\cite{hou2022milan, liu2022exploring, peng2022unified} also explores using CLIP to guide the MIM pre-training. MILAN~\\cite{hou2022milan} and dBOT~\\cite{liu2022exploring} focus on the impact of target representations, that the CLIP containing multi-modality knowledge can provide more benefits to MIM. MaskDistill~\\cite{peng2022unified} works on the design of distillation loss in supervision. Differently, we investigate two aspects orthogonal to these works. We find that adding supervision on visible patches further helps visual learning compared to only supervising the masked patches. Moreover, we explore the relationship between mask ratio and model scales.\nThese two findings provide useful guidelines for MIM pre-training.\n\n\\section{Conclusion and Limitation}\nThis paper studies two critical ingredients in MIM, \\textit{i.e.}, the supervision position and the mask ratio, with CLIP as the supervision target.\nWith our simple pipeline {CAE v2}\\xspace, we reveal two new insights: i) the feature distillation supervision on visible patches can achieve remarkable performance; ii) the optimal mask ratio is positively correlated to the model size.\nFollowing these two guidelines, our {CAE v2}\\xspace achieves superior performance on all scales of models on various downstream tasks.\n\n\\myparagraph{Limitation.} Limited by resources, we do not study on larger models, like ViT-Huge and ViT-Giant. We leave this exploration in the future.\n\n\n\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\subsection{Klein's classification and McKay's correspondence} \nFelix Klein \\cite{Klein} classified\nfinite subgroups $G$ of ${\\mathbf{SL}}_2({\\mathds{C}})$:\nup to conjugation, there are two infinite series and three isolated cases.\n\n\\begin{enumerate}\n\\item The associated quotient singularity ${\\mathds{C}}^2\/G$ is called a \n \\emph{Klein singularity} and the singularities arising this way\n are precisely the \\emph{rational double point singularities}.\n Its minimal resolution of singularities is a union\n of ${\\mathds{P}}^1$'s, whose dual intersection graph $\\Gamma$ \n is a simply-laced Dynkin diagram of finite type, that is,\n of type $A_n$, $D_n$, $E_6$, $E_7$ or $E_8$.\n\\item John McKay \\cite{McKay} associated a finite graph $\\widehat{\\Gamma}$\n to $G \\subset{\\mathbf{SL}}_2({\\mathds{C}})$, whose vertices correspond to the isomorphism classes of the \n simple representations of $G$.\n This graph is a Dynkin diagram of affine type\n $\\widehat{A}_n$, $\\widehat{D}_n$, $\\widehat{E}_6$, $\\widehat{E}_7$ or $\\widehat{E}_8$.\n\\item After these preparations, the \\emph{classical McKay correspondence} consists of the \nfollowing observations:\n\\begin{enumerate}\n\\item The graph $\\Gamma$ is obtained from $\\widehat{\\Gamma}$ by removing the vertex corresponding to the \ntrivial representation.\n\\item There exists a bijection between \nconjugacy classes of $G$, vertices of $\\widehat{\\Gamma}$, and isomorphism classes of simple representations of $G$.\n\\item There exists a bijection between\nfinite subgroups of ${\\mathbf{SL}}_2({\\mathds{C}})$ up to conjugacy, the above Dynkin diagrams of affine type,\nKlein singularities, and the above Dynkin diagrams of finite type.\n\\end{enumerate}\n\\end{enumerate}\n\nBy now, there are various approaches to and a vast literature on this subject, such as \n\\cite{Knorrer, Kostant, McKay, McKay2, Steinberg} and many more.\nAlso, there are now generalisations into very\ndifferent directions: \nhigher dimensional algebraic geometry \\cite{Reid}, K-theory \\cite{GSV},\nderived categories of coherent sheaves \\cite{Bridgeland,KV}, \nrepresentations of quivers \\cite{Kirillov}, non-commutative geometry and Hopf algebras \\cite{Chan},\nand string theory \\cite{Dirichlet} - just to mention a few.\n\n\\subsection{Positive characteristic}\nNow, let $k$ be an algebraically closed field of characteristic $p>0$.\n\n\\subsubsection{Wild McKay correspondence}\nThe classical McKay correspondence as sketched above is still partially \navailable over $k$ if $G$ is assumed to be a finite subgroup of ${\\mathbf{SL}}_2(k)$\nof order prime to $p$ - this is the \\emph{tame} case.\nIf $p$ divides the order of $G$ - this is the \\emph{modular} or \\emph{wild} case -\nthen this correspondence breaks down.\nWe refer to Yasuda's survey \\cite{Yasuda} about conjectures\nand partial results concerning such wild McKay correspondences.\n\n\\subsubsection{Linearly reductive McKay correspondence}\nIn this article, we show that if $G$ is a finite and\n\\emph{linearly reductive} subgroup scheme of ${\\mathbf{SL}}_{2,k}$, \nthen there is a reasonable version of the classical McKay correspondence.\nFor example, we obtain a McKay correspondence for \\emph{all} rational double point \nsingularities if $p\\geq7$.\nInstead of considering groups, we allow non-reduced \ngroup schemes over $k$, but we require their categories of $k$-linear and finite-dimensional\nrepresentations to be semi-simple.\nWe refer to \\cite{Yasuda2} about conjectures and partial results concerning\na McKay correspondence for the group scheme $\\boldsymbol{\\alpha}_p$, which is not linearly reductive.\n\nThus, let $G$ be a finite and linearly reductive subgroup scheme of ${\\mathbf{SL}}_{2,k}$ \nwith $p\\geq7$.\n(In this introduction, we will exclude small characteristics whenever this is makes\nour discussion easier.)\nLet $x\\in X:=U\/G$ with $U={\\mathds{A}}_k^2$ or $U=\\widehat{{\\mathds{A}}}_k^2$\nbe the associated Klein singularity, \nwhich is a rational double point.\nOne goal of this article is to define a notion of conjugacy class for $G$,\nto construct graphs $\\Gamma$ and $\\widehat{\\Gamma}$, and to \nestablish bijections as above.\n\nLet us make three comments:\n\\begin{enumerate}\n\\item It is interesting in its own that a McKay correspondence can be extended \nfrom finite group schemes of length prime to $p$, that is, the tame case, \nto linearly reductive group schemes.\n\\item What makes this linearly reductive McKay correspondence \nreally interesting is that the bijection in Theorem \\ref{thm: main intro} is \\emph{not}\ntrue when considering finite groups of order prime to $p$ only, \nsee Example \\ref{sec: mup intro}.\n\\item Probably, many more aspects of the classical McKay correspondence \ncan be carried over to the linearly reductive setting, but rather than writing a whole\nmonograph, we decided to establish only some basic bijections.\n\\end{enumerate}\n\n\n\\subsection{Linearly reductive group schemes}\nLet $G$ be a \\emph{finite and linearly reductive group scheme} \nover an algebraically closed field $k$ of characteristic $p\\geq0$.\nBy definition, this means that the category $\\mathrm{Rep}_k(G)$ of\n$k$-linear and finite-dimensional representations of $G$ is semi-simple.\n\nIf $p=0$, then every finite group scheme over $k$ is \\'etale and linearly reductive.\nand in fact, it is\nthe constant group scheme associated to a finite group.\nIn fact, the functor $G\\mapsto G_{\\mathrm{abs}}:=G(k)$ induces an equivalence \nof categories between finite group schemes over $k$ and finite groups.\n\nIf $p>0$, then every linearly reductive group scheme admits a \ncanonical semi-direct product decomposition\n$$\n G\\,\\cong\\,G^\\circ\\rtimes G^{\\mathrm{\\acute{e}t}},\n$$\nwhere $G^\\mathrm{\\acute{e}t}$ is a group scheme of length prime to $p$ (and thus, the constant \ngroup scheme associated to a finite group of order prime to $p$) and where $G^\\circ$\nis infinitesimal and diagonalisable.\nThe latter implies that $G^\\circ$ is a product of group schemes of the form\n$\\boldsymbol{\\mu}_{p^n}$.\nConversely, every such semi-direct product of a diagonalisable group scheme\nwith the constant group scheme associated to a finite group of order prime to $p$\nis linearly reductive.\nThis structure result is usually attributed to Nagata \\cite{Nagata61},\nbut see also \\cite{AOV}, \\cite{Chin}, and \\cite{Hashimoto}.\nIn particular, the class of finite and linearly reductive group schemes over $k$\nstrictly contains the class of constant group schemes associated \nto finite groups of order prime to $p$.\n\n\n\\subsubsection{Abstract groups}\nAssociated to $G$, there is an abstract finite group $G_{\\rm abs}$\nand we refer for the slightly technical definition to Section \\ref{subsec: abstract groups}.\nThe order of $G_{\\rm abs}$ is equal to the length of $G$.\nFor example, if $G$ is \\'etale over $k$, then we have\n$G_{\\rm abs}\\cong G(k)$.\nThe assignment $G\\mapsto G_{\\mathrm{abs}}$ establishes an equivalence\nof categories \n\\begin{equation}\n\\label{eq: catequivalence}\n\\left\\{ \\begin{array}{l}\n\\mbox{finite linearly reductive}\\\\ \\mbox{group schemes over $k$} \n\\end{array}\\right\\}\n\\quad\\leftrightarrow\\quad\n\\left\\{ \\begin{array}{l}\n\\mbox{finite groups with a}\\\\ \\mbox{unique $p$-Sylow subgroup} \n\\end{array}\\right\\},\n\\end{equation}\nsee \\cite{LMM} and Lemma \\ref{lem: lrequivalence}.\n\n\n\\subsubsection{Canonical lifts}\nIn \\cite{LMM}, we showed that if $G$ is a finite and linearly\nreductive group scheme, then there exists a lift of $G$ over the\nring of Witt vectors $W(k)$.\nWe note that $G^\\circ$ and $G^\\mathrm{\\acute{e}t}$ even lift uniquely to $W(k)$\nand we define the \\emph{canonical lift} $G_{\\rm can}\\to \\mathrm{Spec}\\: K$ of $G$ to\nbe the unique lift that is a semi-direct product of the lifts\nof $G^\\circ$ and $G^{\\mathrm{\\acute{e}t}}$. \nAny lift of $G$ to some extension field of $K$ becomes\nisomorphic to $G_{\\mathrm{can}}$ after possibly passing to some\nfurther field extension.\nMoreover, the finite group $G_{\\mathrm{can}}(\\overline{K})$ \nis isomorphic to $G_{\\rm abs}$.\n\n\n\\subsubsection{Representation theory}\n\\label{subsubsec: rep thy}\nBy \\cite{LMM} and Proposition \\ref{prop: liftingrepresentation},\nthere exist canonical equivalences of representation categories \n\\begin{equation}\n\\label{eq: rep thy intro}\n \\mathrm{Rep}_{k}(G) \\,\\to\\, \\mathrm{Rep}_{\\overline{K}}(G_{\\mathrm{can},\\overline{K}})\n \\,\\to\\, \\mathrm{Rep}_{\\mathds{C}}(G_{\\mathrm{abs}})\n\\end{equation}\nthat are compatible with degrees, direct sums, tensor products, duals, \nand simplicity.\nThese equivalences induce isomorphisms of rings\n$$\n K_k(G) \\,\\to\\, K_{\\overline{K}}(G_{\\mathrm{can},\\overline{K}}) \\,\\to\\,\n K_{{\\mathds{C}}}(G_{\\mathrm{abs}}),\n$$\nsee Corollary \\ref{cor: Ktheory}.\nHere, $K_F(G)$ denotes the K-group associated to \n$F$-linear and finite-dimensional $G$-representations.\n\n\n\\subsubsection{Hopf algebras}\nIf $G$ is a finite group scheme over $k$, then the multiplication map\nturns $H^0(G,{\\cal O}_G)$ into a finite-dimensional \\emph{Hopf algebra}\nover $k$.\nWe discuss finite group schemes and among them the linearly reductive\nones from the point of Hopf algebras in \nAppendix \\ref{sec: Hopf}. \n\n\n\\subsection{Linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$}\nLet $k$ be an algebraically closed field of characteristic $p\\geq0$.\nHashimoto \\cite{Hashimoto} extended Klein's classification \\cite{Klein}\nof finite subgroups of ${\\mathbf{SL}}_2({\\mathds{C}})$ up to conjugation to the setting\nof finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$.\nIf $p\\geq7$, then one obtains a list analogous to Klein's classical \nlist.\nIf $p\\in\\{2,3,5\\}$, then some classical cases are missing, \nbut there are no new cases.\n\n\\subsection{McKay graph and McKay correspondence}\n\\label{subsec: McKay graph intro}\nLet $G$ be a finite and linearly reductive subgroup scheme of\n${\\mathbf{SL}}_{2,k}$.\nAs in McKay's original construction \\cite{McKay}, we associate\nan affine Dynkin diagram $\\widehat{\\Gamma}$ to $G$,\nits embedding into ${\\mathbf{SL}}_{2,k}$, and the set of isomorphism classes\nof simple representations of $G$.\nThis is the \\emph{McKay graph} associated to this data.\nIn fact, we will see that it is compatible with the equivalences\ninduced by \\eqref{eq: catequivalence} and \\eqref{eq: rep thy intro}\nand we refer to Section \\ref{subsec: McKay graph} for details.\nWe establish the following version of McKay's theorem \\cite{McKay}\nin positive characteristic.\n\n\\begin{Theorem}[Theorem \\ref{thm: main}]\n\\label{thm: main intro}\n Let $k$ be an algebraically closed field of characteristic $p\\geq0$.\n There exists a bijection between\n non-trivial, finite, and linearly reductive subgroup schemes of\n ${\\mathbf{SL}}_{2,k}$ up to conjugation and affine Dynkin graphs\n of type\n $$\n \\begin{array}{lcl}\n \\widehat{A}_n, \\widehat{D}_n, \\widehat{E}_6, \\widehat{E}_7, \\widehat{E}_8\n &\\mbox{\\quad if \\quad }& p=0\\mbox{\\quad or \\quad}p\\geq7, \\\\\n \\widehat{A}_n, \\widehat{D}_n, \\widehat{E}_6, \\widehat{E}_7 &\\mbox{\\quad if \\quad }& p=5,\\\\\n \\widehat{A}_n, \\widehat{D}_n &\\mbox{\\quad if \\quad }& p=3, \\\\\n \\widehat{A}_n &\\mbox{\\quad if \\quad }&p=2.\n \\end{array}\n $$\n\\end{Theorem}\n\nBy construction, this bijection is compatible with the classical McKay correspondence via\nthe lifting results and the equivalences \\eqref{eq: catequivalence},\n\\eqref{eq: rep thy intro}.\n\n\\begin{Example}\n\\label{sec: mup intro}\nThe linearly reductive group scheme corresponding to \n$\\widehat{A}_n$ is $\\boldsymbol{\\mu}_{n+1}$.\nThis group scheme is reduced, that is, \\'etale, if and only if\n$p$ does not divide $n+1$.\nIn particular, it is crucial to allow non-reduced group schemes\nin order to obtain a bijection as in characteristic zero.\n\\end{Example}\n\n\n\\subsection{Linearly reductive quotient singularities}\nConsider ${\\mathbf{GL}}_{2,k}$ with its usual linear action on \n$U={\\mathds{A}}^2_k$ or $U=\\widehat{{\\mathds{A}}}_k^2$.\nIf $G$ is a finite, linearly reductive, and \\emph{very small}\n(see Definition \\ref{def: very small}) subgroup scheme of ${\\mathbf{GL}}_{2,k}$, then the associated\nquotient singularity $x\\in X:=U\/G$ is a \ntwo-dimensional \\emph{linearly reductive quotient singularity} \nin the sense of \\cite{LMM}.\nBy loc.cit., such a singularity determines \n$G$ together with its embedding $G\\to{\\mathbf{GL}}_{2,k}$ up to isomorphism and\nconjugation, respectively.\n\nIn Section \\ref{subsec: Ishii Ito Nakamura}, we will see \nthat a minimal resolution of singularities of a two-dimensional linearly reductive\nquotient singularity $x\\in X=U\/G$\nis provided by the $G$-Hilbert scheme\n\\begin{equation}\n\\label{eq: intro IIN}\n \\pi \\,:\\, G\\mathrm{-Hilb}(U) \\,\\to\\, U\/G,\n\\end{equation}\nwhich generalises work of Ishii, Ito, and Nakamura \n\\cite{Ishii,IN}.\n\n\\begin{Remark}\nIn dimension two, a linearly reductive quotient singularity\nis the same as an F-regular singularity \\cite{LMM}.\nThus, every two-dimensional F-regular singularity can be \nresolved by a suitable $G$-Hilbert scheme, see \nalso Remark \\ref{rem: f-regular}.\n\\end{Remark}\n\nIf moreover $G$ is a subgroup scheme of ${\\mathbf{SL}}_{2,k}$, \nthen $x\\in X=U\/G$ is called a \\emph{Klein singularity}.\nKlein singularities are rational double point singularities.\nIf $p=0$ or $p\\geq7$, then conversely every rational double point \nis a Klein singularity by Hashimoto \\cite{Hashimoto} and, independently, by \n\\cite{LiedtkeSatriano}.\nIf $p\\in\\{2,3,5\\}$, then not every rational double point \nis a Klein singularity.\n\n\n\\subsection{Canonical lifts and simultaneous resolutions}\nLet $G$ be a very small, finite, and linearly reductive subgroup scheme of ${\\mathbf{GL}}_{2,k}$ \nand let $x\\in X=U\/G$ be the associated linearly reductive quotient singularity.\nAssume $p>0$ and let $W(k)$ be the ring of Witt vectors of $k$.\nIn Section \\ref{subsec: canonical lift singularity}, \nwe will establish the existence of a \n\\emph{canonical lift} \n$$\n \\mathcal{X}_{\\mathrm{can}}\\,\\to\\,\\mathrm{Spec}\\: W(k)\n$$\nof $x\\in X=U\/G$.\nUsing $G$-Hilbert schemes in families, we \nwill see in Section \\ref{subsec: simultaneous resolution} \nthat it admits a simultaneous and minimal resolution of singularities\n$$\n \\widetilde{\\pi} \\,:\\, \\mathcal{Y}\\,\\to\\,\\mathcal{X}_{\\mathrm{can}}\\,\\to\\,\\mathrm{Spec}\\: W(k).\n$$\nWe will prove this resolution to be unique, see \nTheorem \\ref{thm: IN}.\n\n\n\\subsection{McKay correspondence for Klein singularities}\nLet $G$ be a finite and linearly reductive subgroup scheme of ${\\mathbf{SL}}_{2,k}$ \nand let $x\\in X=U\/G$ be the associated Klein singularity.\n\nAs discussed in Section \\ref{subsec: McKay graph intro}, we have\nthe McKay graph $\\widehat{\\Gamma}$ associated to \n$G$, its embedding into ${\\mathbf{SL}}_{2,k}$, and the\nset of isomorphism classes of simple representations of $G$.\n\nLet $\\pi:Y\\to X$ be a minimal resolution of singularities.\nSince $x\\in X$ is a rational double point, the exceptional divisor $\\mathrm{Exc}(\\pi)$ \nof $\\pi$ is a configuration of ${\\mathds{P}}^1$'s, whose\ndual intersection graph $\\Gamma$ is a simply-laced Dynkin diagram.\n\n\\begin{Theorem}[Theorem \\ref{thm: bijection simple and exceptional}]\n\\label{thm: bijection simple intro}\n Let $k$ be an algebraically closed field of characteristic $p\\geq0$.\n Let $G$ be a finite and linearly reductive subgroup scheme of ${\\mathbf{SL}}_{2,k}$\n and let $x\\in X=U\/G$ be the associated Klein singularity.\n Then, there exists a natural bijection of the graph $\\Gamma$ \n with the graph obtained from $\\widehat{\\Gamma}$ by removing the vertex\n corresponding to the trivial representation.\n \\end{Theorem}\n \n Since every rational double point in characteristic $p\\geq7$ \n is a Klein singularity, we obtain the following.\n\n\\begin{Corollary}\nThere exists a linearly reductive McKay correspondence\nfor rational double point singularities in every characteristic $p\\geq7$.\n\\end{Corollary}\n\nTo establish this theorem, we use the Ishii-Ito-Nakamura resolution\nof singularities \\eqref{eq: intro IIN}, as well as Hecke correspondences as in the work\nof Ito and Nakamura \\cite{IN} and Nakajima \\cite{Nakajima, NakajimaLectures}.\n\n\n\\subsection{Generalisations and variants}\nLet $G$ be a very small, finite,\nand linearly reductive subgroup scheme of ${\\mathbf{GL}}_{2,k}$ \nand let $x\\in X=U\/G$ be the associated two-dimensional linearly\nreductive quotient singularity.\nLet $\\pi:Y\\to X$ be the minimal resolution of singularities and \nlet $\\mathrm{Exc}(\\pi)$ be the exceptional divisor of $\\pi$.\n\\begin{enumerate}\n\\item In Theorem \\ref{thm: exc pi}, \nwe associate a representation of $G$ to each point of $\\mathrm{Exc}(\\pi)$.\nThis generalises results of Ishii and Nakamura \\cite{IshiiCrelle, Ishii}.\n\\item In Theorem \\ref{thm: reflexive}, we establish a bijection between \nthe components of $\\textrm{Exc}(\\pi)$ and reflexive \n${\\cal O}_X$-modules.\nProbably, this result should be viewed as a theorem on two-dimensional F-regular\nsingularities, see Remark \\ref{rem: f-regular Artin Verdier}.\nIt generalises work of Artin and Verdier \\cite{AV}, \nWunram \\cite{Wunram}, and Ishii and Nakamura \\cite{Ishii}.\n\\item In Theorem \\ref{thm: derived}, we establish an equivalence of derived\ncategories of coherent sheaves ${\\mathcal{D}}(Y)$ and \n${\\mathcal{D}}^G(U)$ ($G$-equivariant \nsheaves on $U$).\nThis generalises work of Kapranov and Vasserot \\cite{KV},\nBridgeland, King, and Reid \\cite{Bridgeland},\nand Ishii, Nakamura, and Ueda \\cite{IshiiCrelle,Ishii, IshiiUeda}.\nOf course, these articles themselves generalise work of Gonzalez-Sprinberg and Verdier \n\\cite{GSV} from K-theory to derived categories of coherent sheaves.\n\\end{enumerate}\n\n\n\\subsection{Ito-Reid correspondence}\nIf $G$ is a finite subgroup of ${\\mathbf{SL}}_2({\\mathds{C}})$, then \nIto and Reid \\cite{ItoReid} found a natural bijection,\nthe \\emph{Ito-Reid correspondence},\nbetween the conjugacy classes of $G$ and the vertices \nof the McKay graph $\\widehat{\\Gamma}$.\n\n\\begin{Theorem}\n\\label{thm: ito reid intro}\n Let $k$ be an algebraically closed field of characteristic $p\\geq0$.\n There exists an Ito-Reid correspondence for finite and linearly reductive\n subgroup schemes of ${\\mathbf{SL}}_{2,k}$.\n\\end{Theorem}\n\nThis can be proven using lifting results and the Ito-Reid correspondence\nover ${\\mathds{C}}$, see Section \\ref{subsec: Ito-Reid} for details.\n\nThe main difficulty is to define a notion of conjugacy class\nfor finite and linearly reductive group schemes that makes\nsuch a correspondence work:\nLet us recall the ring $K_k(G)$ from Section \\ref{subsubsec: rep thy}.\nIf $G$ is a finite and linearly reductive group scheme over $k$,\nthen we define the set of \\emph{conjugacy classes} of $G$\nto be\n$$\n\\mathrm{Spec}\\: \\left({\\mathds{C}}\\otimes K_k(G)\\right).\n$$\nAt first sight, this might look rather artificial.\nOne should think of it as defining\nconjugacy classes to be ``dual'' to simple representations. \nMoreover, our definition is compatible with \nlifting over $W(k)$ and it induces a bijection of the conjugacy\nclasses of $G$ with the conjugacy classes of $G_{\\rm abs}$.\nWe refer to Section \\ref{subsec: conjugacy classes}\nand Appendix \\ref{subsec: second approach} for details.\n\n\n\\subsection{Conjugacy classes}\n\\label{intro: conjugacy classes}\n\nIn Theorem \\ref{thm: ito reid intro}, we had to find a definition\nof conjugacy classes that makes this theorem true.\nThis begs for the question whether there are other definitions \nor approaches, which is a question that is interesting in its own.\n\nLet $G$ be a finite group scheme (not necessarily linearly reductive)\nover an algebraically closed field $k$.\nIn Appendix \\ref{app: conjugacy class}, we study the following\napproaches to the notion of the set of conjugacy classes of $G$:\n\n\\begin{enumerate}\n\\item The set of conjugacy classes $G(k)\/\\sim$ of the group of $k$-rational points of $G$.\n\\item The spectrum of $F\\otimes K_k(G)$, where $F$ is a field of characteristic zero\nthat contains ``sufficiently many'' roots of unity.\n\\item The scheme that represents the functor of conjugacy classes from schemes over $k$ to sets\ndefined by $S\\mapsto G(S)\/\\sim$,\n\\item the isotypical component of the trivial representation of the \nadjoint representation of $G$.\n\\item The simple subrepresentations of the extended adjoint representations\n$^{\\mathrm{Ad}}A$ and $^{\\mathrm{Ad}}(A^*)$, which are defined using the\nquantum doubles of the Hopf algebra $A:=H^0(G,{\\cal O}_G)$\nand its dual $A^*$.\n\\end{enumerate}\nAll approaches lead essentially to the ``same answer'' in characteristic\nzero.\nHowever, they usually lead to very different notions in positive characteristic.\nOn the other hand, all approaches have their merits and drawbacks.\nFor linearly reductive group schemes, the approach (2) leads to a definition that is\ncompatible with lifting and that leads to an Ito-Reid correspondence.\n\n\n\\subsection{Organisation of this article}\n\n\\begin{enumerate}\n\\item[-] In Section \\ref{sec: linearly reductive}, we recall basic facts about finite and \nlinearly reductive group schemes over algebraically closed fields.\n\\item[-] In Section \\ref{sec: McKay}, we construct the McKay graph $\\widehat{\\Gamma}$.\nThe main result is Theorem \\ref{thm: main}, a McKay correspondence.\n\\item[-] In Section \\ref{sec: lrq singularities}, we recall basic facts about linearly\nreductive quotient singularities.\nWe establish an Ishii-Ito-Nakamura-type resolution of singularities,\nthe canonical lift of such a singularity, and a unique simultaneous resolution\nof singularities of the canonical lift.\n\\item[-] In Section \\ref{sec: Hecke}, we revisit the Ishii-Ito-Nakamura-type resolution $\\pi$\nof singularities of $x\\in X=U\/G$\nand we introduce Hecke correspondences, which leads to \nTheorem \\ref{thm: bijection simple and exceptional},\na bijection between simple and non-trivial representations of $G$ \nand components of $\\pi$.\nWe also study generalisations of this result to two-dimensional linearly reductive \nquotient singularities.\n\\item[-] In Section \\ref{sec: ItoReid}, we establish an Ito-Reid correspondence\nbetween conjugacy classes of $G$ \nand exceptional divisors in the minimal resolution of singularities of $x\\in X=U\/G$.\n\\item[-] In Section \\ref{sec: outlook}, we study derived categories\nof $G$-equivariant sheaves on $U$ and on the minimal \nresolution $\\pi:Y\\to X=U\/G$.\n\\item[-] In Appendix \\ref{sec: Hopf}, we recall results on finite group schemes\nfrom the point of view of Hopf algebras.\nWe recall the adjoint representation, quantum doubles, and the\nextended adjoint representation.\n\\item[-] In Appendix \\ref{app: conjugacy class}, we study several approaches\ntoward the notion of a conjugacy class for finite group schemes.\n\\end{enumerate}\n\n\\begin{VoidRoman}[Acknowledgements]\n I thank Frank Himstedt, Martin Lorenz, Frans Oort, Matt Satriano,\n and Takehiko Yasuda for discussions and comments.\n\\end{VoidRoman}\n\n\n\\section{Linearly reductive group schemes}\n\\label{sec: linearly reductive}\n\nIn this section, we recall a couple of general facts about finite and linearly\nreductive group schemes over algebraically closed fields.\nWe discuss the close relationship between such a group scheme $G$ \nand a certain abstracted group $G_{\\mathrm{abs}}$ associated to it.\nFor the relationship to Hopf algebras, we refer to Appendix \\ref{sec: Hopf}.\n\n\\subsection{Group schemes}\nLet $G$ be a finite group scheme over an algebraically closed \nfield $k$ of characteristic $p\\geq0$.\nSince $k$ is perfect, there is a short exact sequence\nof finite group schemes over $k$\n\\begin{equation}\n\\label{connected etale}\n 1\\,\\to\\,G^\\circ\\,\\to\\,G\\,\\to\\, G^{\\mathrm{\\acute{e}t}} \\,\\to\\,1,\n\\end{equation}\nwhere $G^\\circ$ is the connected component of the identity and where $G^{\\mathrm{\\acute{e}t}}$\nis an \\'etale group scheme over $k$.\nThe reduction $G_{\\mathrm{red}}\\to G$ provides a canonical splitting of \\eqref{connected etale}\nand we obtain a canonical semi-direct product decomposition $G\\cong G^\\circ\\rtimes G^\\mathrm{\\acute{e}t}$.\nSince $k$ is algebraically closed, $G^{\\mathrm{\\acute{e}t}}$ is the constant group scheme associated\nto the finite group $G(k)=G^{\\mathrm{\\acute{e}t}}(k)$ of $k$-rational points.\nMoreover, $G^\\circ$ is an infinitesimal group scheme of length equal to \nsome power of $p$.\nIn particular, if $p=0$ or if the length of $G$ is prime to $p$, \nthen $G^\\circ$ is trivial and then, $G$ is \\'etale. \n\nIf $M$ is a finitely generated abelian group, then the group algebra\n$k[M]$ carries a Hopf algebra structure and the associated commutative\ngroup scheme is denoted $D(M):=\\mathrm{Spec}\\: k[M]$.\nBy definition, such group schemes are called \\emph{diagonalisable}.\nFor example, we have \n$D({\\mathbf{C}}_n)\\cong\\boldsymbol{\\mu}_n$, where ${\\mathbf{C}}_n$ denotes the cyclic group of order $n$.\nWe have that $\\boldsymbol{\\mu}_n$ is \\'etale over $k$ if and only if\n$p\\nmid n$.\n\nA finite group scheme $G$ over $k$ is said to be \\emph{linearly reductive} \nif every $k$-linear and finite-dimensional representation of $G$ is semi-simple.\nIf $p=0$, then all finite group schemes over $k$ are \\'etale and\nlinearly reductive.\nIf $p>0$, then, by a theorem that is often attributed to\nNagata \\cite[Theorem 2]{Nagata61}\n(but see also \\cite[Proposition 2.10]{AOV}, \\cite{Chin}, \nand \\cite[Section 2]{Hashimoto}), \na finite group scheme over $k$ is\nlinearly reductive if and only if it is an extension of a finite and \\'etale\ngroup scheme, whose length is prime to $p$, \nby a diagonalisable group scheme. \n\n\\subsection{Abstract groups and canonical lifts}\n\\label{subsec: abstract groups}\nLet $G$ be a finite and linearly reductive group scheme\nover an algebraically closed field $k$ of characteristic $p>0$.\nFollowing \\cite[Section 2]{LMM}, we study the finite group\n$$\nG_{\\mathrm{abs}} \\,:=\\, \n\\left(\\underline{((G^\\circ)^{D}(k))}_{{\\mathds{C}}}\\right)^{D}({\\mathds{C}}) \\,\\rtimes\\, G^\\mathrm{\\acute{e}t}(k).\n$$\nHere, $-^D$ denotes $\\underline{\\mbox{Hom}}(-,{\\mathds{G}}_m)$, the \\emph{Cartier dual}, of\na commutative and finite group scheme.\nIf $G$ is \\'etale, then $G_{\\mathrm{abs}}=G(k)$ and if $G=\\boldsymbol{\\mu}_{p^n}$,\nthen $G_{\\mathrm{abs}}={\\mathbf{C}}_{p^n}$.\nIn any case, the order of $G_{\\mathrm{abs}}$ is equal to the length of $G$.\n\n\\begin{Definition}\nThe finite group $G_{\\mathrm{abs}}$\nis called the \\emph{abstract finite group} associated to $G$.\n\\end{Definition}\n\n\\begin{Lemma} \\label{lem: lrequivalence}\nThe functor\n$$\nG \\,\\mapsto\\, G_{\\mathrm{abs}}\n$$\nestablishes an equivalence of categories between the category of finite and \nlinearly reductive group schemes over $k$ \nand the category of finite groups with a normal and abelian $p$-Sylow subgroup.\n\\end{Lemma}\n\n\\begin{proof}\n\\cite[Lemma 2.1]{LMM}.\n\\end{proof}\n\n\\begin{Remark}\nA finite group $G$ with a normal and abelian $p$-Sylow subgroup $P$ is the same\nas a finite group with a unique and abelian $p$-Sylow subgroup.\nIn this case, the Schur-Zassenhaus theorem implies that\n$G$ is isomorphic to a semi-direct product $P\\rtimes G\/P$.\n\\end{Remark}\n\nNext, we study lifts to characteristic zero:\nlet $W(k)$ be the ring of Witt vectors of $k$, let $K$ be its field\nof fractions, and let $\\overline{K}$ be an algebraic closure\nof $K$.\nBy \\cite[Proposition 2.4]{LMM},\nthere exist lifts of $G$ as finite and flat\ngroup scheme over $W(k)$.\nMore precisely, $G^\\circ$ and $G^\\mathrm{\\acute{e}t}$ even lift uniquely to $W(k)$,\nbut their extension class usually does not, see also\n\\cite[Example 2.6]{LMM}.\nHowever, there is a unique lift ${\\cal G}_{\\mathrm{can}}$\nof $G$ to $W(k)$ that is characterised by being \na semi-direct product of the unique lift of $G^\\circ$ with the unique lift of \n$G^{\\mathrm{\\acute{e}t}}$.\n\n\\begin{Definition}\\label{def: canonical lift}\nThe lift $\\mathcal{G}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$ is called\nthe \\emph{canonical lift} of $G$.\nWe set $G_{\\mathrm{can}}:=\\mathcal{G}_K\\to \\mathrm{Spec}\\: K$\nand also call it canonical lift.\n\\end{Definition}\n\nEvery other lift of $G$ to some extension of $R\\supseteq W(k)$ differs from\n${\\cal G}_{\\mathrm{can},R}$ by a twist and thus, there is only one \\emph{geometric lift}\nof $G$ to $\\overline{K}$ up to isomorphism, namely $G_{\\mathrm{can},\\overline{K}}$,\nsee \\cite[Section 2.2]{LMM}.\n\nSince $\\overline{K}$ is algebraically closed and\nof characteristic zero, $G_{\\mathrm{can},\\overline{K}}$ is the constant\ngroup scheme associated to the finite group\n$G_{\\mathrm{can}}(\\overline{K})$.\nIn fact, we have $G^\\circ\\cong\\prod_i \\boldsymbol{\\mu}_{p^{n_i}}$ for some $n_i$'s and if we set\n$N:=\\max\\{n_i\\}_i$, fix a primitive $p^N$.th root of unity $\\zeta_{p^N}$ and set \n$K_N:=K(\\zeta_{p^N})$, \nthen $G_{\\mathrm{can},K_N}$ is the constant group scheme associated \nto the finite group $G_{\\mathrm{can}}(K_N)$ and we have\n$G_{\\mathrm{can}}(K_N)=G_{\\mathrm{can}}(\\overline{K})$.\n\n\\begin{Lemma}\n There exist isomorphism of finite groups\n $$\n G_{\\mathrm{abs}} \\,\\cong\\, G_{\\mathrm{can}}(K_N) \\,=\\,G_{\\mathrm{can}}(\\overline{K}).\n $$\nIn particular, there exist isomorphisms\n $$\n G_{\\mathrm{can},K_N} \\,\\cong\\,\n \\left(\\underline{G_{\\mathrm{abs}}}\\right)_{K_N}\n \\mbox{ \\quad and \\quad }\n G_{\\mathrm{can},\\overline{K}} \\,\\cong\\,\n \\left(\\underline{G_{\\mathrm{abs}}}\\right)_{\\overline{K}}\n $$\n of finite group schemes over $K_N$ and $\\overline{K}$, respectively.\\qed\n\\end{Lemma}\n\n\\subsection{Representation theory}\n\\label{subsec: representation comparison}\nLet $k$ be an algebraically closed field of characteristic $p>0$,\nand let $W(k)$, $K$, and $\\overline{K}$ be as in the previous section.\nThe equivalence of Lemma \\ref{lem: lrequivalence}\ncan be extended to representations and K-theory.\nIf $G$ is a finite group or a finite group scheme over $k$,\nwe denote by ${\\rm Rep}_k(G)$ the category of its\n$k$-linear and finite-dimensional representations.\nLet us first recall \\cite[Corollary 2.11]{LMM}.\n\n\\begin{Proposition} \\label{prop: liftingrepresentation}\n Let $G$ be a finite and linearly reductive group scheme\n over $k$.\n Then, there exist canonical equivalences of categories\n $$\n \\mathrm{Rep}_{k}(G) \n \\,\\to\\, \\mathrm{Rep}_{K_N}(G_{\\mathrm{can},K_N})\n \\,\\to\\, \\mathrm{Rep}_{\\overline{K}}(G_{\\mathrm{can},\\overline{K}})\n \\,\\to\\, \\mathrm{Rep}_{\\mathds{C}}(G_{\\mathrm{abs}}),\n $$\n which are compatible with degrees, direct sums, tensor products, duals, \n and simplicity.\\qed\n\\end{Proposition}\n\n\\begin{Remark}\n In particular, this allows us to define \\emph{characters} or even a\n \\emph{character table} of $G$ via \n Proposition \\ref{prop: liftingrepresentation} and $G_{\\mathrm{abs}}$.\n\\end{Remark}\n\nLet $K_k(G)$ be $K$-group associated to ${\\rm Rep}_k(G)$.\nIn fact, $K_k(G)$ has a natural structure of a commutative ring with one, \nwhere the sum (resp. product) structure comes from direct sums (resp. tensor products)\nof representations.\nA straight forward application of Proposition \\ref{prop: liftingrepresentation}\nis the following.\n\n\\begin{Corollary}\n \\label{cor: Ktheory}\n There exist isomorphisms of rings\n $$\n K_k(G) \\,\\to\\, \n K_{K_N}(G_{\\mathrm{can},K_N}) \\,\\to\\,\n K_{\\overline{K}}(G_{\\mathrm{can},\\overline{K}}) \\,\\to\\,\n K_{{\\mathds{C}}}(G_{\\mathrm{abs}}).\\qed\n $$\n\\end{Corollary}\n\n\n\\section{McKay graph and McKay correspondence}\n\\label{sec: McKay}\n\nIn this section, we introduce the \\emph{McKay graph}\nassociated to a finite and linearly reductive subgroup scheme $G$\nover an algebraically closed field $k$ of characteristic $p\\geq0$\nand a representation $\\rho:G\\to{\\mathbf{GL}}_{n,k}$.\nThis induces a bijection between certain affine Dynkin diagrams\nand finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$.\nAs an application, we establish a \\emph{linearly reductive McKay correspondence}.\n\n\\subsection{McKay graph}\n\\label{subsec: McKay graph}\nLet $G$ be a finite and linearly reductive group scheme\nover an algebraically closed field $k$ of characteristic $p\\geq0$.\nLet $\\{\\rho_i\\}_i$ be the finite set of isomorphism classes of $k$-linear \nand simple representations of $G$.\nFollowing tradition, we assume that $\\rho_0$ is the trivial representation.\nWe fix a representation $\\rho:G\\to{\\mathbf{GL}}(V)$.\nIf $G$ is a subgroup scheme of ${\\mathbf{SL}}_{n,k}$ or ${\\mathbf{GL}}_{n,k}$, \nthen $\\rho$ is usually the linear representation corresponding to the embedding\nof $G$ into this linear algebraic group.\nBy assumption, $\\mathrm{Rep}_k(G)$ is semi-simple.\nTherefore, there exist unique integers $a_{ij}\\in{\\mathds{Z}}_{\\geq0}$ for each $i$,\nsuch that we have isomorphisms of $k$-linear representations\n$$\n \\rho\\otimes\\rho_i \\,\\cong\\, \\bigoplus_{j}\\, \\rho_j^{\\oplus a_{ij}}.\n$$\nAssociated to this data, we define the \\emph{McKay graph}, \ndenoted $\\Gamma(G,\\{\\rho_i\\},\\rho)$:\n\\begin{enumerate}\n\\item[-] The vertices are the $\\{\\rho_{i}\\}_i$.\n(Some sources exclude the\nthe trivial representation $\\rho_0$.)\n\\item[-] There are $a_{ij}$ edges from the vertex corresponding to\n$\\rho_i$ to the vertex corresponding to $\\rho_j$.\n\\end{enumerate}\nWe now establish a couple of elementary properties of\nthis graph, which are well-known in the classical case\nand which immediately carry over to the linearly reductive\nsituation.\nWe leave the proof of the first lemma to the reader.\n\n\\begin{Lemma}\n We have\n $$\n a_{ij} \\,=\\, \\dim_k\\, \\mathrm{Hom}(\\rho_i,\\rho_j\\otimes\\rho).\n $$\n In particular, if $\\rho$ is self-dual, that is, $\\rho\\cong\\rho^\\vee$,\n then $a_{ij}=a_{ji}$ for all $i,j$.\n In this case, we can consider $\\Gamma(G,\\{\\rho_i\\},\\rho)$ \n as an undirected graph.\\qed\n\\end{Lemma}\n\n\\begin{Lemma}\n\\label{lem: selfdual}\nLet $\\rho:G\\to{\\mathbf{SL}}_{2,k}$ be a homomorphism of group schemes\nover $k$, considered as a\n2-dimensional representation.\nThen, $\\rho$ is self-dual.\n\\end{Lemma}\n\n\\begin{proof}\nBeing a 2-dimensional representation, \n$\\rho^\\vee$ is isomorphic to $\\rho\\otimes\\det(\\rho)$ and\nthe lemma follows.\n\\end{proof}\n\n\\begin{Lemma}\n\\label{lem: faithful}\nLet $\\rho:G\\to{\\mathbf{GL}}_{n,k}$ be a faithful representation.\nThen, every irreducible representation of $G$\noccurs as subrepresentation of $\\rho^{\\otimes m}$ for\nsome suitable $m$.\n\\end{Lemma}\n\n\\begin{proof}\nThis is well-known for finite groups.\nUsing Proposition \\ref{prop: liftingrepresentation}, it \ncarries over to finite and linearly reductive group schemes.\n\\end{proof}\n\n\\begin{Corollary}\nIf $\\rho$ is a faithful representation, then the graph\n$\\Gamma(G,\\{\\rho_i\\},\\rho)$ is connected.\n\\end{Corollary}\n\n\\begin{proof}\nIt suffices to note that the number of paths in $\\Gamma$\nof length $m$ that connect the vertices corresponding to $\\rho_i$ and $\\rho_j$\nis equal to the multiplicity of $\\rho_i$ in $\\rho_j\\otimes\\rho^{\\otimes m}$,\nsee, for example, the proof of \\cite[Theorem 8.13]{Kirillov}.\n\\end{proof}\n\nLet $k$ be an algebraically closed field of characteristic $p>0$,\nand let $W(k)$, $K$, and $\\overline{K}$ be as in Section \\ref{subsec: abstract groups}.\nThere, we also discussed the\ncanonical lift $G_{\\mathrm{can}}$ of $G$ over $K$ and we saw that there exists\nan isomorphism of finite groups\n$G_{\\mathrm{abs}}\\cong G_{\\mathrm{can}}(\\overline{K})$.\nProposition \\ref{prop: liftingrepresentation} implies the following result.\n\n\\begin{Proposition}\n\\label{prop: McKay Quiver lift}\nLet $G$ be a finite and linearly reductive group scheme over $k$.\nLet $\\{\\rho_i\\}_i$ be the set of isomorphism classes of \nsimple representations of $G$.\nLet $\\rho$ be a finite-dimensional representation of $G$.\n\\begin{enumerate}\n \\item \n Proposition \\ref{prop: liftingrepresentation} yields\n sets of representations $\\{\\rho_{\\mathrm{can},i}\\}_i$ and $\\{\\rho_{\\mathrm{abs},i}\\}_i$\n of $G_{\\mathrm{can}}$ and $G_{\\mathrm{abs}}$, respectively, which are the \n sets of isomorphism classes of simple representations of $G_{\\mathrm{can}}$\n and $G_{\\mathrm{abs}}$, respectively.\n \\item \n Proposition \\ref{prop: liftingrepresentation} yields\n representations $\\rho_{\\mathrm{can}}$ and $\\rho_{\\mathrm{abs}}$\n of $G_{\\mathrm{can}}$ and $G_{\\mathrm{abs}}$, respectively.\n \\end{enumerate}\nThis data leads to a bijection of the three McKay graphs\n $$\n \\Gamma(G,\\{\\rho_i\\}_i,\\rho),\\quad\n \\Gamma(G_{\\mathrm{can}},\\{\\rho_{\\mathrm{can},i}\\}_i,\\rho_{\\mathrm{can}}),\\quad\n \\Gamma(G_{\\mathrm{abs}},\\{\\rho_{\\mathrm{abs},i}\\}_i,\\rho_{\\mathrm{abs}}).\\qed\n $$\n\\end{Proposition}\n\n\\subsection{McKay correspondence}\nWe now run through the classical McKay correspondence \\cite{McKay} \nin our setting, where we follow \\cite[Section 8.3]{Kirillov}.\nLet $k$ be an algebraically closed field of characteristic $p\\geq0$\nand let $G$ be a finite and linearly reductive subgroup scheme\nof ${\\mathbf{SL}}_{2,k}$.\n\\begin{enumerate}\n \\item The K-group $K_k(G)$ carries a symmetric bilinear form \n $$\n ([V],[W])_0 \\,:=\\, \\dim_k\\,\\mathrm{Hom}_G(V,W).\n $$\n \\item We consider the closed embedding $G\\to{\\mathbf{SL}}_{2,k}$ as a \n 2-dimensional \n representation \n $\\rho:G\\to{\\mathbf{SL}}_{2,k}\\to{\\mathbf{GL}}_{2,k}$\n and define an operator \n $$\n A\\,:\\,K_k(G)\\,\\to\\, K_k(G),\\mbox{\\qquad} [V]\\mapsto [V]\\otimes\\rho\\,.\n $$\n Since $\\rho$ is self-dual by Lemma \\ref{lem: selfdual}, it follows\n that $A$ is symmetric with respect to $(-,-)_0$.\n \\item Using $\\rho$, we define a symmetric bilinear form on \n $K_k(G)\\otimes_{\\mathds{Z}}{\\mathds{R}}$ by\n $$\n ([V],[W]) \\,:=\\, \\left([V],\\, (2-A)[W]\\right)_0,\n $$\n which is positive semi-definite.\n The class $\\delta\\in K_k(G)$ of the regular representation \n of $G$ generates the radical of $(-,-)$.\n \\item Let $\\{\\rho_i\\}$ be the set of isomorphism classes of simple\n representations of $G$.\n \\begin{enumerate}\n \\item\n The McKay graph $\\widehat{\\Gamma}=\\Gamma(G,\\{\\rho_i\\}_i,\\rho)$\n is an affine Dynkin diagram\n of type $\\widehat{A}_n$, $\\widehat{D}_n$ with $n\\geq4$, \n $\\widehat{E}_6$, $\\widehat{E}_7$\n or $\\widehat{E}_8$.\n \\item\n After removing the vertex corresponding to the trivial representation\n $\\rho_0$ from $\\widehat{\\Gamma}$, \n we obtain a finite Dynkin diagram \n of type $A_n$, $D_n$ with $n\\geq4$, $E_6$, $E_7$\n or $E_8$, respectively.\n \\end{enumerate}\n\\end{enumerate}\nIf $G$ is a finite subgroup of ${\\mathbf{SL}}_2({\\mathds{C}})$, then these statements are part\nof the classical McKay correspondence, see \\cite{McKay} or \n\\cite[Section 8.3]{Kirillov}.\nIn our setting of finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$,\nthe above claims immediately follow from the classical McKay correspondence \ntogether with the lifting results Proposition \\ref{prop: liftingrepresentation}\nand Proposition \\ref{prop: McKay Quiver lift}.\nWe then obtain the following analog of \nMcKay's theorem \\cite{McKay}.\n\n\\begin{Theorem}\n\\label{thm: main}\n Let $k$ be an algebraically closed field of characteristic $p\\geq0$.\n There exists a bijection between\n finite, non-trivial, and linearly reductive subgroup schemes of\n ${\\mathbf{SL}}_{2,k}$ up to conjugation and affine Dynkin diagrams\n of type\n $$\n \\begin{array}{lcl}\n \\widehat{A}_n, \\widehat{D}_n, \\widehat{E}_6, \\widehat{E}_7, \\widehat{E}_8\n &\\mbox{\\quad if \\quad }& p=0\\mbox{\\quad or \\quad}p\\geq7, \\\\\n \\widehat{A}_n, \\widehat{D}_n, \\widehat{E}_6, \\widehat{E}_7 &\\mbox{\\quad if \\quad }& p=5,\\\\\n \\widehat{A}_n, \\widehat{D}_n &\\mbox{\\quad if \\quad }& p=3, \\mbox{ and} \\\\\n \\widehat{A}_n &\\mbox{\\quad if \\quad }&p=2.\n \\end{array}\n $$\n\\end{Theorem}\n\n\\begin{proof}\nIf $p=0$, then this is part of the classical McKay correspondence.\nIf $p>0$, then this follows from the linearly reductive McKay correspondence just discussed together \nwith Hashimoto's classification \\cite[Theorem 3.8]{Hashimoto}\nof finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$.\n\\end{proof}\n\n\\begin{Remark}\nThe linearly reductive group schemes corresponding to $\\widehat{E}_6$, $\\widehat{E}_7$, and\n$\\widehat{E}_8$ are \\'etale and correspond to finite groups of order prime to $p$.\nThe linearly reductive group scheme corresponding to $\\widehat{A}_n$ (resp.\n$\\widehat{D}_n$) is \\'etale if and only if $p\\nmid (n+1)$ (resp. $p\\nmid (n-2)$).\nWe refer to \\cite[Proposition 4.2]{LiedtkeSatriano} for details.\nThus, even if $p$ is sufficiently large, then one does not obtain a bijection in \nTheorem \\ref{thm: main} with finite groups of order to prime to $p$ only.\n\\end{Remark}\n\n\\begin{Remark}\nSteinberg \\cite{Steinberg} established many properties of the McKay graph\nand the McKay correspondence for finite subgroups of ${\\mathbf{SU}}_2({\\mathds{C}})$\n\\emph{without} using classification lists.\nIt seems plausible to obtain a proof of Theorem \\ref{thm: main} along these \nlines without using lifting results or classification lists.\n\\end{Remark}\n\n\n\\section{Linearly reductive quotient singularities}\n\\label{sec: lrq singularities}\n\nIn this section, we recall \\emph{linearly reductive quotient singularities}\nin the sense of \\cite{LMM} and some general results, including\nthe \\emph{canonical lift} of such a singularity over the ring of Witt vectors.\nIn dimension two, we establish a minimal resolution of singularities using $G$-Hilbert schemes\nas in the work of Ishii, Ito, and Nakamura \\cite{IshiiCrelle, Ishii, IN}.\nWe also show that the canonical lift admits a unique minimal and simultaneous\nresolution of singularities.\nFinally, we discuss \\emph{Klein singularities} and their relation to\nrational double point singularities.\n\n\\subsection{Quotient singularities}\nLet $k$ be an algebraically closed field of characteristic $p\\geq0$.\nIf $G$ is a finite and linearly reductive group scheme over $k$,\nif $V$ is a finite-dimensional $k$-vector space,\nand if $\\rho:G\\to{\\mathbf{GL}}(V)$ is a linear representation, \nthen we define the \\emph{$\\lambda$-invariant} of $\\rho$ \nas in \\cite[Definition 2.7]{LMM} to be\n$$\n\\lambda(\\rho) \\,:=\\, \\max_{\\{{\\mathrm{id}}\\}\\neq\\boldsymbol{\\mu}_n\\subseteq G} \\dim V^{\\boldsymbol{\\mu}_n},\n$$\nwhere $V^{\\boldsymbol{\\mu}_n}$ denotes the subspace of $\\boldsymbol{\\mu}_n$-invariants.\nAs explained in \\cite[Remark 2.8]{LMM}, the representation\n$\\rho$ is faithful if and only if $\\lambda(\\rho)\\neq\\dim V$.\nMoreover, $\\rho$ contains no pseudo-reflections if and only\nif $\\lambda(\\rho)\\leq\\dim V-2$ and in this case,\nthe representation $\\rho$ is said to be \\emph{small}.\n\n\\begin{Definition}\n\\label{def: very small}\n The representation $\\rho$ is called \n \\emph{very small} if $\\lambda(\\rho)=0$.\n\\end{Definition}\n\nNote that in dimension two the notions of small and very small coincide.\nWe refer to \\cite[Section 3]{LMM} for the classification of \nfinite and linearly reductive group schemes that admit a very small\nrepresentation.\nBy \\cite[Proposition 6.5]{LMM}, $\\lambda(\\rho)$ is equal to the dimension\nof the non-free locus of the induced $G$-action on the spectrum of the\nformal power series ring $k[[V^\\vee]]=(k[V^\\vee])^\\wedge$.\nFollowing \\cite[Definition 6.4]{LMM} and using the linearisation result\n\\cite[Proposition 6.3]{LMM}, we define:\n\n\\begin{Definition}\n \\label{def: lrq singularity}\n A \\emph{linearly reductive quotient singularity} over $k$\n is an isolated singularity that is analytically isomorphic to\n $\\mathrm{Spec}\\: k[[V^\\vee]]^G$, where $G$ is a finite and linearly reductive group scheme over\n $k$ and where $\\rho:G\\to{\\mathbf{GL}}(V)$ is a very small representation.\n\\end{Definition}\n\n\\begin{Remark}\nWe set $U:=\\mathrm{Spec}\\: k[V^\\vee]$ \nor $U:=\\mathrm{Spec}\\: k[[V^\\vee]]$ and simply write\n$x\\in X=U\/G$ with the assumptions of\nDefinition \\ref{def: lrq singularity} implicitly understood.\n\\end{Remark}\n\nProperties of these singularities have been studied in \\cite{LMM}:\nfor their local \\'etale fundamental groups, class groups, and\nF-signatures, we refer to \\cite[Section 7]{LMM}.\n\nBy \\cite[Theorem 8.1]{LMM}, a linearly reductive quotient singularity \ndetermines the finite and linearly reductive group scheme $G$\ntogether with the very small representation $\\rho:G\\to{\\mathbf{GL}}(V)$\nuniquely up to isomorphism and conjugacy, respectively.\nIt thus makes sense to refer to $\\rho(G)\\subset{\\mathbf{GL}}(V)$\nas the \\emph{finite and linearly reductive subgroup scheme of ${\\mathbf{GL}}(V)$ \nassociated to the linearly reductive quotient singularity}\n$x\\in X=U\/G$.\nIn particular, the classification of linearly reductive quotient singularities \nin dimension $d$ is ``the same'' as the classification of very small, finite, \nand linearly reductive subgroup schemes of ${\\mathbf{GL}}_{d,k}$ up to conjugacy.\nWe refer to \\cite[Section 3]{LMM} for details and this classification.\n\n\\subsection{Minimal resolution of singularities}\nIf $x\\in X$ is a normal surface singularity, then it \nadmits a unique minimal resolution of singularities\n$$\n \\pi\\,:\\, Y\\,\\to\\,X.\n$$\nIf $x\\in X$ is moreover a rational singularity, then the \nexceptional locus of $\\pi$ is a union\nof ${\\mathds{P}}^1$'s, whose dual intersection graph $\\Gamma$ contains no cycles.\nThe graph $\\Gamma$ is called the \\emph{type} of $x\\in X$.\nIf the type determines the singularity up to analytic isomorphism,\nthen the singularity is said to be \\emph{taut}.\n\nOver ${\\mathds{C}}$, taut singularities have been classified by \nLaufer \\cite{Laufer}.\nFor example, two-dimensional finite quotient singularities \nover ${\\mathds{C}}$ are taut.\nSince two-dimensional linearly reductive quotient singularities over algebraically\nclosed fields of positive characteristic are \nF-regular (see \\cite[Proposition 7.1]{LMM}), they are taut by Tanaka's\ntheorem \\cite{Tanaka}.\n\n\\begin{Remark}\n\\label{rem: type quotient singularity}\nVery small, finite, and linearly reductive subgroup schemes of ${\\mathbf{GL}}_{2,k}$\nhave been classified in \\cite[Theorem 3.4]{LMM}, extending Brieskorn's classification \n\\cite[Satz 2.9]{Brieskorn} of small subgroups of ${\\mathbf{GL}}_2({\\mathds{C}})$.\nThe types of the associated quotient singularities in terms of this classification are \ngiven by \\cite[Satz 2.11]{Brieskorn}.\n\nFinite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$ are automatically\nvery small and they\nhave been classified by Hashimoto \\cite[Theorem 3.8]{Hashimoto}, extending \nKlein's classification \\cite{Klein} of finite subgroups of ${\\mathbf{SL}}_2({\\mathds{C}})$\nand we refer to \\cite{Hashimoto, LiedtkeSatriano} for the types\nof the associated quotient singularities\n(see also Theorem \\ref{thm: classification Klein} below).\n\\end{Remark}\n\n\n\\subsection{The Ishii-Ito-Nakamura resolution}\n\\label{subsec: Ishii Ito Nakamura}\nIn \\cite{IN}, Ito and Nakamura showed that if \n$G$ is a finite subgroup of ${\\mathbf{SL}}_2({\\mathds{C}})$, then a minimal\nresolution of singularities of ${\\mathds{C}}^2\/G$ is provided\nby the $G$-Hilbert scheme $G\\mathrm{-Hilb}({\\mathds{C}}^2)$.\nIshii \\cite{IshiiCrelle} extended this to quotient singularities\n${\\mathds{C}}^2\/G$ for $G$ a finite and very small subgroup of ${\\mathbf{GL}}_2({\\mathds{C}})$\nand Ishii and Nakamura \\cite{Ishii} extended this to quotient\nsingularities $U\/G$ for $G$ a finite and very small subgroup\nof ${\\mathbf{GL}}_2(k)$ of order prime to $p$.\n\nLet $k$ be an algebraically closed field of characteristic $p\\geq0$.\n Let $G$ be a very small, finite, and linearly reductive subgroup scheme of \n ${\\mathbf{GL}}_{2,k}$.\n Set $U:={\\mathds{A}}^2_k$ or $\\widehat{{\\mathds{A}}}_k^2$ \n and let $x\\in X:=U\/G$ be the associated linearly reductive\n quotient singularity.\n By \\cite{Blume}, there exists a $G$-Hilbert scheme\n$G\\mbox{-Hilb}(U)$ over $k$ that parametrises zero-dimensional\n$G$-invariant subschemes $Z\\subset U$ \n(so-called \\emph{clusters}), such that the $G$-representation on \n$H^0(Z,{\\cal O}_Z)$ is the regular representation.\nTaking a cluster $Z$ to its $G$-orbit\n(see, for example, \\cite[Remark 3.3]{Blume}) \ninduces a morphism\n$$\n\\pi\\,:\\,Y\\,:=\\,G\\mbox{-Hilb}(U) \\,\\to\\, U\/G\\,=\\,X\n$$\nover $k$.\n\n\\begin{Theorem}\n\\label{thm: IshiiItoNakamura}\n The morphism $\\pi$ is a minimal resolution of singularities.\n\\end{Theorem}\n\n\\begin{proof}\nIf $p=0$, then this is shown in \\cite{Ishii}, extending \\cite{IN}.\nIf $p>0$ and $G$ is of length prime to $p$, then this is shown in \\cite{Ishii}.\nHowever, this proof also works if $G$ is linearly reductive and $p$ divides the length\nof $G$.\n\\end{proof}\n\n\\begin{Remark}\n\\label{rem: f-regular}\nBy \\cite[Theorem 11.5]{LMM}, every two-dimensional F-regular singularity\n$x\\in X$ is a linearly reductive quotient singularity, say $x\\in X=U\/G$ for\nsome finite and linearly reductive subgroup scheme $G$ of ${\\mathbf{GL}}_{2,k}$.\nBy \\cite[Theorem 8.1]{LMM}, $G$ and its embedding into ${\\mathbf{GL}}_{2,k}$ are unique\nup to isomorphism and conjugacy, respectively.\nBy Theorem \\ref{thm: IshiiItoNakamura}, $G\\mbox{-Hilb}(U)\\to U\/G=X\\ni x$\nis the minimal resolution of singularities.\nIn particular, every two-dimensional F-regular singularity can be resolved\nby a suitable $G$-Hilbert scheme.\n\\end{Remark}\n\n\\subsection{Canonical lifts}\n\\label{subsec: canonical lift singularity}\nLet $k$ be an algebraically closed field of characteristic $p>0$,\nlet $W(k)$ be the ring of Witt vectors, let $K$ be its field of fractions, and let\n$\\overline{K}$ be an algebraic closure.\nLet $x\\in X=U\/G$ be a $d$-dimensional \nlinearly reductive singularity, where $U={\\mathds{A}}^d_k$ or $U=\\widehat{{\\mathds{A}}}_k^d$.\nNext, we set ${\\cal U}:={\\mathds{A}}^d_{W(k)}$ or $\\widehat{{\\mathds{A}}}^d_{W(k)}$, respectively.\nIn Section \\ref{subsec: abstract groups}, we recalled the canonical lift \n${\\cal G}_{\\mathrm{can}}$ of $G$ over $W(k)$.\nBy \\cite[Proposition 2.9]{LMM}, there exists a unique lift of the $G$-action\nfrom $U$ to ${\\cal U}$.\nFrom this, we obtain a flat family \n\\begin{equation}\n\\label{eq: lrq lift}\n \\mathcal{X}_{\\mathrm{can}}\\,:=\\,{\\cal U}\/{\\cal G}_{\\mathrm{can}} \\,\\to\\, \\mathrm{Spec}\\: W(k)\\,.\n\\end{equation}\nof linearly reductive quotient singularities over $W(k)$, whose special fibre\nover $k$ is $x\\in X=U\/G$.\n\n\\begin{Definition}\n The family \\eqref{eq: lrq lift} is called the \\emph{canonical lift} of the linearly reductive \n quotient singularity $x\\in X=U\/G$.\n \\end{Definition}\n \nBy the Lefschetz principle, the geometric generic fibre of $\\mathcal{X}_{\\mathrm{can}}$\ncan be identified with a finite quotient singularity of the form ${\\mathds{C}}^d\/G_{\\mathrm{abs}}$, \nwhere $G_{\\mathrm{abs}}$ is the abstract group associated to $G$\nand the embedding of $G_{\\mathrm{abs}}\\to{\\mathbf{GL}}_d({\\mathds{C}})$\ncorresponds to the embedding $G\\to{\\mathbf{GL}}_{d,k}$ provided by\nProposition \\ref{prop: liftingrepresentation}.\nThe canonical lift is unique in the following sense:\n\n\\begin{Proposition}\n We keep the notations and assumptions.\n Let $W(k)\\subseteq R$ be a finite extension of complete DVRs and let $\\mathcal{X}\\to\\mathrm{Spec}\\: R$ \n be a lift of $x\\in X$ that is of the form ${\\cal V}\/{\\cal G}\\to\\mathrm{Spec}\\: R$ for some flat lift\n ${\\cal G}$ of $G$ to $R$ and ${\\cal V}\\cong{\\cal U}\\times_RS$.\n Then, there exists a finite extension $R\\subseteq S$ of complete DVRs, such that\n \\begin{enumerate}\n \\item There exists an isomorphism \n $$\n {\\cal G}_{\\mathrm{can}} \\times_{W(k)}S \\,\\cong\\, {\\cal G}\\times_RS\n $$\n of group schemes over $S$.\n \\item The very small representation $\\rho:G\\to{\\mathbf{GL}}_{d,k}$ corresponding to the $G$-action on $U$\n lifts uniquely to ${\\cal G}_{\\mathrm{can}}$ and ${\\cal G}$, respectively, and they become conjugate\n over $S$. \n \\item There is an isomorphism \n $$\n \\mathcal{X}_{\\mathrm{can}}\\times_{W(k)}S \\,\\cong\\, \\mathcal{X}\\times_RS\n $$\n of deformations of $x\\in X$ over $S$.\n \\end{enumerate}\n\\end{Proposition}\n\n\\begin{proof}\nSince ${\\cal G}$ and ${\\cal G}_{\\mathrm{can},R}$ are lifts of $G$\nto $R$, they become isomorphic after passing to a finite extension $R\\subseteq S$,\nsee also the discussion in Section \\ref{subsec: abstract groups}.\nThere, we also saw that the linear representation $\\rho:G\\to{\\mathbf{GL}}_{d,k}$ \nlifts uniquely to ${\\cal G}$ and ${\\cal G}_{\\mathrm{can}}$, respectively,\nand that they become conjugate over $S$.\nFrom this, we deduce an isomorphism\n$\\mathcal{X}_{\\mathrm{can}}\\times_{W(k)}S\\cong\\mathcal{X}\\times_RS$.\n\\end{proof}\n\n\\begin{Remark}\nIf $d\\geq3$, then \\cite[Corollary 10.10]{LMM} shows that\na $d$-dimensional linearly reductive quotient singularity $x\\in X$ admits\nprecisely one lift over $W(k)$, namely the canonical lift.\nIf $d=2$, then linearly reductive quotient singularities usually have positive dimensional deformation\nspaces and admit many non-isomorphic\nlifts to $W(k)$, see \\cite[Section 12]{LMM}.\n\\end{Remark}\n\n\n\\subsection{Simultaneous resolution of singularities}\n\\label{subsec: simultaneous resolution}\nIf $\\mathcal{X}\\to S$ is a deformation of a rational double point singularity $x\\in X$,\nthen it admits a simultaneous resolution of singularities, but usually only after some\nfinite base-change $S'\\to S$, see \\cite{ArtinBrieskorn}.\nIn the case where $\\mathcal{X}\\to S$ is a family of rational surface singularities, then\nsuch a finite $S'\\to S$ exists if $S$ maps to the so-called \\emph{Artin component}\ninside the versal deformation space of $x\\in X$.\nMoreover, due to the existence of flops, \nthese simultaneous resolutions (if they exist) are not unique in general.\n\nIn the special case where $x\\in X$ is a two-dimensional linearly reductive\nquotient singularity over some algebraically closed field $k$ of\ncharacteristic $p>0$ and where $\\mathcal{X}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$ is the canonical\nlift of $x\\in X$, we will now show that there exists a simultaneous and minimal resolution of singularities over $W(k)$\nand that it is unique.\nThis simultaneous resolution can be most elegantly constructed using \nthe Ishii-Ito-Nakamura resolution from Section \\ref{subsec: Ishii Ito Nakamura}\nin families.\n\nLet $k$ be an algebraically closed field of characteristic $p>0$.\nLet $G$ be a finite and linearly reductive group scheme over $k$ \nand let $\\rho:G\\to{\\mathbf{GL}}_{2,k}$ be a very small representation.\nLet ${\\cal G}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$\nbe the canonical lift of $G$ and let \n$\\widetilde{\\rho}:{\\cal G}_{\\mathrm{can}}\\to{\\mathbf{GL}}_{2,W(k)}$ be the lift of $\\rho$ to $W(k)$.\nLet ${\\cal U}:={\\mathds{A}}^2_{W(k)}$ and let \n$$\n \\mathcal{X}_{\\mathrm{can}} \\,:=\\, {\\cal U}\/{\\cal G}_{\\mathrm{can}} \\,\\to\\, \\mathrm{Spec}\\: W(k)\n$$\nbe the canonical lift of $x\\in X$.\nBy \\cite{Blume}, there exists a ${\\cal G}_{\\mathrm{can}}$-Hilbert scheme\n$$\n{\\cal G}_{\\mathrm{can}}\\mbox{-Hilb}({\\cal U}) \\,\\to\\,\\mathrm{Spec}\\: W(k)\n$$\nthat parametrises ${\\cal G}_{\\mathrm{can}}$-invariant subschemes $Z\\subset{\\cal U}$ \nthat are finite and flat over $W(k)$ (so-called ${\\cal G}_{\\mathrm{can}}-$\\emph{clusters})\nand such that the ${\\cal G}_{\\mathrm{can}}$-representation on \n$H^0(Z,{\\cal O}_Z)$ is the regular representation.\nTaking such a cluster $Z$ to its ${\\cal G}_{\\mathrm{can}}$-orbit\n(see, for example, \\cite[Remark 3.3]{Blume}) \ninduces a morphism\n\\begin{equation}\n\\label{eq: IIN in families}\n\\widetilde{\\pi}\\,:\\,\n\\mathcal{Y}\\,:=\\,{\\cal G}_{\\mathrm{can}}\\mbox{-Hilb}({\\cal U}) \\,\\to\\, {\\cal U}\/{\\cal G}_{\\mathrm{can}}\\,=\\,\\mathcal{X}_{\\mathrm{can}}\n\\end{equation}\nover $W(k)$.\n\n\\begin{Theorem}\n\\label{thm: IN}\n Keeping the assumptions and notations\n $$\n \\widetilde{\\pi} \\,:\\, \\mathcal{Y} \\,\\to\\, \\mathcal{X}\\,\\to\\,\\mathrm{Spec}\\: W(k)\n $$\n is a simultaneous minimal resolution of singularities of the canonical lift\n $\\mathcal{X}\\to\\mathrm{Spec}\\: W(k)$ of the linearly reductive quotient singularity\n $x\\in X=U\/G$.\n \\begin{enumerate}\n \\item The simultaneous resolution $\\widetilde{\\pi}$ is unique up to isomorphism.\n \\item The exceptional locus of $\\widetilde{\\pi}$ is a union of ${\\mathds{P}}^1_{W(k)}$'s\n meeting transversally\n and we denote by $\\Gamma$ its dual intersection graph.\n \\item The special fibre and the generic\n fibre are linearly reductive quotient singularities of type $\\Gamma$\n and $\\widetilde{\\pi}$ identifies the components of the exceptional loci\n of $\\widetilde{\\pi}_k$ and $\\widetilde{\\pi}_K$.\n \\end{enumerate}\n\\end{Theorem}\n\n\\begin{proof}\nWe recall that we defined $G_{\\mathrm{can}}:={\\cal G}_{\\mathrm{can},K}$, which\nwill simplify the notation in the following.\nThe generic and special fibre of \\eqref{eq: IIN in families}\nover $K$ and $k$ are isomorphic to\n$$\n\\begin{array}{cccccc}\n&G_{\\mathrm{can}}\\mbox{-Hilb}({\\cal U}_K) &\\to& {\\cal U}_K\/G_{\\mathrm{can}} &\\to&\\mathrm{Spec}\\: K\\\\\n\\mbox{and \\quad}&\nG\\mbox{-Hilb}(U) &\\to& U\/G &\\to&\\mathrm{Spec}\\: k,\n\\end{array}\n$$\nrespectively, by \\cite[Remark 3.1]{Blume}.\nNow, \n$G_{\\mathrm{can},\\overline{K}}\\mbox{-Hilb}({\\cal U}_{\\overline{K}}) \\to {\\cal U}_{\\overline{K}}\/G_{\\mathrm{can},\\overline{K}}$\nand\n$G\\mbox{-Hilb}(U) \\to U\/G$,\nare minimal resolutions of singularities by Theorem \\ref{thm: IshiiItoNakamura}.\nThus, \n$G_{\\mathrm{can}}\\mbox{-Hilb}({\\cal U}_K) \\to {\\cal U}_K\/G_{\\mathrm{can}}$\nis a resolution of singularities, which is minimal since it is minimal over $\\overline{K}$.\nThus, $\\mathcal{Y}\\to\\mathcal{X}\\to\\mathrm{Spec}\\: W(k)$ is a simultaneous minimal resolution \nof singularities.\n\nThe exceptional fibres $\\widetilde{\\pi}_K$ and $\\widetilde{\\pi}_k$ of $\\widetilde{\\pi}$ over $K$ and $k$ \nare unions of ${\\mathds{P}}^1$'s that intersect transversally.\nThe types, that is dual resolution graphs, associated to $\\widetilde{\\pi}_K$ and $\\widetilde{\\pi}_k$\nare the same (see also Remark \\ref{rem: type quotient singularity}) \nand we denote this graph by $\\Gamma$.\nIn particular, the exceptional fibres of $\\widetilde{\\pi}_K$ and $\\widetilde{\\pi}_k$ have\nthe same numbers of irreducible components.\nIn particular, the specialisation maps of N\\'eron-Severi lattices\n$$\n\\mathrm{NS}(\\mathcal{Y}_K) \\,\\leftarrow\\, \\mathrm{NS}(\\mathcal{Y}) \\,\\to\\, \\mathrm{NS}(\\mathcal{Y}_k)\n$$\nare isometries of lattices.\nThese identify the components of the exceptional loci of $\\widetilde{\\pi}_K$\nand $\\widetilde{\\pi}_k$.\nMoreover, given such a component of the exceptional locus of $\\widetilde{\\pi}_k$,\nit is isomorphic to ${\\mathds{P}}^1_k$ and it extends uniquely to a ${\\mathds{P}}^1_{W(k)}$ \nin the exceptional locus of $\\widetilde{\\pi}$.\nThis establishes claims (2) and (3).\n\nIt remains to prove claim (1):\nLet $\\mathcal{Y}'\\to\\mathcal{X}\\to \\mathrm{Spec}\\: W(k)$ be a simultaneous resolution\nof singularities that coincides with the minimal resolution on special and generic fibres,\nrespectively.\nLet $\\alpha:\\mathcal{Y}'_K\\,\\to\\,\\mathcal{Y}_K$ be an isomorphism over $\\mathcal{X}_K$\nand choose a relatively (to $\\mathcal{X}$)\nample invertible sheaf $\\mathcal{L}$ on $\\mathcal{Y}_K$.\nThen, $\\alpha^*\\mathcal{L}$ is relatively ample on $\\mathcal{Y}_K$.\nSince the types of the singularities of $\\mathcal{X}_K$ and $\\mathcal{X}_k$\nare the same, the fibres of\n$\\mathcal{Y}'_K\\to\\mathcal{X}_K$ and $\\mathcal{Y}'_k\\to\\mathcal{Y}_k$\ncontain the same number of exceptional divisors.\nThus, the specialisation\nmap of N\\'eron-Severi lattices $\\mathrm{NS}(\\mathcal{Y}'_K)\\to\\mathrm{NS}(\\mathcal{Y}'_k)$\nis an injective map between two lattices of the same rank and the same discriminant,\nwhence an isometry.\nSimilarly, the specialisation\nmap of N\\'eron-Severi lattices $\\mathrm{NS}(\\mathcal{Y}_K)\\to\\mathrm{NS}(\\mathcal{Y}_k)$\nis an isometry.\nIn particular, we can identify the components of the exceptional loci \nof $\\mathcal{Y}_K$ and $\\mathcal{Y}_k$.\n(If the types of $\\mathcal{X}_k$ and $\\mathcal{X}_K$ differ, then this specialisation map is \nusually only injective, there are usually more $(-2)$-curves in the special fibre and this is also the\nplace where the difficulties with flops begin.)\nIn particular, $\\mathcal{L}$ and $\\alpha^*\\mathcal{L}$ extend to relative ample\ninvertible sheaves on $\\mathcal{Y}$ and $\\mathcal{Y}'$, respectively.\nBy \\cite[Theorem 5.14]{Kovacs}, the isomorphism $\\alpha$\nextends to an isomorphism $\\mathcal{Y}'\\to\\mathcal{Y}$ over $\\mathcal{X}$ and \nthe claimed uniqueness follows.\n\\end{proof}\n\n\n\\subsection{Rational double point singularities and Klein singularities}\nWe specialise Definition \\ref{def: lrq singularity} to the following case.\n\n\\begin{Definition}\n\\label{def: Klein singularity}\n A \\emph{Klein singularity} is a linearly reductive quotient singularity\n as in Definition \\ref{def: lrq singularity} with $\\dim V=2$\n and $\\det\\rho=1$, that is, $\\rho$ is a homomorphism of $G$ \n to ${\\mathbf{SL}}_{2,k}$.\n\\end{Definition}\n\nIn particular, a Klein singularity is a two-dimensional\nand linearly reductive quotient singularity, it is a rational surface singularity,\nand since $\\det(\\rho)=1$, it is Gorenstein.\nQuite generally, rational and Gorenstein surface singularities\nare precisely the \\emph{rational double point singularities} \n\\cite{ArtinRational}.\nWe refer to \\cite{Durfee} or \\cite{Seminaire} for more background\non surface singularities and rational double point singularities.\n\nIf $x\\in X$ is a rational double point, then its type $\\Gamma$ is a \nsimply-laced finite Dynkin graph.\nIn characteristic zero, every rational double point singularity is taut.\nIn positive characteristic, a Klein singularity is F-regular and thus, taut.\nIn the case of rational double points,\nthis also follows from Artin's explicit classification \\cite{ArtinRDP}.\nOn the other hand, rational double point singularities that are not F-regular \nneed not be taut.\nWe have the following relation between rational double point singularities\nand Klein singularities in positive characteristic, see \n\\cite[Theorem 11.2]{LMM}.\n\n\\begin{Theorem}\n\\label{thm: classification Klein}\nLet $k$ be an algebraically closed field of characteristic $p\\geq0$.\nLet $x\\in X$ be a normal surface singularity over $k$.\n\\begin{enumerate}\n\\item If $p=0$ or $p\\geq7$, then $x\\in X$ is a Klein singularity if and only if $x\\in X$ is a rational double point singularity.\n\\item If $p>0$, then $x\\in X$ is a Klein singularity if and only if $x\\in X$ is a rational double point singularity and F-regular. \nThe finite Dynkin graphs of these singularities are of type\n$$\n \\begin{array}{lcl}\n A_n, D_n, E_6, E_7, E_8\n &\\mbox{\\quad if \\quad }& p\\geq7, \\\\\n A_n, D_n, E_6, E_7 &\\mbox{\\quad if \\quad }& p=5,\\\\\n A_n, D_n &\\mbox{\\quad if \\quad }& p=3, \\\\\n A_n &\\mbox{\\quad if \\quad }&p=2.\n \\end{array}\n $$\n\\end{enumerate}\n\\end{Theorem}\n\nA Klein singularity $x\\in X=U\/G$ is a linearly reductive quotient singularity and thus,\ndetermines $G$ and $\\rho:G\\to{\\mathbf{SL}}_{2,k}$ up to isomorphism and conjugacy, respectively.\nThus, the classification of Klein singularities boils down to the classification\nof finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{2,k}$ and we refer to\nRemark \\ref{rem: type quotient singularity} for this classification.\n\n\\begin{Remark}\nTheorem \\ref{thm: main} and Theorem \\ref{thm: classification Klein} rely on the\nclassification of finite and linearly reductive subgroup schemes\nof ${\\mathbf{SL}}_{2,k}$.\nIt is therefore no surprise that the classification lists coincide.\n\\end{Remark}\n\n\\section{Hecke correspondences}\n\\label{sec: Hecke}\n\nLet $x\\in X=U\/G$ be a Klein singularity over an algebraically closed field $k$ \nof characteristic $p>0$.\nIn Section \\ref{subsec: abstract groups}, we discussed the\ncanonical lift $\\mathcal{X}\\to \\mathrm{Spec}\\: W(k)$ of $x\\in X$ and \nin Section \\ref{subsec: simultaneous resolution} we established\na simultaneous minimal resolution of singularities of the canonical lift.\nMore precisely, this simultaneous resolution was constructed\nusing $G$-Hilbert schemes extending previous work of Ishii, Ito, and Nakamura\n\\cite{IshiiCrelle, Ishii, IN}.\n\nIn this section, we refine this resolution of singularities as in\nthe work of Ito and Nakamura \\cite{IN} and Nakajima \\cite{Nakajima, NakajimaLectures}:\nwe eventually obtain a bijection between the components\nof the minimal resolution of singularities of the Klein singularity $x\\in X=U\/G$\nand the simple and non-trivial representations of $G$ using special\n \\emph{Hecke correspondences}.\n\n\\subsection{The Ito-Nakamura resolution revisited}\nWe first slightly extend the setup of Section \\ref{subsec: Ishii Ito Nakamura}.\nLet $k$ be an algebraically closed field of characteristic $p>0$ and\nlet $x\\in X=U\/G$ be a Klein singularity as in Definition \\ref{def: Klein singularity},\nthat is, we have $U={\\mathds{A}}^2_k$ and $G$ a very small, finite, and linearly reductive\nsubgroup scheme of ${\\mathbf{SL}}_{2,k}$.\n\nLet $\\{\\rho_i\\}_{i\\in I}$ be the set of isomorphism classes of simple representations of $G$.\nGiven a finite-dimensional representation $\\rho$ of $G$, there exist\nnon-negative integers $\\nu_i\\in{\\mathds{Z}}_+$, such that $\\rho$ is isomorphic\nto $\\bigoplus_i\\rho_i^{\\oplus \\nu_i}$ and we combine these into a multi-index\n$\\nu=(\\{\\nu_i\\})\\in{\\mathds{Z}}_{\\geq0}^I$.\nWe set $\\dim(\\nu):=\\sum_i\\nu_i\\dim\\rho_i$, which is the dimension\nof the representation associated to $\\nu$.\n\nFor any integer $n\\geq1$, the $G$-action on $U$ induces\na $G$-action on the Hilbert scheme $\\mathrm{Hilb}^n(U)$, which \nparametrises zero-dimensional subschemes of length $n$ of $U$. \nWe consider the fixed point scheme\n$$\n\\mathrm{Hilb}^{n,G}(U) \\,:=\\,\n\\left( \\mathrm{Hilb}^n(U) \\right)^G,\n$$\nthat is, the largest subscheme of $\\mathrm{Hilb}^n(U)$\non which $G$ acts trivially.\nIt parametrises $G$-invariant and zero-dimensional subschemes of\nlength $n$ of $U$.\nGiven $\\nu=(\\{\\nu_i\\})_i\\in{\\mathds{Z}}_{\\geq0}^I$ with $\\dim(\\nu)=n$, we define\n$$\n H^\\nu\\,=\\,\\left\\{\n Z\\,\\in\\,\\mathrm{Hilb}^{n,G}(U)\\,|\\,\n H^0(Z,{\\cal O}_Z)\\,\\cong\\,\n \\bigoplus \\rho_i^{\\oplus \\nu_i}\\mbox{ as $G$-representation}\n \\right\\},\n$$\nwhich defines a subscheme of $\\mathrm{Hilb}^{n,G}(U)$.\nAdapting \\cite[Lemma 12.4]{Kirillov}, which follows \\cite[Section 9]{IN},\nto our situation, we have the following.\n\n\\begin{Lemma}\nWe keep the assumptions and notations and let \n$n\\geq1$ be an integer.\n\\begin{enumerate}\n \\item The scheme $\\mathrm{Hilb}^{n,G}(U)$ is smooth over $k$.\n \\item We have a decomposition\n $$\n \\mathrm{Hilb}^{n,G}(U) \\,=\\, \\bigsqcup_{\\nu}\\, H^\\nu,\n $$\n where the disjoint union runs over all multi-indices $\\nu\\in{\\mathds{Z}}_{\\geq0}^I$ \n of dimension $n$.\n Each $H^\\nu$ is a smooth subscheme of $\\mathrm{Hilb}^{n,G}(U)$.\n\\end{enumerate}\n\\end{Lemma}\n\n\\begin{proof}\nSince $U$ is two-dimensional, $\\textrm{Hilb}^n(U)$ is smooth over $k$ by\n\\cite[Theorem 2.4]{Fogarty}.\nTo show Claim (1), it remains to show that the fixed point scheme for the\n$G$-action is also smooth,\nwhich follows from Lemma \\ref{lem: fixed scheme} below.\nThen, Claim (2) is obvious.\n\\end{proof}\n\n\\begin{Lemma}\n\\label{lem: fixed scheme}\nLet $k$ be an algebraically closed field,\nlet $X$ be a scheme that is smooth over $k$,\nlet $G$ be a finite and linearly reductive group scheme over $k$,\nand assume that $G$ acts on $X$.\nThen, the fixed point scheme $X^G\\subseteq X$ is smooth over $k$.\n\\end{Lemma}\n\n\\begin{proof}\nLet $x\\in X^G$.\nPassing to the completion of the local ring ${\\cal O}_{X,x}$,\nusing that $X$ is smooth over $k$, and passing to\ncoordinates such that the $G$-action is linear\n(this is always possible by \\cite[proof of Corollary 1.8]{Satriano}),\nwe may assume that \n$$\n\\widehat{{\\cal O}}_{X,x} \\,\\cong\\, k[[u_1,...,u_d]]\n$$\nand that the $G$-action is linear.\nIn this description it is easy to see that the $G$-invariant subscheme\nof $\\mathrm{Spec}\\:\\widehat{{\\cal O}}_{X,x}$ is smooth\n(see also the proof of \\cite[Lemma 9.1]{IN}), which implies that\n$X^G$ is smooth near $x$.\n\\end{proof}\n\nAn important special case is the \\emph{regular representation} of $G$,\nwhere we have $\\nu_i=\\dim\\rho_i$ for all $i$ and in this case, \nwe will write $\\delta$ for the corresponding multi-index.\nThe dimension of $\\delta$ is equal to the length of $G$.\nIn this case, we have \n$$\n \\pi\\,:\\, H^\\delta \\,=\\, G\\mbox{-Hilb}(U) \\,\\to\\, U\/G \\,=\\,X,\n$$\nwhich we already studied in Section \\ref{subsec: Ishii Ito Nakamura}.\nThere, we saw that $\\pi$ is a minimal resolution\nof singularities of the Klein singularity $x\\in X=U\/G$.\n\n\n\\subsection{Hecke correspondences}\nWe keep the assumptions and notations of the previous section.\nWe let $n$ be the length of $G$ and let $\\delta$ be the multi-index\ncorresponding to the regular representation.\nFor $i\\in I$, we set $\\alpha_i:=(0,...,0,1,0...)$ with the non-zero entry in \nthe $i$.th position.\nWe define\n$$\n B_i \\,:=\\, \\left\\{ J_1 \\,\\in\\, H^{\\delta-\\alpha_i},\\quad J_2\\,\\in\\, H^\\delta\\quad |\\quad J_2\\subseteq J_1 \\right\\}\n \\,\\subseteq\\, H^{\\delta-\\alpha_i}\\times H^\\delta,\n$$\nwhich is a \\emph{Hecke correspondence},\nsee \\cite{Nakajima}, \\cite[Section 6.1]{NakajimaLectures} or \\cite[Section 12.4]{Kirillov}.\nWe let\n$$\n E_i \\,:=\\, \\mathrm{pr}_2(B_i)\\,\\subseteq\\, H^\\delta \\,=\\, G\\mathrm{-Hilb}(U)\n$$\nbe the image under the projection $\\mathrm{pr}_2$ onto the second factor.\n\nIn characteristic zero, the following result is due to Nakajima \\cite{Nakajima}\nand independently to Ito and Nakamura \\cite{IN}, see also \n\\cite[Section 12.4]{Kirillov}.\n\n\\begin{Theorem}\n\\label{thm: bijection simple and exceptional}\n Let $k$ be an algebraically closed field of characteristic $p\\geq0$.\n Let $G$ be a finite and linearly reductive subgroup scheme of ${\\mathbf{SL}}_{2,k}$,\n let $x\\in X:=U\/G$ be the associated Klein singularity, and \n let $\\pi:H^\\delta\\to X$ be the Ito-Nakamura resolution.\n \\begin{enumerate}\n \\item The assignment $i\\mapsto E_i$ defines a bijection\n between the set $\\{\\rho_i\\}_i$ of isomorphism classes of \n non-trivial simple representations of $G$ \n and the set of irreducible components of the exceptional divisor $\\mathrm{Exc}(\\pi)$ \n of $\\pi$.\n \\item If $i\\neq j$, then \n $$\n B_i\\cdot B_j \\,=\\, A_{ij}\n $$\n where the numbering of the $\\{\\rho_i\\}_i$ is as in\n \\cite[Section 10]{IN}.\n \\end{enumerate}\nWe thus obtain a bijection between the McKay graph\n of $G$ and the dual resolution graph of $\\pi$.\n\\end{Theorem}\n\n\\begin{proof}\nThe case-by-case analysis of \\cite[Section 12]{IN} carries directly over\nfrom finite subgroups of ${\\mathbf{SL}}_2({\\mathds{C}})$ to finite and linearly reductive subgroup\nschemes of ${\\mathbf{SL}}_{2,k}$.\n\\end{proof} \n\nHere is a second proof of this theorem that uses our lifting results:\n\n\\begin{proof}\nIf $p=0$, then the Lefschetz principle allows to reduce to $k={\\mathds{C}}$,\nwhere the theorem is due to Ito, Nakajima, and Nakamura \\cite{IN, Nakajima}.\n\nWe now assume $p>0$.\nLet $\\mathcal{X}_{\\mathrm{can}}={\\cal U}\/{\\cal G}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$\nbe the canonical lift of $x\\in X=U\/G$\n(see Section \\ref{subsec: canonical lift singularity})\nand let $\\widetilde{\\pi}:\\mathcal{Y}\\to\\mathcal{X}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$ \nbe the simultaneous resolution of singularities\n(see Section \\ref{subsec: simultaneous resolution}).\nMoreover, we have the abstract group $G_{\\mathrm{abs}}$\nand the canonical lift $G_{\\mathrm{can}}$\nassociated to $G$ (see Section \\ref{subsec: abstract groups})\nand we have a bijection of simple representations of \n$G_{\\mathrm{abs}}$, $G_{\\mathrm{can}}$, and $G$\nby Proposition \\ref{prop: liftingrepresentation}.\n\nThe generic fibre $\\mathcal{X}_K$ is isomorphic\nto ${\\cal U}_K\/G_{\\mathrm{can}}$ and using the Lefschetz principle,\nthe geometric generic fibre can be identified with ${\\mathds{C}}^2\/G_{\\mathrm{abs}}$.\nOver ${\\mathds{C}}$, we have Theorem \\ref{thm: bijection simple and exceptional}\nfor $G_{\\mathrm{abs}}$, ${\\mathds{C}}^2\/G_{\\mathrm{abs}}$, and its minimal resolution of singularities\nby \\cite{IN, Nakajima}.\n\nBy the Lefschetz principle, we have it \nfor $G_{\\mathrm{can}}$, \n$\\mathcal{X}_K={\\cal U}_{K}\/G_{\\mathrm{can}}$ \nand $\\widetilde{\\pi}_K$.\nUsing the comparison results \nProposition \\ref{prop: liftingrepresentation} and Theorem \\ref{thm: IN},\nwe obtain it for\n$\\mathcal{X}_k={\\cal U}_k\/{\\cal G}_{\\mathrm{can},k}$ \nand $\\widetilde{\\pi}_k$, that is,\nfor $X=U\/G$, $G$ and $\\pi$.\n\\end{proof}\n\n\\subsection{Local McKay correspondence}\n\\label{subsec: local McKay}\nWe now extend work of Ishii and Nakamura \\cite{IshiiCrelle, Ishii} to our linearly \nreductive setting.\nLet $G$ be a very small, finite, and linearly reductive subgroup scheme of ${\\mathbf{GL}}_{2,k}$\nand let $x\\in X=U\/G$ be the associated two-dimensional linearly reductive quotient\nsingularity.\nUntil the end of this section, we do not require it to be a Klein singularity.\nNote that in dimension two, linearly reductive quotient singularities\nare the same as F-regular singularities, see Remark \\ref{rem: f-regular}.\n\nLet $\\{\\rho_i\\}_i$ be the set of simple representations of $G$.\nLet $\\rho_0$ be the trivial representation and we choose our numbering of the $\\rho_i$'s \nto be the one of \\cite[Theorem 3.6]{Ishii}.\nLet $\\pi:Y\\to X$ be its minimal resolution\nof singularities and let $\\{E_i\\}$ be the irreducible components of the\nexceptional divisor $\\textrm{Exc}(\\pi)$ of $\\pi$.\nThen, there is exists a connection between the $\\{\\rho_i\\}$ and the\nexceptional divisors of $\\pi$ as follows - this is yet another version of the \nMcKay correspondence.\n\n\\begin{Theorem}\n\\label{thm: exc pi}\n Keeping assumptions and notations, let ${\\mathfrak m}\\subset{\\cal O}_U$ be the maximal\n ideal corresponding to the origin, let $y\\in Y$ be a closed point, let $Z_y$ be\n the $G$-invariant cluster of $U$ corresponding to $y$, and let ${\\cal I}_{Z_y}\\subset{\\cal O}_U$\n be its ideal sheaf.\n Then, the $G$-representation on ${\\cal I}_{Z_y}\/{\\mathfrak m}{\\cal I}_{Z_y}$ is given by\n $$\n \\left\\{\n \\begin{array}{ll}\n \\rho_i\\,\\oplus\\,\\rho_0 &\\mbox{ \\quad if }y\\in E_i\\,\\backslash\\,\\bigcup_{j\\neq i}E_j\\\\\n \\rho_i\\,\\oplus\\,\\rho_j\\,\\oplus\\,\\rho_0 & \\mbox{ \\quad if }y\\in E_i\\cap E_j\\mbox{ \\quad with \\quad }i\\neq j.\n \\end{array}\n \\right. \n $$\n\\end{Theorem}\n\n\\begin{proof}\nFor $k={\\mathds{C}}$, this is \\cite[Theorem 7.1]{IshiiCrelle}.\nFor $k$ algebraically closed of arbitrary characteristic and $G$ is a very small and finite\nsubgroup of ${\\mathbf{GL}}_{2}(k)$, this is \\cite[Theorem 3.6]{Ishii}.\nHowever, these proofs also work if $G$ is a very small and finite and linearly reductive\nsubgroup scheme of ${\\mathbf{GL}}_{2,k}$.\n\\end{proof}\n\n\\begin{Remark}\nIn \\cite{Ishii}, the $G$-quiver structure of $G\\mathrm{-Hilb}(U)$\nwas studied in the case where $G$ is a very small and finite subgroup of\n${\\mathbf{GL}}_{2}(k)$. \nWe leave the extension of these results to the linearly reductive case to the reader.\n\\end{Remark}\n\n\n\\subsection{Reflexive sheaves on the minimal resolution}\nWe end this section by shortly digressing on work of Wunram \\cite{Wunram},\nIshii and Nakamura \\cite{Ishii}, which generalises work of\nArtin and Verdier \\cite{AV}.\nWe keep the assumptions and notations from Section \\ref{subsec: local McKay}.\nLet $F\\subset Y$ be the \\emph{fundamental divisor} of $\\pi$,\nsee \\cite{ArtinRational}.\n\n\\begin{Theorem}\n\\label{thm: reflexive}\nKeeping assumptions and notations, there exists a bijection between\n\\begin{enumerate}\n\\item the set of irreducible components $\\{E_i\\}$ of $\\mathrm{Exc}(\\pi)$ and\n\\item the set of non-trivial indecomposable full ${\\cal O}_Y$-modules $\\{{\\cal M}_i\\}$,\nspecial in the sense that $H^1(Y,{\\cal M}_i^{\\vee})=0$\n\\end{enumerate}\nThis correspondence ${\\cal M}_i\\mapsto E_i$ is defined by\n$$\n c_1({\\cal M}_i)\\cdot E_j \\,=\\, \\delta_{i,j}\\,.\n$$\nThe rank of ${\\cal M}_i$ is equal to $c_1({\\cal M}_i)\\cdot F$, the \nmultiplicity of $E_i$ in $F$.\\qed\n\\end{Theorem}\n\n\\begin{proof}\nFor $k={\\mathds{C}}$, this is \\cite{Wunram}.\nFor $k$ algebraically closed of arbitrary characteristic and $G$ a very small and finite\nsubgroup of ${\\mathbf{GL}}_{2,k}$, this is \\cite[Theorem 2.8]{Ishii}.\nHowever, these proofs also work if $G$ is a very small, finite, and linearly reductive\nsubgroup scheme of ${\\mathbf{GL}}_{2,k}$.\n\\end{proof}\n\nHere, a \\emph{full} ${\\cal O}_Y$-module is as defined in \\cite[Definition 2.4]{Ishii}.\nBy \\cite[Corollary 2.5]{Ishii}, the assignment ${\\cal M}\\mapsto\\pi_*{\\cal M}$\nsets up a bijection between the set of (indecomposable) \nfull ${\\cal O}_Y$-modules with the set of (indecomposable) \nreflexive ${\\cal O}_X$-modules.\nIn particular, the above theorem yields a bijection between the set of irreducible\ncomponents of $\\mathrm{Exc}(\\pi)$ and the set of reflexive\n${\\cal O}_X$-modules.\n\n\\begin{Remark}\n\\label{rem: f-regular Artin Verdier}\nThe group scheme $G$ plays no r\\^{o}le in this theorem and the discussion thereafter.\nProbably, these results should be viewed as results \non two-dimensional F-regular \nsingularities (see Remark \\ref{rem: f-regular}).\n\\end{Remark}\n\n\n\\section{Conjugacy classes and Ito-Reid correspondence}\n\\label{sec: ItoReid}\n\nFor a finite group, the number of isomorphism classes of\ncomplex semi-simple representations is equal to the number \nof conjugacy classes.\nThus, one can \\emph{choose} a bijection between these two sets.\nIn particular, for a finite subgroup of ${\\mathbf{SL}}_2({\\mathds{C}})$\none can \\emph{choose} a bijection between the vertices of its \nMcKay graph and the conjugacy classes of $G$.\nIn \\cite{ItoReid}, Ito and Reid gave a \\emph{canonical}\nbijection between these two sets.\nIn this section, we extend this \\emph{Ito-Reid correspondence}\nto finite and linearly reductive subgroup schemes of\n${\\mathbf{SL}}_{2,k}$.\nThe main difficulty is to define a suitable notion of \\emph{conjugacy classes}\nfor finite and linearly reductive group schemes.\n\n\n\\subsection{Conjugacy classes}\n\\label{subsec: conjugacy classes}\n\nGiven a finite group $G_{\\mathrm{abs}}$, a\nrepresentation $\\rho:G_{\\mathrm{abs}}\\to{\\mathbf{GL}}_n({\\mathds{C}})$,\nand an element $g\\in G_{\\mathrm{abs}}$, we have the \ntrace $\\mathrm{tr}(\\rho(g))\\in{\\mathds{C}}$.\nThis pairing $(\\rho,g)\\mapsto \\mathrm{tr}(\\rho(g))$\ninduces a non-degenerate pairing between\nisomorphism classes of \nsemi-simple representations of $G_{\\mathrm{abs}}$\nand conjugacy classes.\nThus, conjugacy classes can be thought of as being\n``dual'' to semi-simple representations, which\ncan be used to give an unusual definition of\nconjugacy classes.\nThis definition generalises to \nfinite and linearly reductive group schemes.\nMore precisely, we make the following definition,\nwhich is inspired by a result\nof Serre \\cite[Section 11.4]{Serre} and which we discuss\nin detail in Appendix \\ref{subsec: second approach}.\n\n\\begin{Definition}\n \\label{def: conjugacy class}\n Let $G$ be a finite and linearly reductive group scheme over an\n algebraically closed field $k$ of characteristic $p\\geq0$.\n Then, the \\emph{set of conjugacy classes} is defined to be\n $\\mathrm{Spec}\\: {\\mathds{C}}\\otimes K_k(G)$.\n\\end{Definition} \n\nWe discuss several approaches to conjugacy classes for finite group schemes \nin Appendix \\ref{app: conjugacy class} -\nthere, we hope to convince the reader that Definition \\ref{def: conjugacy class}\nis the best for the purposes of this article.\nFor example, by Proposition \\ref{prop: conjugacy lift and bijection}\nit is compatible with canonical lifts and there is a natural\nbijection with the set of conjugacy classes of the abstract group \n$G_{\\mathrm{abs}}$ associated to $G$.\nConcerning the choice of field ${\\mathds{C}}$ in Definition \\ref{def: conjugacy class},\nwe refer to Remarks \\ref{rem: conjugacy classes}.\n\n\\subsection{An explicit bijection}\nFor a finite and linearly reductive subgroup scheme $G$ of ${\\mathbf{SL}}_{2,k}$,\nwe have an associated embedding of the finite group\n$G_{\\rm abs}$ into ${\\mathbf{SL}}_2({\\mathds{C}})$.\nLet $\\{\\rho_i\\}_i$ be the set of isomorphism classes of\nsemi-simple representations of $G$ and \nlet $\\{\\rho_{\\mathrm{abs},i}\\}_i$ be the corresponding\nset for $G_{\\mathrm{abs}}$ obtained by Proposition \\ref{prop: liftingrepresentation}.\nThe associated McKay graphs\n$\\Gamma(G,\\rho,\\{\\rho_i\\}_i)$ and \n$\\Gamma(G_{\\mathrm{abs}},\\rho_{\\mathrm{abs}}, \\{\\rho_{\\mathrm{abs},i}\\}_i)$\ncoincide by Proposition \\ref{prop: McKay Quiver lift}\nand we denote both by $\\widehat{\\Gamma}$.\nNext, the group $G_{\\mathrm{abs}}$ admits a presentation of the form\n$$\n G_{\\rm abs} \\,=\\, \\left\\langle A,B,C \\,|\\, A^r=B^s=C^t=ABC \\right\\rangle\n$$\nfor suitable non-negative integers $r,s,t$.\nThe non-trivial conjugacy classes of $G_{\\rm abs}$ can be uniquely represented\nby the following elements\n$$\nABC,\\quad A^i \\mbox{ with }1\\leq i\\leq r-1,\n \\quad B^i \\mbox{ with }1\\leq i\\leq s-1,\n \\quad C^i \\mbox{ with }1\\leq i\\leq t-1.\n$$\nThis allows to give an explicit bijection between the \nconjugacy classes of $G_{\\mathrm{abs}}$ and the vertices of \n$\\widehat{\\Gamma}$.\nWe refer to \\cite[Section 2]{Kirillov} for details.\nUsing Proposition \\ref{prop: conjugacy lift and bijection}, we obtain \nan explicit bijection between the conjugacy classes of $G$\nand the vertices of $\\widehat{\\Gamma}$.\n\n\n\\subsection{The Ito-Reid correspondence}\n\\label{subsec: Ito-Reid}\nIf $G$ is a finite subgroup of ${\\mathbf{SL}}_2({\\mathds{C}})$, then Ito\nand Reid \\cite{ItoReid} (see also \\cite{Reid}) \nconstructed a \\emph{canonical} bijection.\nLet us run through \\cite[Section 2]{Reid} and show that this\ncan be carried over to our linearly reductive setting:\nlet $k$ be an algebraically closed field of characteristic $p>0$\nand let $G$ be a finite and linearly reductive subgroup scheme\nof ${\\mathbf{SL}}_{2,k}$.\n\\begin{enumerate}\n\\item Associated to $G$, we have the abstract group $G_{\\mathrm{abs}}$.\nBy Proposition \\ref{prop: liftingrepresentation}, an embedding\n$G\\to{\\mathbf{SL}}_{n,k}$ yields an embedding $G_{\\mathrm{abs}}\\to{\\mathbf{SL}}_n({\\mathds{C}})$.\nBy Proposition \\ref{prop: conjugacy lift and bijection}, we can identify\nconjugacy classes of $G$ (in the sense of Definition \\ref{def: conjugacy class}) \nand conjugacy classes of $G_{\\mathrm{abs}}$, which allows\nus to define the \\emph{age} of a conjugacy class of $G$ \nvia the corresponding notion for $G_{\\mathrm{abs}}$ as, for example,\nin \\cite[Section 2]{Reid}.\nA conjugacy class of age 1 is called \\emph{junior} and if $n=2$,\nthen all conjugacy classes are junior.\n\\end{enumerate}\n\n\\begin{Remark}\nUsing the ``toric mechanism'' mentioned in \\cite[Section 2]{Reid},\none can define the age directly and without referring to lifts,\nbut we will not pursue this here.\n\\end{Remark}\n\n\\begin{enumerate}\n\\setcounter{enumi}{1}\n\\item Let $x\\in X:=U\/G$ be the associated Klein singularity. \nWe have the canonical lift $\\mathcal{X}_{\\mathrm{can}}\\to \\mathrm{Spec}\\: W(k)$ and the \nsimultaneous resolution of singularities \n$\\widetilde{\\pi}:\\mathcal{Y}\\to\\mathcal{X}_{\\mathrm{can}}\\to\\mathrm{Spec}\\: W(k)$ \nby Theorem \\ref{thm: IN}.\nPassing to geometric generic fibres and using the Lefschetz principle,\nwe obtain the minimal resolution of singularities of ${\\mathds{C}}^2\/G_{\\mathrm{abs}}$.\nThe special fibre of $\\widetilde{\\pi}$, the geometric generic fibre\nof $\\widetilde{\\pi}$ and the minimal resolution of ${\\mathds{C}}^2\/G_{\\mathrm{abs}}$\nare \\emph{crepant} in the sense of \\cite{YPG} and we can identify the\nexceptional divisors of these three resolutions with each other,\nsee Theorem \\ref{thm: IN}.\nThis way, we obtain an Ito-Reid correspondence between\njunior conjugacy classes of $G$ and crepant divisors of the resolution\n\\cite[Theorem 2.1]{Reid}.\n\\end{enumerate}\n\n\\begin{Remark}\nIt seems reasonable that one can extend this correspondence\nto finite and linearly reductive subgroup schemes of ${\\mathbf{SL}}_{n,k}$ with $n\\geq3$,\nbut we will not pursue this here.\n\\end{Remark}\n\n\n\\section{Derived categories}\n\\label{sec: outlook}\n\nLet $G$ be a very small, finite, and linearly reductive subgroup scheme of ${\\mathbf{GL}}_{2,k}$,\nlet $x\\in X:=U\/G$ be the associated linearly reductive\nquotient singularity, and let $\\pi:Y\\to X$ be its minimal resolution of singularities.\nGonzalez-Sprinberg and Verdier \\cite{GSV} gave an interpretation of the McKay correspondence\nas an isomorphism between the K-groups $K^G(U)$ and $K(Y)$.\nKapranov and Vasserot \\cite{KV} and Bridgeland, King, and Reid \\cite{Bridgeland} generalised this\nto an equivalence of derived categories ${\\mathcal{D}}^G(U)$ and ${\\mathcal{D}}(Y)$. \nIn this section, we extend this to our setting, following\nIshii, Ito, Nakamura, and Ueda \\cite{IshiiCrelle, Ishii, IshiiUeda}.\n\nWe have a commutative diagram\n $$\n \\xymatrix{\n U\\times_k Y \\ar[r]^{\\pi_U}\\ar[d]^{\\pi_Y} &U\\ar[d]^\\varpi\\\\\n Y \\ar[r]^\\pi & X.\n }\n $$\nBy Theorem \\ref{thm: IshiiItoNakamura}, the minimal resolution $\\pi$ can be constructed by\nthe Ishii-Ito-Nakamura resolution\n$$\n G\\mathrm{-Hilb}(U) \\,\\to\\, U\/G\n$$\nWe let ${\\mathcal{Z}}$ be the universal cluster over $G\\mathrm{-Hilb}(U)$,\nwe identify $G\\mathrm{-Hilb}(U)$ with $Y$\nand then, we have a commutative diagram\n $$\n \\xymatrix{\n {\\mathcal{Z}} \\ar[r]^{q}\\ar[d]^{p} &U\\ar[d]^\\varpi\\\\\n Y \\ar[r]^\\pi & X.\n }\n $$\nLet ${\\mathcal{D}}(Y)$ be the derived category of coherent\nsheaves on $Y$.\nLet ${\\mathcal{D}}^G(U)$ be the derived category of\n$G$-equivariant coherent sheaves on $U$.\nFollowing \\cite{IshiiCrelle} and \\cite{Ishii}, we define two functors\n$$\n\\begin{array}{ccccc}\n \\Psi &:& {\\mathcal{D}}^G(U) &\\to& {\\mathcal{D}}(Y)\\\\\n \\Phi &:& {\\mathcal{D}}(Y) &\\to& {\\mathcal{D}}^G(U)\n\\end{array}\n$$\nby\n$$\n\\Psi(-) \\,:=\\, [p_*\\, \\mathbf{L}q^*\\, (-)]^G\n$$\nand\n$$\n\\Phi(-) \\,:=\\, \\mathbf{R}\\pi_{U,*} \\left(\n{\\cal O}_{{\\mathcal{Z}}}^\\vee \\,\\otimes^{\\mathbf{L}}\\,\\pi_Y^*(-\\otimes\\rho_0)\\,\\otimes^{\\mathbf{L}}\\, \\pi_U^*K_{U}\\right) [2],\n$$\nwhere ${\\cal O}_{{\\mathcal{Z}}}^\\vee:=\\mathbf{R}Hom({\\cal O}_{{\\mathcal{Z}}},{\\cal O}_{Y\\times U})$ denotes the dual of ${\\cal O}_{{\\mathcal{Z}}}$,\nwhere $-\\otimes\\rho_0:{\\mathcal{D}}(Y)\\to{\\mathcal{D}}^G(Y)$ denotes the functor that attaches the trivial $G$-action, and where\n$K_U$ denotes the canonical sheaf of $U$.\nWe refer to \\cite[Section 3.1]{Ishii} for details, conventions, and notations.\n\n\\begin{Theorem}\n\\label{thm: derived}\n Keeping assumptions and notations, \n $\\Phi$ is fully faithful and $\\Psi$ is a left adjoint of $\\Phi$.\n\\end{Theorem}\n\n\\begin{proof}\nFor $G$ is a very small subgroup of ${\\mathbf{GL}}_2({\\mathds{C}})$, this is \\cite[Section 6]{IshiiCrelle} and \n\\cite[Proposition 1.1 and Lemma 2.9]{IshiiUeda}.\nFor $k$ algebraically closed of arbitrary characteristic and $G$ is a very small and finite\nsubgroup of ${\\mathbf{GL}}_{2}(k)$, this is \\cite[Theorem 3.2]{Ishii}.\nHowever, these proofs also work if $G$ is a very small, finite, and linearly reductive\nsubgroup scheme of ${\\mathbf{GL}}_{2,k}$.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}