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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{symplectic groupoid} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory} [[!include infinity-Lie theory - contents]] \hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry} [[!include symplectic geometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{lie_integration_and_poisson_manifolds}{Lie integration and Poisson manifolds}\dotfill \pageref*{lie_integration_and_poisson_manifolds} \linebreak \noindent\hyperlink{symplectic_realization}{Symplectic realization}\dotfill \pageref*{symplectic_realization} \linebreak \noindent\hyperlink{in_geometric_quantization_of_poisson_manifolds}{In geometric quantization of Poisson manifolds}\dotfill \pageref*{in_geometric_quantization_of_poisson_manifolds} \linebreak \noindent\hyperlink{as_lie_integration_of_poisson_lie_algebroid}{As Lie integration of Poisson Lie algebroid}\dotfill \pageref*{as_lie_integration_of_poisson_lie_algebroid} \linebreak \noindent\hyperlink{a_reduced_phase_space_of_open_poisson_sigmamodel}{A reduced phase space of open Poisson sigma-model}\dotfill \pageref*{a_reduced_phase_space_of_open_poisson_sigmamodel} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{OfLiePoissonStructure}{Of Lie-Poisson structure}\dotfill \pageref*{OfLiePoissonStructure} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A [[Poisson manifold]] may be thought of as a [[Poisson Lie algebroid]], a [[Lie algebroid]] with extra structure: called an [[n-symplectic manifold]] for $n = 1$. By [[Lie integration]] this Lie algebroid should integrate to a [[Lie groupoid]] with extra structure. Symplectic groupoids are supposed to be these objects that integrate [[n-symplectic manifold]] aka [[Poisson manifold]]s in this sense. The [[category algebra|groupoid algebra]] of these symplectic groupoids are [[C-star algebra]]s that may be regarded as the [[quantization]] of the original [[Poisson manifold]]. This is described in the references below. \hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition} The original definition of (\hyperlink{Weinstein}{Weinstein}) is this: \begin{defn} \label{WeinsteinsDefintion}\hypertarget{WeinsteinsDefintion}{} A \textbf{symplectic Lie groupoid} is a [[Lie groupoid]] $\mathbf{X}_\bullet$ whose manifold of [[morphisms]] $\mathbf{X}_1$ is equipped with a [[symplectic manifold|symplectic structure]] whose symplectic form $\omega \in \Omega^2_{closed}(\mathbf{X}_1)$ is \emph{multiplicative} in that the alternating sum of its canonical [[pullback of a differential form|pullbacks]] to the space $\mathbf{X}_2$ of composable morphisms vanishes: \begin{displaymath} 0 = \delta \omega = pr_1^* \omega - compose^* \omega + pr_2^* \omega \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} The manifold of [[objects]] $\mathbf{X}_0$ of a symplectic Lie groupoid $\mathbf{X}_\bullet$, def. \ref{WeinsteinsDefintion}, carries the structure of a [[Poisson manifold]] which is unique, up to [[isomorphism]], with the property that the target map $t \colon \mathbf{X}_1 \to \mathbf{X}_0$ is a [[homomorphism]] of [[Poisson manifolds]] (canonically regarding the [[symplectic manifold]] $(\mathbf{X}_1, \omega)$ as a Poisson manifold). The [[Poisson manifolds]] that arise this way as $\mathbf{X}_0$ of a symplectic Lie groupoid are called \textbf{integrable Poisson manifolds}. \end{remark} \begin{remark} \label{}\hypertarget{}{} Reformulated more abstractly, def. \ref{WeinsteinsDefintion} says that the differential form $\omega$, when extended to a triple \begin{displaymath} (0, \omega, 0) \in \oplus_{k = 0,1,2} \Omega^{3-k}(\mathbf{X}_{k}) \end{displaymath} is a [[cocycle]] of degree 3 in the [[de Rham complex]] of $\mathbf{X}$, identified with the [[simplicial de Rham complex]] of the [[nerve]] $X_\bullet$ of $X$. \end{remark} This observation leads to the following generalization \begin{defn} \label{}\hypertarget{}{} A \textbf{pre-quasi symplectic groupoid} is a [[Lie groupoid]] $\mathbf{X}$ equipped with a [[differential 2-form]] $\omega_2 \in \Omega^2(\mathbf{X}_1)$ and a differential 3-form $\omega_3 \in \Omega^3(\mathbf{X}_0)$ such that \begin{displaymath} (0, \omega_2, \omega_3) \in \oplus_{k = 0,1,2} \Omega^{3-k}(\mathbf{X}_{k}) \end{displaymath} is a [[cocycle]] in the [[simplicial de Rham complex]] of $\mathbf{X}_\bullet$, hence such that \begin{displaymath} \delta \omega_2 = 0 \end{displaymath} \begin{displaymath} d\omega_2 + \delta \omega_3 = 0 \,, \end{displaymath} where $\delta = \sum_{k} (-1)^k \partial_k^*$ is the alternating sum of the [[pullback of a differential form|pullbacks]] along the face maps of the [[nerve]] $\mathbf{X}_\bullet$. \end{defn} This appears as (\hyperlink{Xu}{Xu, def. 2.1}, \hyperlink{LGX}{LG-Xu, def. 2.1}). This structure is called a \textbf{twisted presymplectic groupoid} in (\hyperlink{BCWZ}{BCWZ, def. 2.1}). \begin{remark} \label{}\hypertarget{}{} Since therefore a (pre-)symplectic groupoid is really a Lie groupoid equipped with a cocycle in degree-3 [[de Rham cohomology]] (instead of degree 2 as for a symplectic manifold), it is really rather an object in [[2-plectic geometry]]. \end{remark} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{lie_integration_and_poisson_manifolds}{}\subsubsection*{{Lie integration and Poisson manifolds}}\label{lie_integration_and_poisson_manifolds} Every [[Lie groupoid]] [[Lie integration|integrating]] a [[Poisson Lie algebroid]] is naturally a symplectic Lie groupoid. Picking always the unique source-simply connected integrating [[Lie groupoid]] produces a [[functor]] \begin{displaymath} \Sigma : PoissonManifolds \to SymplecticGroupoids \,. \end{displaymath} When the [[Poisson manifold]] we start with happens to be a [[symplectic manifold]], then its symplectic Lie groupoid is always the [[fundamental groupoid]] of $X$: \begin{displaymath} (X,\pi)\;\; symplectic \;\;\Rightarrow\;\; \Sigma(X,\pi) \simeq_{iso} \Pi(X) \,. \end{displaymath} When $X$ is [[simply connected topological space|simply connected]] such that $\Pi(X)$ is the [[codiscrete category|codiscrete groupoid]] $Pair(X)$ we have that the symplectic form on $Mor(\Pi(X)) = X \times X$ is $\omega \otimes (-\omega)$, for $\omega$ the [[symplectic manifold|symplectic form]] on $X$. Conversely, for every symplectic groupoid $\mathbf{X}$ there is a unique [[Poisson manifold]] structure on its manifold $\mathbf{X}_0$ of objects such that the [[codomain]] map $t \colon \mathbf{X}_1 \to \mathbf{X}_0$ is a [[homomorphism]] of [[Poisson manifolds]]. (For instance \hyperlink{Racaniere}{Racaniere, theorem 6.3}) One says also that $\mathbf{X}$ \textbf{integrates the Poisson manifold} $\mathbf{X}_0$. \hypertarget{symplectic_realization}{}\subsubsection*{{Symplectic realization}}\label{symplectic_realization} The source map of a symplectic groupoid over a Poisson manifold constitutes a \emph{[[symplectic realization]]} of this Poisson manifold, hence its canonical desingularization via [[Lie integration]]. See at \emph{[[symplectic realization]]} for more. \hypertarget{in_geometric_quantization_of_poisson_manifolds}{}\subsubsection*{{In geometric quantization of Poisson manifolds}}\label{in_geometric_quantization_of_poisson_manifolds} In the \emph{groupoid approach to quantization} symplectic groupoids are used to discuss [[geometric quantization]] not just of [[symplectic manifold]]s but more generally of [[Poisson manifold]]s. See [[geometric quantization of symplectic groupoids]]. \hypertarget{as_lie_integration_of_poisson_lie_algebroid}{}\subsubsection*{{As Lie integration of Poisson Lie algebroid}}\label{as_lie_integration_of_poisson_lie_algebroid} (\ldots{}) \hypertarget{a_reduced_phase_space_of_open_poisson_sigmamodel}{}\subsubsection*{{A reduced phase space of open Poisson sigma-model}}\label{a_reduced_phase_space_of_open_poisson_sigmamodel} The symplectic groupoid of a Poisson manifold is also the [[reduced phase space]] of the open sector of the corresponding [[Poisson sigma-model]]. (\hyperlink{CattaneoFelder01}{Cattaneo-Felder 01}) \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{OfLiePoissonStructure}{}\subsubsection*{{Of Lie-Poisson structure}}\label{OfLiePoissonStructure} \begin{example} \label{}\hypertarget{}{} Let $G$ be a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$ and consider the [[dual vector space]] $\mathfrak{g}^*$ equipped with its [[Lie-Poisson structure]]. Then the [[action groupoid]] $\mathfrak{g}^* //G$ of the [[coadjoint action]] carries a multiplicative symplectic form $\omega$ induced by the identification of the manifold of [[morphisms]] with the [[cotangent bundle]] of the group, $G \times \mathfrak{g}^* \simeq T^* G$, induced by right translation from the [[Poincare form]] on the [[cotangent bundle]]. This makes $(\mathfrak{g}^* //G, \omega)$ a symplectic groupoid which [[Lie integration|Lie integrates]] the [[Lie-Poisson structure]] on $\mathfrak{g}^*$. \end{example} This appears for instance as (\hyperlink{Weinstein91}{Weinstein 91, example 3.2}, \hyperlink{BursztynCrainic}{Bursztyn-Crainic 05, example 4.3}). \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[symplectic manifold]] \item [[symplectic Lie n-algebroid]] \item [[symplectic infinity-groupoid]] \end{itemize} [[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]] \hypertarget{references}{}\subsection*{{References}}\label{references} The notion of symplectic groupoids was apparently proposed independently by Karas\"e{}v, [[Alan Weinstein|Weinstein]], and Zakrzewski, all motiviated from the problem of [[quantization]]. \begin{itemize}% \item [[Alan Weinstein]], \emph{Symplectic groupoids and Poisson manifolds}, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 101--104. \end{itemize} \begin{itemize}% \item [[Alan Weinstein]], \emph{Symplectic groupoids, geometric quantization, and irrational rotation algebras} in \emph{Symplectic geometry, groupoids, and integrable systems} (Berkeley, CA, 1989), 281--290, Springer, New York, (1991) MR1104934. \item [[Alan Weinstein]], \emph{Tangential deformation quantization and polarized symplectic groupoids}, in \emph{Deformation theory and symplectic geometry} (Ascona, 1996), 301--314, Kluwer (1997) \href{http://www.ams.org/mathscinet-getitem?mr=1480730}{MR1480730} \item M. V. Karas"ev, \emph{The Maslov quantization conditions in higher cohomology and analogs of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds II}, Selecta Mathematica Sovietica 8, pp. 235--257, 1989. \item S. Zakrzewski, Quantum and classical pseudogroups I, II, Commun. Math. Phys. 134 (1990) \end{itemize} See also the references at \emph{[[geometric quantization of symplectic groupoids]]} . Lecture notes include \begin{itemize}% \item S\'e{}bastien Racani\`e{}re, \emph{Lie algebroids, Lie groupoids and Poisson geometry} (\href{http://empg.maths.ed.ac.uk/Activities/GCY/Racaniere.pdf}{pdf}) \end{itemize} The notion of pre-quasi-symplectic groupoids is introduced and the intepretation of symplectic groupoids in [[higher geometry]] is made fairly explicit in \begin{itemize}% \item [[Ping Xu]], \emph{Momentum Maps and Morita Equivalence}, J. Diff. Geom (\href{http://arxiv.org/abs/math/0307319}{arXiv:math/0307319}) \end{itemize} \begin{itemize}% \item [[Camille Laurent-Gengoux]], [[Ping Xu]], \emph{Quantization of pre-quasi-symplectic groupoids and their Hamiltonian spaces} in \emph{The Breadth of Symplectic and Poisson Geometry} Progress in Mathematics, 2005, Volume 232, 423-454 (\href{http://arxiv.org/abs/math/0311154}{arXiv:math/0311154}) \end{itemize} These ``pre-quasi-symplectic groupoids'' had been called ``twisted presymplectic groupoids'' in \begin{itemize}% \item [[Henrique Bursztyn]], [[Marius Crainic]], [[Alan Weinstein]], [[Chenchang Zhu]], \emph{Integration of twisted Dirac brackets} (\href{http://arxiv.org/abs/math/0303180}{arXiv:math/0303180}) \end{itemize} The identification with reduced phase spaces of the open Poisson sigma-model is in \begin{itemize}% \item [[Alberto Cattaneo]], [[Giovanni Felder]], \emph{Poisson sigma models and symplectic groupoids}, in \emph{Quantization of Singular Symplectic Quotients}, (ed. [[Klaas Landsman]], M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198 (Birkh\"a{}user, 2001), 61--93. (\href{http://arxiv.org/abs/math/0003023}{arXiv:math/0003023}) \end{itemize} Further developments include \begin{itemize}% \item [[Alan Weinstein]], \emph{Noncommutative geometry and geometric quantization} in P. Donato et al. (eds.) \emph{Symplectic geometry and Mathematical physics}, Birkh\"a{}user 1991 \item [[Ping Xu]], \emph{Morita equivalence and symplectic realizations of Poisson manifolds}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 4, 25 no. 3 (1992) (\href{http://www.numdam.org/item?id=ASENS_1992_4_25_3_307_0}{NUMDAM}) \item [[Henrique Bursztyn]], [[Marius Crainic]], \emph{Dirac structures, momentum maps and quasi-Poisson manifolds} (\href{http://www.preprint.impa.br/FullText/Bursztyn__Fri_Dec_23_11_24_19_BRDT_2005.html/alanfestimpa.pdf}{pdf}) \item F. Bonechi, N. Ciccoli, N. Staffolani, M. Tarlini, \emph{The quantization of the symplectic groupoid of the standard Podles sphere} (\href{http://arxiv.org/abs/1004.3163}{arXiv:1004.3163}) \end{itemize} The [[formal groupoid]] version of symplectic groupoids is discussed in \begin{itemize}% \item [[Alberto S. Cattaneo|Alberto Cattaneo]], Benoit Dherin, [[Giovanni Felder]], \emph{Formal symplectic groupoid}, Comm. Math. Phys. \textbf{253} (2005), no. 3, 645--674 \href{http://arxiv.org/abs/math/0312380}{math.SG/0312380} \href{HTTP://dx.doi.org/10.1007/s00220-004-1199-z}{doi}; \emph{Formal Lagrangian operad}, \href{http://arxiv.org/abs/math/0505051}{math.SG/0505051} \end{itemize} [[!redirects symplectic Lie groupoid]] [[!redirects symplectic groupoids]] [[!redirects symplectic Lie groupoids]] \end{document}