Central \(S^ 1\)-extensions of symplectic groupoids and the Poisson classes. (Q701236)

The results of the paper under review concern the notion of central extension of a Lie groupoid \((\Gamma\rightrightarrows\Gamma_0,\alpha,\beta)\) by an Abelian Lie group~\(A\). The main tool in this connection is provided by an appropriate cohomology \(H^*_{\text{es}}(\Gamma;A)\) introduced by \textit{A.~Weinstein} and \textit{P.~Xu} [J. Reine Angew. Math. 417, 159--189 (1991; Zbl 0722.58021)]. Thus, Theorem~2.2 of the present paper establishes the general result that, if the Lie groupoid \(\Gamma\) is generated by arbitrarily small neighborhoods of the identity, then the space \(H^2_{\text{es}}(\Gamma;A)\) parameterizes the isomorphism classes of central extensions of \(\Gamma\) by the Abelian Lie group \(A\). The special situation when \(A=S^1\) is then studied. In this case, if \({\mathcal A}\) is the Lie algebroid of \(\Gamma\), then there exists a natural homomorphism \(\Psi\colon H^2_{\text{es}}(\Gamma;A)\to H^2({\mathcal A};{\mathbb R})\), where the latter cohomology is Lie algebroid cohomology with coefficients in~\({\mathbb R}\). If moreover \(\Gamma\) is equipped with a symplectic form~\(\omega\), then the corresponding Poisson tensor naturally corresponds to a cohomology class in \(H^2({\mathcal A};{\mathbb R})\). The main result of the paper (Theorem 3.2) is that, if \(((\Gamma,\omega)\rightrightarrows\Gamma_0),\alpha,\beta)\) is a symplectic \(\alpha\)-connected, \(\alpha\beta\)-transversal or \(\alpha\)-simply connected groupoid, then there exists at most one central \(S^1\)-extension \(E\) of \(\Gamma\) whose cohomology class in \(H^2_{\text{es}}(\Gamma;A)\) is mapped under \(\Psi\) to the cohomology class in \(H^2({\mathcal A};{\mathbb R})\) corresponding to the Poisson tensor of~\(\omega\). Moreover, the existence of \(E\) can be established under certain quantizability hypotheses (Corollary~3.3), and in this case \(E\) is a contact groupoid (Corollary~3.4).