Coste, A.; Dazord, P.; Weinstein, A. Symplectic groupoids. (Groupoïdes symplectiques.) (French) Zbl 0668.58017 Publ. Dép. Math., Nouv. Sér., Univ. Claude Bernard, Lyon 1987, Fasc. 2A, 1-62 (1987). The paper is devoted to the geometric aspects of the theory of symplectic groupoids and algebroids, which was inspired by certain problems in symplectic mechanics. Roughly speaking, a Lie groupoid \(\Gamma\), whose manifold of units is denoted by \(\Gamma_ 0\), is a smooth groupoid in the sense of Ehresmann and a Lie algebroid, which is a vector bundle over \(\Gamma_ 0\) with a bracket operation, is an “infinitesimal version” of a Lie groupoid. A Lie groupoid \(\Gamma\) endowed with a symplectic 2-form is called a symplectic groupoid, if the graph of its multiplication is a Lagrangian submanifold in \((-\Gamma)\times \Gamma \times \Gamma\). On the other hand, every Poisson manifold \((\Gamma_ 0,\Lambda_ 0)\) induces a Lie algebroid structure on \(T^*\Gamma_ 0\to \Gamma_ 0\) and every Lie algebroid of such a type is called a symplectic algebroid. The main result of the paper reads that a Lie algebroid is the algebroid of a local symplectic groupoid if and only it it is a symplectic algebroid. This assertion has several concrete applications in symplectic geometry. Reviewer: I.Kolář Cited in 7 ReviewsCited in 105 Documents MSC: 53D17 Poisson manifolds; Poisson groupoids and algebroids 58H05 Pseudogroups and differentiable groupoids Keywords:symplectic manifold; Lie groupoid; Lie algebroid; symplectic groupoids and algebroids; symplectic mechanics PDFBibTeX XML Full Text: EuDML