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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

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
58H05 Pseudogroups and differentiable groupoids
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