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Zbl 0668.58017
Coste, A.; Dazord, P.; Weinstein, A.
Groupoides symplectiques. (Symplectic groupoids). (French)
[B R] Publ. Dép. Math., Nouv. Sér., Univ. Claude Bernard, Lyon 2/A, 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 $\Gamma\sb 0$, is a smooth groupoid in the sense of Ehresmann and a Lie algebroid, which is a vector bundle over $\Gamma\sb 0$ 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 $(\Gamma\sb 0,\Lambda\sb 0)$ induces a Lie algebroid structure on $T\sp*\Gamma\sb 0\to \Gamma\sb 0$ 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.
[O.Kolář]

MSC 2000:: *37J99 Finite-dimensional Hamiltonian etc. systems
58H99 Pseudogroups and general structures on manifolds

Keywords: symplectic manifold; Lie groupoid; Lie algebroid; symplectic groupoids and algebroids; symplectic mechanics

Cited in: Zbl 1067.58016 Zbl 0849.58075 Zbl 0704.58018 Zbl 0701.58025

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