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Groupoides symplectiques. (Symplectic groupoids). (French) Zbl 0668.58017
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 ${{\Gamma }}_{0}$, 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 $\left(-{\Gamma }\right)×{\Gamma }×{\Gamma }$. On the other hand, every Poisson manifold $\left({{\Gamma }}_{0},{{\Lambda }}_{0}\right)$ 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: O.Kolář

##### MSC:
 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 58H99 Pseudogroups, differentiable groupoids and general structures on manifolds