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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 Γ, whose manifold of units is denoted by Γ 0 , is a smooth groupoid in the sense of Ehresmann and a Lie algebroid, which is a vector bundle over Γ 0 with a bracket operation, is an “infinitesimal version” of a Lie groupoid. A Lie groupoid Γ endowed with a symplectic 2- form is called a symplectic groupoid, if the graph of its multiplication is a lagrangian submanifold in (-Γ)×Γ×Γ. On the other hand, every Poisson manifold (Γ 0 ,Λ 0 ) induces a Lie algebroid structure on T * Γ 0 Γ 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:
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
58H99Pseudogroups, differentiable groupoids and general structures on manifolds