iv : d g - ga / 9 60 50 03 v 1 5 M ay 1 99 6 Flux homomorphism on symplectic groupoids ∗
نویسنده
چکیده
For any Poisson manifold P , the Poisson bracket on C∞(P ) extends to a Lie bracket on the space Ω(P ) of all differential one-forms, under which the space Z(P ) of closed one-forms and the space B(P ) of exact one-forms are Lie subalgebras. These Lie algebras are related by the exact sequence: 0 −→ R −→ C∞(P ) d −→ Z(P ) f −→ H(P,R) −→ 0, where H(P,R) is considered as a trivial Lie algebra, and f is the map sending each closed oneform to its cohomology class. The goal of the present paper is to lift this exact sequence to the group level for compact Poisson manifolds under certain integrability condition. In particular, we will give a geometric description of a Lie group integrating the underlying Poisson algebra C∞(P ). The group homomorphism obtained by lifting f is called the flux homomorphism for symplectic groupoids, which can be considered as a generalization, in the context of Poisson manifolds, of the usual flux homomorphism of the symplectomorphism groups of symplectic manifolds introduced by Calabi.
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