1996/05/05 by Ping Xu, Xu, Ping
Mathematics · #22A22 #58B25 #58F05 (Primary) 22E65 #58H05 (Secondary) #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #dg-ga #math.DG #msc:22A22 #msc:22E65 #msc:58B25 #msc:58F05 #msc:58H05
paper · pdf · doi:10.48550/arxiv.dg-ga/9605003
LaTex, 24 pages, to appear in Math. Z
arxiv created 1996/05/05 · openalex publication_date 1996/05/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An anologue of the Calabi invariant for Poisson manifolds is considered. For any Poisson manifold P, the Poisson bracket on C∞(P) extends to a Lie bracket on the space Ω1(P) of all differential one-forms, under which the space Z1(P) of closed one-forms and the space B1(P) of exact one-forms are Lie subalgebras. These Lie algebras are related by the exact sequence: 0\lon \reals \lon C∞(P)\stackreld\lon Z1(P)\stackrelf\lon H1(P, \reals)\lon 0, where H1(P,\reals) is considered as a trivial Lie algebra, and f is the map sending each closed one-form 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) .