An approach to the tangential Poisson cohomology based on examples in duals of Lie algebras

2002/07/27 by Angela Gammella
Mathematics · #math.DG #math.QA #msc:20G10 #msc:53D55 #msc:53D17 #msc:17B30

paper · pdf

published as Pacific Journal of Mathematics, Vol. 203, No.2, 2002 · 35 pages, Latex 2e

arxiv created 2002/07/27 · arxiv updated 2009/11/30

Abstract

We study the tangential Poisson cohomology (TP-cohomology) of regular Poisson manifolds, first defined by Lichnerowicz using contravariant tensor fields. We show that for a regular Poisson manifold M, the TP-cohomology coincides with the leafwise de Rham (or Cech) cohomology of the symplectic foliation of M. Its computation in various degrees leads to open, non trivial problems. To get a better understanding of these difficulties, we study explicitly many examples coming from nilpotent and 3-dimensional (real) Lie algebras. For the latter, we compare the TP-cohomology and the usual Poisson cohomology (P-cohomology).

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