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

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

2002-07-24

arxiv.org/abs/math/0207215v2

Pacific Journal of Mathematics, Vol. 203, No.2, 2002

Mathematics

Differential Geometry

35 pages, Latex 2e

Scientific paper

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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