Abstract

In this paper, we are interested in the tangential Poisson cohomology (TP-cohomology) of regular Poisson manifolds, a cohomology which was 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. Computing the spaces of such a cohomology leads actually to open and quite nontrivial problems. To get a better understanding of these dificulties, 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).

Authors