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

Angela Gammella

Vol. 203 (2002), No. 2, 283–320

DOI: 10.2140/pjm.2002.203.283

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 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).

Milestones

Received: 19 May 2000

Revised: 16 February 2001

Published: 1 April 2002