Poisson (Co)Homology

Abstract

A Poisson bracket on a commutative associative algebra , or a Poisson structure on a manifold M, leads in a natural way to cohomology spaces, derived from the multi-derivations of (multivector fields on M), and to homology spaces, derived from the Kähler differentials of (differential forms on M). These spaces give information on the derivations, normal forms, deformations and several invariants of the Poisson structure. In some specific, but important, cases they are related to classically known cohomology spaces, like de Rham cohomology or Lie algebra cohomology, as will be shown in subsequent chapters. In general, Poisson cohomology is finer, but is also more difficult to compute. We construct in this chapter the various complexes which lead to these homologies and cohomologies, we describe a few natural operations in Poisson cohomology and homology and we show that the Poisson cohomology and homology spaces of a Poisson manifold are, under certain conditions, isomorphic to each other. We describe in particular the modular class, which is a cohomology class associated to the Poisson structure; its vanishing is shown to imply the existence of an isomorphism between Poisson homology and cohomology.