field. Accordingly, the Poisson cohomology class does not depend on the original choice of the volume form , and it is called the modular class of the Poisson manifold. A Poisson manifold is called unimodular if its modular class vanishes. Notice that this happens if and only if there exists a volume form such that the modular vector field vanishes, i.e. for every ; in other words, is invariant under the flow of any Hamiltonian vector field. For instance: symplectic structures are always unimodular, since the Liouville form is invariant under all Hamiltonian vector fields; for linear Poisson structures the modular class is the infinitesimal modular character of , since the modular vector field associated to the standard Lebesgue measure on is the constant vector field on . Then is unimodular as Poisson manifold if and only if it is unimodular as Lie algebra; For regular Poisson structures the modular class is related to the Reeb class of the underlying symplectic foliation (an element of the first leafwise cohomology group, which obstructs the existence of a volume normal form invariant by vector fields tangent to the foliation). Poisson homology Poisson cohomology was introduced in 1977 by Lichnerowicz himself; a decade later, Brylinski introduced a homology theory for Poisson manifolds, using the operator . Several results have been proved relating Poisson homology and cohomology. For instance, for orientable unimodular Poisson manifolds, Poisson homology turns out to be isomorphic to Poisson cohomology: this was proved independently by Xu and Evans-Lu-Weinstein. Poisson maps A smooth map between Poisson manifolds is called a if it respects the Poisson structures, i.e. one of the following equivalent conditions holds (compare with the equivalent definitions of Poisson structures above): the Poisson brackets and satisfy for every and smooth functions the bivector fields and are -related, i.e. the Hamiltonian vector fields associated to every smooth function are -related, i.e. the differential is a Dirac morphism. An anti-Poisson map satisfies analogous conditions with a minus sign on one side. Poisson manifolds are the objects of a category , with Poisson maps as morphisms. If a Poisson map is also a diffeomorphism, then we call a Poisson-diffeomorphism. Examples Given the product Poisson manifold , the canonical projections , for , are Poisson maps. The inclusion mapping of a symplectic leaf, or of an open subspace, is a Poisson map. Given two Lie algebras and , the dual of any Lie