Poisson Cohomology of Trivial Extension Algebras

Let B be the trivial extension of a Poisson algebra A by a left Poisson module M. Then there is a Poisson structure on B. We define a morphism of graded algebras θ: HP(B) → HP(A) from the Poisson cohomology ring of B to that of A. In the case that both A and M are finite dimensional as vector space, there is a long exact sequence computing the Poisson cohomology of B. We study the connecting homomorphism and then obtain a necessary and sufficient condition for each θ^n to be surjective. Finally, we study the kernel of θ^1 explicitly.