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 $$\Theta : {\text {HP}}(B)\rightarrow {\text {HP}}(A)$$ 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 $$\Theta ^n$$ to be surjective. Finally, we study the kernel of $$\Theta ^1$$ explicitly.