Poisson structures on $C[X_1,... ,X_n$] associated with rigid Lie algebras

Abstract

We present the classical Poisson-Lichnerowicz cohomology for the Poisson algebra of polynomials $C[X_1,... ,X_n]$ using exterior calculus. After presenting some non-homogenous Poisson brackets on this algebra, we compute Poisson cohomological spaces when the Poisson structure corresponds to a bracket of a rigid Lie algebra.