On Lie-Rinehart-Poisson algebras structures

Abstract

We define the Schouten-Nijenhuis bracket on the algebra of the module of Kahler differentials. We give the main features of Poisson manifolds by using the universal property of derivation. We prove the equivalence between a Lie-Rinehart algebra structure and a Poisson structure and we recover Lichnerowicz's notion of Poisson manifold. We show that a symplectic Lie-Rinehart algebra structure induce a nondegenerate Poisson structure and conversely.