On the structure of split non-commutative Poisson algebras

Abstract

A non-commutative Poisson algebra is a Lie algebra endowed with an, not necessarily commutative, associative product in such a way that the Lie and associative products are compatible via the Leibniz identity. If we part from a split Lie algebra, with respect to a maximal abelian subalgebra H, we obtain a so-called split non-commutative Poisson algebra. This article is devoted to the study of the structure of split non-commutative Poisson algebras, by showing that any of such an algebra 𝒫 is of the form with U a linear subspace of H and any a well-described ideal of 𝒫, satisfying if j ≠ k. Under certain conditions the simplicity of 𝒫 is characterized and it is shown that 𝒫 is the direct sum of the family of its simple ideals.