Non-commutative Poisson algebras with a set grading

Abstract: In this paper we study of the structure of non-commutative Poisson algebras with an arbitrary set $\ss.$ We show that any of such an algebra $\pp$ decomposes as $$\pp=\uu\oplus\sum_{[\lambda]\in(\Lambda_\ss\setminus{0})/\sim}\pp_{[\lambda]},$$ where $\uu$ is a linear subspace complement of $\span_{\bbbf}{ [\pp_{\mu}, \pp_{\eta}]+\pp_{\mu}\pp_{\eta} : \mu, \eta\in[\lam]}\cap\pp_0$ in $\pp_0$ and any $\pp_{[\lambda]}$ a well-described graded ideals of $\pp,$ satisfying $[\pp_{[\lambda]}, \pp_{[\mu]}]+\pp_{[\lambda]} \pp_{[\mu]}=0$ if $[\lambda]\neq[\mu].$ Under certain conditions, the simplicity of $\pp$ is characterized and it is shown that $\pp$ is the direct sum of the family of its graded simple ideals.