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 $$\pp=\uu\oplus\sum_{[λ]\in(Λ_ß\setminus\{0\})/\sim}\pp_{[λ]},$$ where $\uu$ is a linear subspace complement of $\span_{\bbbf}\{ [\pp_μ, \pp_η]+\pp_μ\pp_η : μ, η\in[\lam]\}\cap\pp_0$ in $\pp_0$ and any $\pp_{[λ]}$ a well-described graded ideals of $\pp,$ satisfying $[\pp_{[λ]}, \pp_{[μ]}]+\pp_{[λ]} \pp_{[μ]}=0$ if $[\lambda ]\ne [\mu ].$ 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.

19 pages. arXiv admin note: text overlap with arXiv:2303.13832