On some relationships between the center and the derived subalgebra in Poisson (2-3)-algebras

TL;DR

This paper extends the classical Schur theorem to Poisson (2-3)-algebras, establishing conditions under which the algebra's structure implies finiteness properties of its derived components.

Contribution

It proves an analogue of Schur's theorem specifically for Poisson (2-3)-algebras, a new result in the study of these algebraic structures.

Findings

Proved an analogue of Schur's theorem for Poisson (2-3)-algebras.

Established conditions linking the center and derived subalgebra in these algebras.

Extended classical results from group theory to a new class of algebraic structures.

Abstract

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $G/\zeta(G)$ of a group $G$ is finite, then its derived subgroup $[G,G]$ is also finite. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, $n$-groups, associative algebras, Lie algebras, Lie $n$-algebras, Lie rings, Leibniz algebras. In 2021, L.A. Kurdachenko, O.O. Pypka and I.Ya. Subbotin proved an analogue of Schur theorem for Poisson algebras: if the center of the Poisson algebra $P$ has finite codimension, then $P$ includes an ideal $K$ of finite dimension such that $P/K$ is abelian. In this paper, we continue similar studies for another algebraic structure. An analogue of Schur theorem for Poisson (2-3)-algebras is proved.