Identities in group rings, enveloping algebras and Poisson algebras

TL;DR

This survey reviews key developments in the study of identities in group rings, enveloping algebras, and Poisson algebras, highlighting classical results, recent advances, and methods used to analyze algebraic identities.

Contribution

It provides a comprehensive overview of the state of research on identities in various algebraic structures, including recent results on Poisson symmetric algebras and their properties.

Findings

Classical group ring identities characterized by Passman.

Lie algebra identities related to universal enveloping algebras.

Poisson algebra identities and conditions for abelian Lie algebras.

Abstract

This is a short survey of works on identical relations in group rings, enveloping algebras, Poisson symmetric algebras and other related algebraic structures. First, the classical work of Passman specified group rings that satisfy nontrivial identical relations. This result was an origin and motivation of close research projects. Second, Latyshev and Bahturin determined Lie algebras such that their universal enveloping algebra satisfies a non-trivial identical relation. Next, Passman and Petrogradsky solved a similar problem in case of restricted enveloping algebras. Third, Farkas started to study identical relations in Poisson algebras. On the other hand, Shestakov proved that the symmetric algebra $S(L)$ of an arbitrary Lie algebra $L$ satisfies the identity $\{x,\{y,z\}\}\equiv 0$ if, and only if, $L$ is abelian. We survey further results on existence of identical relations in (truncated) Poisson symmetric algebras of Lie algebras. In particular, we report on recent results on (strong) Lie nilpotency and (strong) solvability of (truncated) Poisson symmetric algebras and related nilpotency classes. Also, we discuss constructions and methods to achieve these results.