Lie properties of crossed products

TL;DR

This paper investigates the Lie algebra properties of crossed product algebras formed from a group and a field, aiming to characterize when these algebras exhibit specific Lie nilpotent and Engel properties.

Contribution

It provides a characterization of crossed product algebras that are upper or lower Lie nilpotent and Lie (n,m)-Engel, based on their Lie properties.

Findings

Characterization of upper Lie nilpotent crossed products

Characterization of lower Lie nilpotent crossed products

Identification of conditions for Lie (n,m)-Engel properties

Abstract

Let $F^\lambda_{\sigma} [G]$ be a crossed product of a group $G$ and the field $F$. We study the Lie properties of $F^\lambda_{\sigma} [G]$ in order to obtain a characterization of those crossed products which are upper (lower) Lie nilpotent and Lie $(n,m)$-Engel.