Almost Lie nilpotent varieties of associative algebras

TL;DR

This paper classifies non-prime almost Lie nilpotent varieties of associative algebras over fields of positive characteristic, extending previous work done for characteristic zero.

Contribution

It provides a complete list of non-prime almost Lie nilpotent varieties over fields of positive characteristic, building on Mal'cev's classification in characteristic zero.

Findings

Identifies all non-prime almost Lie nilpotent varieties in positive characteristic

Extends classification from characteristic zero to positive characteristic

Provides structural insights into Lie nilpotent associative algebras

Abstract

We consider associative algebras over a field. An algebra variety is said to be {\em Lie nilpotent} if it satisfies a polynomial identity of the kind $[x_1, x_2, ..., x_n] = 0$ where $[x_1,x_2] = x_1x_2 - x_2x_1$ and $[x_1, x_2, ..., x_n]$ is defined inductively by $[x_1, x_2, ..., x_n]=[[x_1, x_2, ..., x_{n-1}],x_n]$. By Zorn's Lemma every non-Lie nilpotent variety contains a minimal such variety, called {\em almost Lie nilpotent}, as a subvariety. A description of almost Lie nilpotent varieties for algebras over a field of characteristic 0 was made up by Yu.Mal'cev. We find a list of non-prime almost Lie nilpotent varieties of algebras over a field of positive characteristic.