Nil-generated algebras and group algebras whose units satify a Laurent polynomial identity

TL;DR

This paper investigates conditions under which algebras with units satisfying Laurent polynomial identities exhibit nonmatrix identities, focusing on nil-generated and group algebras over infinite fields in positive characteristic.

Contribution

It establishes new criteria linking Laurent polynomial identities of units to the existence of polynomial identities in nil-generated and group algebras.

Findings

Nil-generated algebras with LPI units have nonmatrix identities

Group algebras over infinite fields in characteristic p > 0 have nonmatrix identities under certain conditions

Existence of polynomial identities in group algebras with arbitrary LPI for units

Abstract

Let A be an algebra whose group of units U(A) satisfies a Laurent polynomial identity (LPI). We establish conditions on these polynomials in such a way that nil-generated algebras and group algebras with torsion groups over infinite fields in characteristic p > 0 have nonmatrix identities. We also determine, in the determine in the context of group algebras with arbitrary LPI for the group of units, the existence of polynomial identities.