Group algebras whose units satisfy a Laurent Polynomial Identity

TL;DR

This paper investigates conditions under which the units of a group algebra satisfy polynomial identities, extending Hartley's Conjecture to Laurent polynomial identities with small support, and establishing new criteria for polynomial identity satisfaction.

Contribution

It extends Hartley's Conjecture by linking Laurent polynomial identities with polynomial identities in group algebras, especially with small support.

Findings

Units satisfying certain Laurent polynomial identities imply polynomial identities in the algebra.

If the Laurent polynomial identity has support of at most 3, then the algebra satisfies a polynomial identity.

The results generalize previous conjectures and provide new criteria for algebraic identities in group algebras.

Abstract

Let $KG$ be the group algebra of a torsion group $G$ over a field $K$. We show that if the units of $KG$ satisfy a Laurent polynomial identity which is not satisfied by the units of the relative free algebra $K[\alpha,\beta : \alpha^2=\beta^2=0]$ then $KG$ satisfies a polynomial identity. This extends Hartley Conjecture which states that if the units of $KG$ satisfies a group identity then $KG$ satisfies a polynomial identity. As an application of our results we prove that if the units of $KG$ satisfies a Laurent polynomial identity with a support of cardinality at most 3 then $KG$ satisfies a polynomial identity.