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arXiv · 1712.04849

Group algebras whose units satisfy a Laurent Polynomial Identity

Osnel Broche·Jairo Z. Gonçalves·Ángel del Río
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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.

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Osnel Broche, Jairo Z. Gonçalves, Ángel del Río. 2017-12-13. Group algebras whose units satisfy a Laurent Polynomial Identity. https://arxiv.org/abs/1712.04849

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