Laurent polynomial identities on symmetric units of group algebras

Abstract

Let F be an infinite field of characteristic $p\ne 2$, G be a group, and $*$ be an involution of G extended linearly to an involution of the group algebra FG. In the literature, group identities on units $\mathcal{U}(\mathit{\text{FG}})$ and on symmetric units ${\mathcal{U}}^{+}(\mathit{\text{FG}})=\{\alpha \in \mathcal{U}(\mathit{\text{FG}})|{\alpha }^{*}=\alpha \}$ have been considered. Here, we investigate normalized Laurent polynomial identities (as a generalization of group identities) on ${\mathcal{U}}^{+}(\mathit{\text{FG}})$ under the conditions that either p > 2 or F is algebraically closed. For instance, we show that if G is torsion and ${\mathcal{U}}^{+}(\mathit{\text{FG}})$ satisfies a normalized Laurent polynomial identity, then ${\mathcal{U}}^{+}(\mathit{\text{FG}})$ satisfies a group identity and FG satisfies a polynomial identity.

KEYWORDS:

• Group algebra
• group identity
• involution
• polynomial identity

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16R50
• 16S34
• 16U60
• 16W10

Acknowledgments

The authors would like to thank the referee for valuable comments and suggestions.

Data availability statement

Data sharing is not applicable to this article as no datasets or any other materials were generated or analyzed during the current study.

Disclosure statement

The authors have no competing interests to declare that are relevant to the content of this article. The authors declare no conflict of interest.

Additional information

Funding

The research work of the second-named author (M. Ramezan-Nassab) was in part supported by a grant from IPM (grant no. 1402160023).