Normal group algebras

Abstract

Let $\mathbb{F}G$ denote the group algebra of the group G over the field $\mathbb{F}$ with $\text{char}\left(\mathbb{F}\right)\ne 2.$ Given both a homomorphism $\sigma :G\to \{\pm 1\}$ and a group involution $\ast :G\to G,$ an oriented involution of $\mathbb{F}G$ is defined by $\alpha =\Sigma {\alpha }_{g}g\mapsto {\alpha }^{⊛}=\Sigma {\alpha }_g\sigma \left(g\right)g^{\ast }.$ In this article, we determine the conditions under which the group algebra $\mathbb{F}G$ is normal, that is, conditions under which $\mathbb{F}G$ satisfies the $⊛$-identity $\alpha {\alpha }^{⊛}={\alpha }^{⊛}\alpha .$ We prove that $\mathbb{F}G$ is normal if and only if the set of symmetric elements under $⊛$ is commutative.

Keywords:

• Group algebras
• involutions
• normal rings
• polynomial identity

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W10
• 16S34
• 16R50

Acknowledgments

The authors are grateful to their advisor Professor César Polcino Milies, who introduced them to the fascinating topic of group rings and of course for his timely suggestions. In particular, they are indebted to him for his idea in the construction of the special element g₀ in Lemma 4.6. Some results in this article are part of the first author’s Ph.D. thesis [12], at Instituto de Matemática e Estatística of the Universidade de São Paulo. This article was written while the authors visited the Universidad Industrial de Santander and Universidad de Nariño, and they thank the members of these institutions for their warm hospitality. Last but not least, the authors thank to the referee for the careful reading and useful suggestions that helped to improve the final version of this article.

Additional information

Funding

This research was partially supported by Vicerrectoria de Investigaciones, Postgrados y Relaciones Internacionales of the Universidad de Nariño.