Normal group algebras

Abstract

Let $\mathbb{F}G$ denote the group algebra of the group $G$ over the field $\mathbb{F}$ with $char(\mathbb{F})\ne 2$. 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 $\alpha =\Sigma {\alpha }_{g}g\mapsto {\alpha }^{⊛}=\Sigma {\alpha }_g\sigma (g)g^{\ast }$. In this paper, 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.

12 pages