Normal group algebras

2019/02/25 by Holguín-Villa, Alexander, Castillo, John H.
#16R50 #16S34 #16W10 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1902.09620

Abstract

Let \mathbbFG denote the group algebra of the group G over the field \mathbbF with char(\mathbbF)≠ 2. Given both a homomorphism σ:G→ \±1\ and a group involution ∗: G→ G, an oriented involution of \mathbbFG is defined by α=Σα_gg ↦ α^\circledast=Σα_gσ(g)g^∗. In this paper, we determine the conditions under which the group algebra \mathbbFG is normal, that is, conditions under which \mathbbFG satisfies the \circledast-identity αα^\circledast=α^\circledastα. We prove that \mathbbFG is normal if and only if the set of symmetric elements under \circledast is commutative.

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