Symmetric Elements and Identities in Group Algebras

Abstract

Let FG be the group ring of a group G over a field F of characteristic p ≥ 0. Let * be the natural involution, γ = ∑γ(g)g → γ^* = ∑γ(g)g^-1. Let us denote by

$${{\left( {KG} \right)}^{+}}=\left\{ {\gamma \in FG:\gamma *=\gamma } \right\}\text{and}{{\left( {KG} \right)}^{-}}=\left\{ {\gamma \in FG:\gamma *=-\gamma } \right\},$$

the sets of symmetric and skew symmetric elements respectively. We investigate whether certain identities on these and similar subsets control identities on the whole ring.

Work supported by NSERC grant A-5300.