Oriented group involutions in group algebras: a survey

Abstract

Let \(\circledast :{\mathbb {F}}G\rightarrow {\mathbb {F}}G\) denote the involution obtained as a linear extension of an involution of G, twisted by the homomorphism \(\sigma :G\rightarrow \{\pm 1\}\). In this survey we gather some results concerning to the Lie properties of symmetric and skew-symmetric elements and the corresponding group identities satisfied by the set of symmetric units, and when these identities determine the structure of the whole group algebra \({\mathbb {F}}G\) [resp. unit group \({\mathcal {U}}({\mathbb {F}}G)\)].