Group algebras of finite groups as Lie algebras

TL;DR

This paper studies the Lie algebra structure of group algebras of finite groups, showing that certain involution-related subalgebras are reductive and decomposing them into simple factors linked to group representations.

Contribution

It introduces a decomposition of involution-associated Lie subalgebras of group algebras into simple factors using group representations, revealing their reductive nature.

Findings

Involutive subalgebras are reductive

Decomposition into simple factors based on representations

Connection between algebra structure and group representations

Abstract

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of $G$.