On Clifford Algebras and Related Finite Groups and Group Algebras

TL;DR

This paper explores the algebraic structure of real Clifford algebras, linking their properties and periodicity to the central product structure of Salingaros vee groups viewed as 2-groups.

Contribution

It unifies various perspectives on Clifford algebras, demonstrating how their properties derive from the structure of associated 2-groups and their group algebra representations.

Findings

Clifford algebras can be viewed as twisted group rings.

The periodicity of Clifford algebras relates to the structure of Salingaros vee groups.

Algebraic properties of Clifford algebras follow from 2-group central product structures.

Abstract

Albuquerque and Majid have shown how to view Clifford algebras $\cl_{p,q}$ as twisted group rings whereas Chernov has observed that Clifford algebras can be viewed as images of group algebras of certain 2-groups modulo an ideal generated by a nontrivial central idempotent. Ablamowicz and Fauser have introduced a special transposition anti-automorphism of $\cl_{p,q}$, which they called a "transposition", which reduces to reversion in algebras $\cl_{p,0}$ and to conjugation in algebras $\cl_{0,q}$. The purpose of this paper is to bring these concepts together in an attempt to investigate how the algebraic properties of real Clifford algebras, including their periodicity of eight, are a direct consequence of the central product structure of Salingaros vee groups viewed as 2-groups.