On Clifford Algebras and Related Finite Groups and Group Algebras

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.