Clifford algebras as twisted group algebras and the Arf invariant

TL;DR

This paper explores the relationships between quadratic forms over GF(2), Clifford algebras, and twisted group algebras, offering new insights and proofs by connecting these mathematical structures through the Arf invariant.

Contribution

It introduces a novel perspective on real Clifford algebras using the Arf invariant of quadratic forms over GF(2), providing new proofs of classical results.

Findings

Revealed connections between quadratic forms, Clifford algebras, and twisted group algebras.

Reinterpreted real Clifford algebras via the Arf invariant.

Provided new proofs of classical theorems in the theory of Clifford algebras.

Abstract

Some connections between quadratic forms over the field of two elements, Clifford algebras of quadratic forms over the real numbers, real graded division algebras, and twisted group algebras will be highlighted. This allows to revisit real Clifford algebras in terms of the Arf invariant of the associated quadratic forms over the field of two elements, and give new proofs of some classical results.