Complex structure of a real Clifford algebra

TL;DR

This paper investigates the complex structure of real Clifford algebras viewed as modules over themselves, revealing conditions under which a basis-independent complex structure exists based on the volume element.

Contribution

It provides a new perspective on real Clifford algebras by analyzing their complex structures as modules, not just as matrix algebras.

Findings

Complex structure exists only when the volume element squares to -1.

The complex structure is uniquely given by right multiplication with the volume element, up to sign.

The paper clarifies the conditions for basis-independent complex structures in Clifford algebras.

Abstract

The classification of real Clifford algebras in terms of matrix algebras is well--known. Here we consider the real Clifford algebra ${\mathcal Cl}(r,s)$ not as a matrix algebra, but as a Clifford module over itself. We show that ${\mathcal Cl}(r,s)$ possesses a basis independent complex structure only when the square of the volume element $\omega$ is -1, in which case it is uniquely given up to sign by right multiplication with $\omega$.