Generating sets for Clifford Algebras

TL;DR

This paper develops an explicit algorithm to generate sets of commuting involutions in Clifford algebras, enabling the construction of minimal representations and spinor spaces with concrete examples.

Contribution

It introduces a method to explicitly construct generating sets of involutions for all Clifford algebras, facilitating minimal representation construction.

Findings

Algorithm for generating involutions in Clifford algebras

Explicit construction of minimal representations and spinor spaces

Examples demonstrating the calculation of minimal representations

Abstract

The aim of this paper is to find generating sets of commuting involutions and use them to explicitly construct minimal representations of Clifford algebras $Cl(n)_{p,q}$. By results of [HL] and [LW], we know the dimension of such minimal representations, which is linked to the maximal number of commuting involutions in the algebra, dependent only on $p$ and $q$. We provide an algorithm to construct these generating sets of involutions explicitly for all Clifford algebras $Cl(n)_{p,q}$ and provide some examples. Involutions yield mutually non-annihilating idempotents whose product gives a projection map $P^+$ with image $Cl(n)P^+$ being a minimal left ideal, which is a spinor space. Using the projections, we find minimal representations of Clifford algebras by combining matrices of left multiplication endomorphisms in the spinor spaces. Finally, we provide examples showing calculations of minimal representations.