Introduction

Given a quadratic form \(Q\) defined on a vector space \(V\) over a field \(F\), the Clifford algebra of \(Q\) is an associative \(F\)-algebra \(C\) with a vector space homomorphism \(f : V\to C\) such that \(f(v)^2 = Q(v)\) for all \(v\in V\). Furthermore, the triple \((C,V,f)\) has the universal property that if \(A\) is any associative algebra with a homomorphism \(g : V \to A\) such that \(g(v)^2 = Q(v)\) for all \(v\in V\), then there is a unique algebra homomorphism \(h : C \to A\) such that \(hf = g\). It can be shown that \(f\) is injective and therefore we may identify \(V\) with its image in \(C\). If the dimension of \(V\) is \(n\), then the dimension of \(C\) is \(2^n\). We shall refer to \(V\) as the quadratic space of \(C\).

The primary references for quadratic forms and Clifford algebras are [Chevalley, 1997] and [Artin, 1957]. A more recent account, with applications is [Lounesto, 2001].