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Introduction

This chapter describes Clifford algebras over fields. A Clifford algebra is an associate algebra defined by a quadratic form and so the first part of this chapter describes functions for the creation and manipulation of quadratic spaces; that is, vector spaces with an associated quadratic form. At the moment many of the functions defined for quadratic spaces require the field to be finite and therefore there is not much support for Clifford algebras over infinite fields.

The Clifford algebra of a quadratic form Q defined on a vector space V is an associative algebra C with a vector space homomorphism f : V to C such that f(v)2 = Q(v) for all v∈V which satisfies 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∈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. Furthermore, if the dimension of V is n, then the dimension of C is 2n.

Clifford algebras are represented in Magma as structure constant algebras. The primary reference for quadratic forms and Clifford algebras is [Che97].

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012