Hurwitz pairs and Clifford valued inner products

Abstract

After an overview of Hurwitz pairs we are showing how to actually construct them and discussing whether, for a given representation, all Hurwitz pairs of the same type are equivalent. Finally modules over a Clifford algebra are considered with compatible inner products; the results being then aplied to Hurwitz pairs. Introduction. Hurwitz pairs appear under different disguises in mathematical physics and are often introduced in a rather ad hoc way. It has been proved by several authors however that a Hurwitz pair is nothing else than an irreducible representation of a Clifford algebra with an inner product compatible with the main antiautomorphism (see e.g. [10] and [9]), and this links the notion of Hurwitz pair to the study of sesquilinear forms on spinor spaces. In the first section of this paper an overview is given of this relation. This result however does not show how to actually construct Hurwitz pairs, nor does it tell whether, for a given representation, all Hurwitz pairs of the same type are equivalent. The answer to this question is yes, and both problems are dealt with in the second section. In the final section modules over a Clifford algebra are considered with a compatible inner product (whithout the demand of irreducibility). It is proved that such inner products can be considered as the trace of Clifford valued inner products, a result which is then applied to Hurwitz pairs. Definitions and notations Clifford algebras. Let R be the real n-dimensional space, where n = p + q, endowed with the nondegenerate inner product B(·, ·) of signature (p, q). We assume that 1991 Mathematics Subject Classification: Primary 15A66, Secondary 30G35. Post-doctoral researcher, N.F.W.O., Belgium. The paper is in final form and no version of it will be published elsewhere.