Spinors

Abstract

AbstractAbstract In this chapter we complete the construction of the fundamental representations for the orthogonal Lie algebras using Clifford algebras and their irreducible representations on spaces of spinors. We show that the orthogonal Lie algebras are isomorphic to Lie algebras of quadratic elements in Clifford algebras. From these isomorphisms and the action of Clifford algebras on spinors we obtain the fundamental representations that were missing from Chapter 5, namely those with highest weights ϖ l for so(2l + 1, \(\mathbb{C}\)) and ϖ l−1, ϖ l for so(2l, \(\mathbb{C}\)). We then show that these Lie algebra representations correspond to regular representations of the spin groups, which are twofold covers of the orthogonal groups. With the introduction of the spin groups and the spin representations we finally achieve the property proved for the special linear groups and symplectic groups in Chapter 5, namely that every finite-dimensional representation of so(n, \(\mathbb{C}\)) is the differential of a unique regular representation of Spin(n, \(\mathbb{C}\)). The chapter concludes with a description of the real forms of the spin groups.KeywordsIrreducible RepresentationClifford AlgebraSpin RepresentationReal FormRegular RepresentationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.