Clifford Algebras and Spinor Groups

Abstract

In this chapter, we generalise the quaternions by studying the real Clifford algebras, and our account of these is heavily influenced by the classic paper of Atiyah, Bott & Shapiro [3]; Porteous [23, 24] also provides an accessible description, as does Curtis [7] but there are some errors and omissions in that account. Lawson & Michelsohn [19] provides a more sophisticated introduction which shows how central Clifford algebras have become to modern geometry and topology; they also appear in Quantum Theory in connection with the Dirac operator. There is also a theory of Clifford Analysis in which the field of complex numbers is replaced by a Clifford algebra and a suitable class of Clifford analytic functions generalising complex analytic functions is studied; motivation for this is provided by the above applications. The groups of units in Clifford algebras contain the spinor groups which we define and also show how they provide double coverings of the special orthogonal groups.