Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups

One of the main goals of these notes is to explain how rotations in R are induced by the action of a certain group Spin(n) on R, in a way that generalizes the action of the unit complex numbers U(1) on R, and the action of the unit quaternions SU(2) on R (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and R). The group Spin(n), called a spinor group, is defined as a certain subgroup of units of an algebra Cln, the Clifford algebra associated with R. Since the spinor groups are certain well chosen subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups. These notes provide a tutorial on Clifford algebra and the groups Spin and Pin, including a study of the structure of the Clifford algebra Clp,q associated with a nondegenerate symmetric bilinear form of signature (p, q) and culminating in the beautiful “8-periodicity theorem” of Elie Cartan and Raoul Bott (with proofs).