Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions

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One of the main goals of these notes is to explain how rotations in realsⁿ are induced by the action of a certain group, Spin(n), on realsⁿ, in a way that generalizes the action of the unit complex numbers, U(1), on reals², and the action of the unit quaternions, SU(2), on reals³ (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and realsⁿ). 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 realsⁿ. 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).