Clifford algebras and spinors

This essay will present a brief outline of the theory of Clifford algebras, together with a small amount of material about quadratic forms. I follow loosely the well known book Geometric algebra by Emil Artin, but with elegant modifications that I saw originally in some lecture notes by Raoul Bott, (dating from 1962), subsequently included in [Atiyah-Bott-Shapiro:1964]. In the first section, I’ll recall a few facts about quadratic forms and orthogonal groups that will be needed eventually. In the second, I’ll discuss quaternion algebras, which will play a major role later on. In the third, I’ll look at Clifford algebras, and spin groups. In the fourth, I’ll exhibit a number of examples, mostly in low dimensions. Throughout, unless specified otherwise, F will be an arbitrary field of characteristic other than two, and Q will be a nondegenerate quadratic form on the F -vector space V .