Associated with any Euclidean space ℝi or Minkowski space ℝp,q is a universal Clifford algebra, denoted by and, respectively. Roughly speaking, a Clifford algebra is an associative algebra with unit into which a given Euclidean or Minkowski space may be embedded, in which the corresponding quadratic form may be expressed as the negative of a square. The real numbers ℝ, the complex numbers ℂ, and the quaternions ℝ are the simplest examples.

Our intent in this chapter is to give an elementary, coherent, and largely self-contained account of the theory of Clifford algebras. In sections 1 and 2 we present the definitions basic to all of our work. The balance of section 2 is devoted to three constructive proofs of the existence of universal Clifford algebras: two basis-free constructions using tensor algebras and exterior algebras, and a basis-dependent construction. The reader who is willing to accept the existence of Clifford algebras may wish to proceed directly to the statement of the major structural results in section 3. Sections 4, 5, and 6 explore the interconnections between Clifford algebras and orthogonal groups; the spin representation and spin groups will be studied in detail, with Spin(p, q) and Spin(p, q + 1) both being realized in using the notion of transformers. The reader who is primarily interested in the analytic applications of Clifford algebras may wish to proceed directly to the discussion of the Euclidean case in section 7. Section 8 is a discussion of spin groups as Lie groups. In section 9 we construct various realizations of Spin(p, q), p + q ≤ 6, whereby these groups are explicitly identified with classical Lie groups.