7 - Representing Clifford algebras

Summary

We have seen that the Clifford algebras and the even Clifford algebras of regular quadratic spaces are isomorphic either to a full matrix algebra M_k(D) or to a direct sum M_k(D) ⊕ M_k(D) of two full matrix algebras, where D = R, C or H. These algebras act naturally on a real or complex vector space, or on a left H-module, respectively. In this way, we obtain representations of the Clifford algebras.

In this chapter, we shall construct some of these representations, using tensor products. These representations give useful information about the algebra, and its relation to its even subalgebra. We then construct some explicit representations of low-dimensional Clifford algebras. These are useful in practice, but it is probably not necessary to consider each of them in detail, on the first reading.

Spinors

When A_p,m is simple, we have represented it as M_k(D), where D = R, C or H, so that we can consider A_p,m acting on the left D-module D^k. A left R-module is just a real vector space, and a left C-module is a complex vector space. Since H is a division algebra, the notions of linear independence, basis, and dimension can be defined as easily for a left H-module as for a vector space, and a left H-module is frequently called a vector space overH.