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Here, a di erent construction is adopted in which the V n and V n spaces are kept maximally distinct, so that they generate a 2n-dimensional vector space. This is the more useful approach for the study of the Lie algebra of the general linear group. We shall shortly see how these two separate approaches are reconciled by setting up an isomorphism between operations in the two algebras. We are interested in the group of orthogonal transformations which keep the V n and n V spaces separate. 89) which must hold for all a.

The fQ^ kg and fP^k g operators are discussed more fully in Chapter 4, where they are related to the theory of the general linear group. 38 Chapter 3 Lie Groups and Spin Groups This chapter demonstrates how geometric algebra provides a natural arena for the study of Lie algebras and Lie groups. In particular, it is shown that every matrix Lie group can be realised as a spin group. 4). The generators of a spin group are bivectors, and it is shown that every Lie algebra can be represented by a bivector algebra.

E. 64) This proof is considerably simpler than any that can be carried out in the more restrictive system of Grassmann algebra. 3 Some Further Developments We conclude this chapter with some further observations. We have seen how most aspects of Grassmann algebra and Berezin calculus can be formulated in terms of geometric algebra. It is natural to expect that other elds involving Grassmann variables can also be reformulated (and improved) in this manner. For example, many of the structures studied by de Witt 52] (super-Lie algebras, super-Hilbert spaces) have natural multivector expressions, and the cyclic cohomology groups of Grassmann algebras described by Coquereaux, Jadczyk and Kastler 53] can be formulated in terms of the multilinear function theory developed by Hestenes & Sobczyk 24, Chapter 3].