Clifford Algebra

Abstract

Clifford algebras are associative algebras similar to Grassmann algebras, which we have more or less met in chapter 7. Grassmann algebras are just the algebras of anti-symmetric tensors. Originally, both were developed for facilitating geometrical computations, and indeed Grassmann algebras have been recently rediscovered by computer scientists for performing calculations in computational geometry. In fact, as we will see a little later, the only difference between Clifford algebras and Grassmann algebras is a slight difference in the algebraic structure. The reason for the utility of these algebras is that they contain several representations of the orthogonal group. That is, given a vector space ℝn we can construct the corresponding Clifford or Grassmann algebra containing various representations of O (n). In particular, we have already seen that the Grassmann algebra of anti-symmetric tensors on ℝn contains all the representations ∧kℝn corresponding to the action of O (n) on k-planes. Clifford algebras have an even richer structure and are also known as geometric algebras. They have even been proposed as “a unified language for mathematics and physics”; see Hestenes and Sobczyk [37].KeywordsDouble CoverClifford AlgebraGeometric AlgebraDual VectorGrassmann AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.