Clifford Algebras and Their Applications to Lie Groups and Spinors

Abstract

In these lectures, we discuss some well-known facts about Clifford algebras: matrix representations, Cartan's periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in n dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc. MSC : 15A66, 22E60, 35Q41

A Note on the Representation of Clifford Algebra

“…For different purpose, Clifford algebra has several different definitions, and 5 kinds are listed in references [16,17,28]. The following definition is the most commonly used in theoretical analysis.…”

“…and engineering [8,9,10,11,12,13,14,15]. Theoretically we have some equivalent definitions for Clifford algebras [16,17]. For the present purpose, we use the original definition of Clifford, which is based on the generators of basis.…”

“…Similarly, for C 2,0 we have C = tI + xe 1 + ye 2 + ze 12 with e 2 1 = e 2 2 = e 2 12 = 1, e 1 e 2 = −e 2 e 1 = e 12 , e 2 e 12 = −e 12 e 2 = −e 1 , e 12 For general cases, the matrix representation of Clifford algebra is an old problem with a long history. As early as in 1908, Cartan got the following periodicity of 8 [16,17].…”

“…They have done a number of researches on general representation theory of Clifford algebra [16,17,19,20,21,22,23,24]. Many isomorphic or equivalent relations between Clifford algebra and matrices were provided.…”

“…In the non-degenerate case, if the standard orthogonal basis of V is introduced, we can derive Definition 1. The definition based on the quotient algebra of tensor algebra in V is introduced by Chevalley[16,17], but it is too abstract for common readers. In the author's opinion, the most efficient and convenient definition of Clifford algebra should be as follows.…”