Classification and Representation of the Clifford Algebras

AbstractIn this chapter, the classification and representation of Clifford algebras are introduced and discussed. Some important theorems regarding the structure of Clifford algebras are presented; later on, these are used for the classification and construction of the representations of the Clifford algebras as well. The periodicity theorems and their generalisation are demonstrated and exemplified. The Clifford algebras up to dimension 8 in all metric signatures are explicitly derived; hence, with the aid of the periodicity theorem, all Clifford algebras of finite dimension are classified. In addition, procedures to explicitly build Clifford algebra matrix representations are presented, through the idempotent structure.