Classification of Clifford Algebras - Complex Case

Complex Case

The complex case is particularly simple: every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form

where n = dim V, so there is essentially only one Clifford algebra in each dimension. We will denote the Clifford algebra on Cn with the standard quadratic form by Cℓ_n(C).

There are two separate cases to consider, according to whether n is even or odd. When n is even the algebra Cℓ_n(C) is central simple and so by the Artin-Wedderburn theorem is isomorphic to a matrix algebra over C. When n is odd, the center includes not only the scalars but the pseudoscalars (degree n elements) as well. We can always find a normalized pseudoscalar ω such that ω2 = 1. Define the operators

These two operators form a complete set of orthogonal idempotents, and since they are central they give a decomposition of Cℓ_n(C) into a direct sum of two algebras

where .

The algebras Cℓ_n±(C) are just the positive and negative eigenspaces of ω and the P_± are just the projection operators. Since ω is odd these algebras are mixed by α (the linear map on V defined by v → −v):

.

and therefore isomorphic (since α is an automorphism). These two isomorphic algebras are each central simple and so, again, isomorphic to a matrix algebra over C. The sizes of the matrices can be determined from the fact that the dimension of Cℓ_n(C) is 2n. What we have then is the following table:

The even subalgebra of Cℓ_n(C) is (non-canonically) isomorphic to Cℓ_n−1(C). When n is even, the even subalgebra can be identified with the block diagonal matrices (when partitioned into 2×2 block matrix). When n is odd, the even subalgebra are those elements of C(2m) ⊕ C(2m) for which the two factors are identical. Picking either piece then gives an isomorphism with Cℓ_n−1(C) ≅ C(2m).