Clifford algebra: Encyclopedia II - Clifford algebra - Examples: Real and complex Clifford algebras

Clifford algebra - Examples: Real and complex Clifford algebras

The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms.

Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:

where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted R^p,q. The Clifford algebra on R^p,q is denoted Cℓ_p,q(R). The symbol Cℓ_n(R) means either Cℓ_n,0(R) or Cℓ_0,n(R) depending on whether the author prefers positive definite or negative definite spaces.

A standard orthonormal basis {e_i} for R^p,q consists of n = p + q mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1. The algebra Cℓ_p,q(R) will therefore have p vectors which square to +1 and q vectors which square to −1.

Note that Cℓ_0,0(R) is naturally isomorphic to R since there are no nonzero vectors. Cℓ_0,1(R) is a two-dimensional algebra generated by a single vector e₁ which squares to −1, and therefore is isomorphic to C, the field of complex numbers. The algebra Cℓ_0,2(R) is a four-dimensional algebra spanned by {1, e₁, e₂, e₁e₂}. The latter three elements square to −1 and all anticommute, and so the algebra is isomorphic to the quaternions H. The next algebra in the sequence is Cℓ_0,3(R) is an 8-dimensional algebra isomorphic to the direct sum H ⊕ H. This is the algebra of biquaternions first studied by Clifford.

One can also study Clifford algebras on complex vector spaces. 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). One can show that the algebra Cℓ_n(C) may be obtained as the complexification of the algebra Cℓ_p,q(R) where n = p + q:

Here Q is the real quadratic form of signature (p,q). Note that the complexification does not depend on the signature. The first few cases are not hard to compute. One finds that

where M₂(C) denotes the algebra of 2×2 matrices over C.

It turns out that every one of the algebras Cℓ_p,q(R) and Cℓ_n(C) is isomorphic to a matrix algebra over R, C, or H or to a direct sum of two such algebras. For a complete classification of these algebras see classification of Clifford algebras.

Adapted from the Wikipedia article "Examples: Real and complex Clifford algebras", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki