In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified as rings.
The connection is explained by the geometric model of loop spaces approach to Bott periodicity: their 2-fold/8-fold periodic embeddings of the classical groups in each other (corresponding to isomorphism groups of Clifford algebras), and their successive quotients are symmetric spaces which are homotopy equivalent to the loop spaces of the unitary/orthogonal group.
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 for each dimension.
This is because the complex numbers include i by which −uk2 = +(iuk)2 and so positive or negative terms are equivalent.
We will denote the Clifford algebra on Cn with the standard quadratic form by Cln(C).
When n is even, the algebra Cln(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.
Picking either piece then gives an isomorphism with Cln[0](C) ≅ End(CN).
The Pauli matrices can be used to generate the Clifford algebra by setting γ1 = σ1, γ2 = σ2.
The odd case follows similarly as the tensor product distributes over direct sums.
The real case is significantly more complicated, exhibiting a periodicity of 8 rather than 2, and there is a 2-parameter family of Clifford algebras.
Firstly, there are non-isomorphic quadratic forms of a given degree, classified by signature.
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 Rp,q.
Given a standard basis {ei} as defined in the previous subsection, the unit pseudoscalar in Clp,q(R) is defined as This is both a Coxeter element of sorts (product of reflections) and a longest element of a Coxeter group in the Bruhat order; this is an analogy.
It corresponds to and generalizes a volume form (in the exterior algebra; for the trivial quadratic form, the unit pseudoscalar is a volume form), and lifts reflection through the origin (meaning that the image of the unit pseudoscalar is reflection through the origin, in the orthogonal group).
To compute the square ω2 = (e1e2⋅⋅⋅en)(e1e2⋅⋅⋅en), one can either reverse the order of the second group, yielding sgn(σ)e1e2⋅⋅⋅enen⋅⋅⋅e2e1, or apply a perfect shuffle, yielding sgn(σ)e1e1e2e2⋅⋅⋅enen.
These both have sign (−1)⌊n/2⌋ = (−1)n(n−1)/2, which is 4-periodic (proof), and combined with eiei = ±1, this shows that the square of ω is given by Note that, unlike the complex case, it is not in general possible to find a pseudoscalar that squares to +1.
If n (equivalently, p − q) is odd then the algebra is no longer central simple but rather has a center which includes the pseudoscalars as well as the scalars.