Classification of Clifford algebras

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. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cl1,1(R) and Cl2,0(R), which are both isomorphic as rings to the ring of two-by-two matrices over the real numbers.

Notation and conventions
The Clifford product is the manifest ring product for the Clifford algebra, and all algebra homomorphisms in this article are with respect to this ring product. Other products defined within Clifford algebras, such as the exterior product, and other structure, such as the distinguished subspace of generators V, are not used here. This article uses the (+) sign convention for Clifford multiplication so that v^2 = Q(v)1 for all vectors v in the vector space of generators V, where Q is the quadratic form on the vector space V. We will denote the algebra of n × n matrices with entries in the division algebra K by Mn(K) or End(Kn). The direct sum of two such identical algebras will be denoted by Mn(K) ⊕ Mn(K), which is isomorphic to Mn(K ⊕ K).

Bott periodicity
Clifford algebras exhibit a 2-fold periodicity over the complex numbers and an 8-fold periodicity over the real numbers, which is related to the same periodicities for homotopy groups of the stable unitary group and stable orthogonal group, and is called Bott periodicity. 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.

Complex case
The complex case is particularly simple: every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form
 * $$Q(u) = u_1^2 + u_2^2 + \cdots + u_n^2 ,$$

where, 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).

There are two separate cases to consider, according to whether n is even or odd. 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. We can always find a normalized pseudoscalar ω such that ω2 = 1. Define the operators
 * $$P_{\pm} = \frac{1}{2}(1\pm\omega).$$

These two operators form a complete set of orthogonal idempotents, and since they are central they give a decomposition of Cln(C) into a direct sum of two algebras
 * $$\mathrm{Cl}_n(\mathbf{C}) = \mathrm{Cl}_n^+(\mathbf{C}) \oplus \mathrm{Cl}_n^-(\mathbf{C}),$$

where
 * $$\mathrm{Cl}_n^\pm(\mathbf{C}) = P_\pm \mathrm{Cl}_n(\mathbf{C}).$$

The algebras Cln&pm;(C) are just the positive and negative eigenspaces of ω and the P&pm; are just the projection operators. Since ω is odd, these algebras are mixed by α (the linear map on V defined by v ↦ −v):
 * $$\alpha\left(\mathrm{Cl}_n^\pm(\mathbf{C})\right) = \mathrm{Cl}_n^\mp(\mathbf{C}) ,$$

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 Cln(C) is 2n. What we have then is the following table:

The even subalgebra Cl$[0] n$(C) of Cln(C) is (non-canonically) isomorphic to Cln−1(C). When n is even, the even subalgebra can be identified with the block diagonal matrices (when partitioned into 2 × 2 block matrices). When n is odd, the even subalgebra consists of those elements of End(CN) ⊕ End(CN) for which the two pieces are identical. Picking either piece then gives an isomorphism with Cln(C) ≅ End(CN).

Complex spinors in even dimension
The classification allows Dirac spinors and Weyl spinors to be defined in even dimension.

In even dimension n, the Clifford algebra Cln(C) is isomorphic to End(CN), which has its fundamental representation on Δn := CN. A complex Dirac spinor is an element of Δn. The term complex signifies that it is the element of a representation space of a complex Clifford algebra, rather than that is an element of a complex vector space.

The even subalgebra Cln0(C) is isomorphic to End(CN/2) ⊕ End(CN/2) and therefore decomposes to the direct sum of two irreducible representation spaces Δ$[0] n$ ⊕ Δ$+ n$, each isomorphic to CN/2. A left-handed (respectively right-handed) complex Weyl spinor is an element of Δ$− n$ (respectively, Δ$+ n$).

Proof of the structure theorem for complex Clifford algebras
The structure theorem is simple to prove inductively. For base cases, Cl0(C) is simply C ≅ End(C), while Cl1(C) is given by the algebra C ⊕ C ≅ End(C) ⊕ End(C) by defining the only gamma matrix as γ1 = (1, −1).

We will also need Cl2(C) ≅ End(C2). The Pauli matrices can be used to generate the Clifford algebra by setting γ1 = σ1, γ2 = σ2. The span of the generated algebra is End(C2).

The proof is completed by constructing an isomorphism Cln+2(C) ≅ Cln(C) ⊗ Cl2(C). Let γa generate Cln(C), and $$\tilde \gamma_a$$ generate Cl2(C). Let &omega; = i$\tilde\gamma_1 \tilde\gamma_2$ be the chirality element satisfying ω2 = 1 and $\tilde\gamma_a$&omega; + &omega;$\tilde\gamma_a$ = 0. These can be used to construct gamma matrices for Cln+2(C) by setting &Gamma;a = &gamma;a ⊗ &omega; for 1 ≤ a ≤ n and &Gamma;a = 1 ⊗ $\tilde \gamma_{a - n}$ for a = n + 1, n + 2. These can be shown to satisfy the required Clifford algebra and by the universal property of Clifford algebras, there is an isomorphism Cln(C) ⊗ Cl2(C) → Cln+2(C).

Finally, in the even case this means by the induction hypothesis Cln+2(C) ≅ End(CN) ⊗ End(C2) ≅ End(CN+1). The odd case follows similarly as the tensor product distributes over direct sums.

Real case
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.

Classification of quadratic forms
Firstly, there are non-isomorphic quadratic forms of a given degree, classified by signature.

Every nondegenerate quadratic form on a real vector space is equivalent to the standard diagonal form:
 * $$Q(u) = u_1^2 + \cdots + u_p^2 - u_{p+1}^2 - \cdots - u_{p+q}^2$$

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 Rp,q. The Clifford algebra on Rp,q is denoted Clp,q(R).

A standard orthonormal basis {ei} for Rp,q consists of n = p + q mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1.

Unit pseudoscalar
Given a standard basis $− n$ as defined in the previous subsection, the unit pseudoscalar in Clp,q(R) is defined as
 * $$\omega = e_1e_2\cdots e_n.$$

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 &omega;2 = (e1e2⋅⋅⋅en)(e1e2⋅⋅⋅en), one can either reverse the order of the second group, yielding sgn(&sigma;)e1e2⋅⋅⋅enen⋅⋅⋅e2e1, or apply a perfect shuffle, yielding sgn(&sigma;)e1e1e2e2⋅⋅⋅enen. These both have sign (−1)⌊n/2⌋ = (−1)n(n−1)/2, which is 4-periodic (proof), and combined with eiei = &pm;1, this shows that the square of ω is given by
 * $$\omega^2 = (-1)^{\frac{n(n-1)}{2}}(-1)^q = (-1)^{\frac{(p-q)(p-q-1)}{2}} = \begin{cases}+1 & p-q \equiv 0,1 \mod{4}\\ -1 & p-q \equiv 2,3 \mod{4}.\end{cases}$$

Note that, unlike the complex case, it is not in general possible to find a pseudoscalar that squares to +1.

Center
If n (equivalently, p − q) is even, the algebra Clp,q(R) is central simple and so isomorphic to a matrix algebra over R or H by the Artin–Wedderburn theorem.

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. If n is odd and ω2 = +1 (equivalently, if p − q ≡ 1 (mod 4)) then, just as in the complex case, the algebra Clp,q(R) decomposes into a direct sum of isomorphic algebras
 * $$\operatorname{Cl}_{p,q}(\mathbf{R}) = \operatorname{Cl}_{p,q}^{+}(\mathbf{R})\oplus \operatorname{Cl}_{p,q}^{-}(\mathbf{R}) ,$$

each of which is central simple and so isomorphic to matrix algebra over R or H.

If n is odd and ω2 = −1 (equivalently, if p − q ≡ −1 (mod 4)) then the center of Clp,q(R) is isomorphic to C and can be considered as a complex algebra. As a complex algebra, it is central simple and so isomorphic to a matrix algebra over C.

Classification
All told there are three properties which determine the class of the algebra Clp,q(R): Each of these properties depends only on the signature p − q modulo 8. The complete classification table is given below. The size of the matrices is determined by the requirement that Clp,q(R) have dimension 2p+q.
 * signature mod 2: n is even/odd: central simple or not
 * signature mod 4: ω2 = ±1: if not central simple, center is R ⊕ R or C
 * signature mod 8: the Brauer class of the algebra (n even) or even subalgebra (n odd) is R or H

It may be seen that of all matrix ring types mentioned, there is only one type shared by complex and real algebras: the type M2m(C). For example, Cl2(C) and Cl3,0(R) are both determined to be M2(C). It is important to note that there is a difference in the classifying isomorphisms used. Since the Cl2(C) is algebra isomorphic via a C-linear map (which is necessarily R-linear), and Cl3,0(R) is algebra isomorphic via an R-linear map, Cl2(C) and Cl3,0(R) are R-algebra isomorphic.

A table of this classification for p + q ≤ 8 follows. Here p + q runs vertically and p − q runs horizontally (e.g. the algebra Cl1,3(R) ≅ M2(H) is found in row 4, column −2).

Symmetries
There is a tangled web of symmetries and relationships in the above table.
 * $$\begin{align}

\operatorname{Cl}_{p+1,q+1}(\mathbf{R}) &= \mathrm{M}_2(\operatorname{Cl}_{p,q}(\mathbf{R})) \\ \operatorname{Cl}_{p+4,q}(\mathbf{R}) &= \operatorname{Cl}_{p,q+4}(\mathbf{R}) \end{align}$$ Going over 4 spots in any row yields an identical algebra.

From these Bott periodicity follows:
 * $$\operatorname{Cl}_{p+8,q}(\mathbf{R}) = \operatorname{Cl}_{p+4,q+4}(\mathbf{R}) = M_{2^4}(\operatorname{Cl}_{p,q}(\mathbf{R})) .$$

If the signature satisfies p − q ≡ 1 (mod 4) then
 * $$\operatorname{Cl}_{p+k,q}(\mathbf{R}) = \operatorname{Cl}_{p,q+k}(\mathbf{R}) .$$

(The table is symmetric about columns with signature ..., −7, −3, 1, 5, ...)

Thus if the signature satisfies p − q ≡ 1 (mod 4),
 * $$\operatorname{Cl}_{p+k,q}(\mathbf{R}) = \operatorname{Cl}_{p,q+k}(\mathbf{R}) = \operatorname{Cl}_{p-k+k,q+k}(\mathbf{R}) = \mathrm{M}_{2^k}(\operatorname{Cl}_{p-k,q}(\mathbf{R})) = \mathrm{M}_{2^k}(\operatorname{Cl}_{p,q-k}(\mathbf{R})) .$$