Classification of Clifford algebras

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:

${\displaystyle Q(u)=u_{1}^{2}+\cdots +u_{p}^{2}-u_{p+1}^{2}-\cdots -u_{p+q}^{2}}$

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 R^p,q. The Clifford algebra on R^p,q is denoted Cl_p,q(R).

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 unit pseudoscalar in Cl_p,q(R) is defined as

${\displaystyle \omega =e_{1}e_{2}\cdots e_{n}.}$

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 ${\displaystyle \omega ^{2}=(e_{1}e_{2}\cdots e_{n})(e_{1}e_{2}\cdots e_{n})}$, one can either reverse the order of the second group, yielding ${\displaystyle {\mbox{sgn}}(\sigma )e_{1}e_{2}\cdots e_{n}e_{n}\cdots e_{2}e_{1}}$, or apply a perfect shuffle, yielding ${\displaystyle {\mbox{sgn}}(\sigma )(e_{1}e_{1}e_{2}e_{2}\cdots e_{n}e_{n})}$. These both have sign ${\displaystyle (-1)^{\lfloor n/2\rfloor }=(-1)^{n(n-1)/2}}$, which is 4-periodic (proof), and combined with ${\displaystyle e_{i}e_{i}=\pm 1}$, this shows that the square of ω is given by

${\displaystyle \omega ^{2}=(-1)^{\frac {n(n-1)}{2}}(-1)^{q}=(-1)^{\frac {(p-q)(p-q-1)}{2}}={\begin{cases}+1&p-q\equiv 0,1\mod {4}\\-1&p-q\equiv 2,3\mod {4}.\end{cases}}}$

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

If n (equivalently, p − q) is even, the algebra Cl_p,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 Cl_p,q(R) decomposes into a direct sum of isomorphic algebras

${\displaystyle \operatorname {Cl} _{p,q}(\mathbf {R} )=\operatorname {Cl} _{p,q}^{+}(\mathbf {R} )\oplus \operatorname {Cl} _{p,q}^{-}(\mathbf {R} )}$

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 Cl_p,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.

All told there are three properties which determine the class of the algebra Cl_p,q(R):

• 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

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 Cl_p,q(R) have dimension 2^p+q.

It may be seen that of all matrix ring types mentioned, there is only one type shared between both complex and real algebras: the type M(2^m,C). For example, Cl₂(C) and Cl_3,0(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 Cl_3,0(R) is algebra isomorphic via an R-linear map, Cl₂(C) and Cl_3,0(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 Cl_1,3(R) ≅ M₂(H) is found in row 4, column −2).

There is a tangled web of symmetries and relationships in the above table.

${\displaystyle {\begin{aligned}\operatorname {Cl} _{p+1,q+1}(\mathbf {R} )&=\mathrm {M} _{2}(\operatorname {Cl} _{p,q}(\mathbf {R} ))\\\operatorname {Cl} _{p+4,q}(\mathbf {R} )&=\operatorname {Cl} _{p,q+4}(\mathbf {R} )\end{aligned}}}$

Going over 4 spots in any row yields an identical algebra.

From these Bott periodicity follows:

${\displaystyle \operatorname {Cl} _{p+8,q}(\mathbf {R} )=\operatorname {Cl} _{p+4,q+4}(\mathbf {R} )=M_{2^{4}}(\operatorname {Cl} _{p,q}(\mathbf {R} )).}$

If the signature satisfies p − q ≡ 1 (mod 4) then

${\displaystyle \operatorname {Cl} _{p+k,q}(\mathbf {R} )=\operatorname {Cl} _{p,q+k}(\mathbf {R} ).}$

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

Thus if the signature satisfies p − q ≡ 1 (mod 4),

${\displaystyle \operatorname {Cl} _{p+k,q}(\mathbf {R} )=\operatorname {Cl} _{p,q+k}(\mathbf {R} )=\operatorname {Cl} _{p-k+k,q+k}(\mathbf {R} )=\mathrm {M} _{2^{k}}(\operatorname {Cl} _{p-k,q}(\mathbf {R} ))=\mathrm {M} _{2^{k}}(\operatorname {Cl} _{p,q-k}(\mathbf {R} )).}$