Classification of Clifford algebras

In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras have been completely classified. 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 such algebras, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra isomorphic, as is the case of Cℓ_2,0(R) and Cℓ_1,1(R) which are both isomorphic 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, are not used here. This article uses the (+) sign convention for Clifford multiplication so that

$v^2 = Q(v)\,$

for all vectors v ∈ 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 M_n(K) or M(n,K). The direct sum of two such identical algebras will be denoted by M_n²(K) = M_n(K) ⊕ M_n(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: there 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 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

$P_{\pm} = \frac{1}{2}(1\pm\omega).$

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

$C\!\ell_n(\mathbf{C}) = C\!\ell_n^{+}(\mathbf{C}) \oplus C\!\ell_n^{-}(\mathbf{C})$ where $C\!\ell_n^\pm(\mathbf{C}) = P_\pm C\!\ell_n(\mathbf{C})$.

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):

$\alpha(C\!\ell_n^\pm(\mathbf{C})) = C\!\ell_n^\mp(\mathbf{C})$.

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

n	Cℓn(C)
2m	M(2^m,C)
2m+1	M(2^m,C) ⊕ M(2^m,C)

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 M(2^m,C) ⊕ M(2^m,C) for which the two factors are identical. Picking either piece then gives an isomorphism with Cℓ_n−1(C) ≅ M(2^m,C).

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 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.

Unit pseudoscalar

The unit pseudoscalar in Cℓ_p,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 $\omega^2=(e_1e_2\cdots e_n)(e_1e_2\cdots e_n)$, one can either reverse the order of the second group, yielding $\mbox{sgn}(\sigma)e_1e_2\cdots e_n e_n\cdots e_2 e_1$, or apply a perfect shuffle, yielding $\mbox{sgn}(\sigma)(e_1e_1e_2e_2\cdots e_ne_n)$. These both have sign $(-1)^{\lfloor n/2 \rfloor}=(-1)^{n(n-1)/2}$, which is 4-periodic (proof), and combined with $e_i e_i = \pm 1$, this shows that the square of ω is given by

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

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

Center

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

$C\ell_{p,q}(\mathbf{R}) = C\ell_{p,q}^{+}(\mathbf{R})\oplus C\ell_{p,q}^{-}(\mathbf{R})$

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.

Classification

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.

p−q mod 8	ω^2	Cℓp,q(R) (n = p+q)	p−q mod 8	ω^2	Cℓp,q(R) (n = p+q)
0	+	M(2^n/2,R)	1	+	M(2^(n−1)/2,R)⊕M(2^(n−1)/2,R)
2	−	M(2^n/2,R)	3	−	M(2^(n−1)/2,C)
4	+	M(2^(n−2)/2,H)	5	+	M(2^(n−3)/2,H)⊕M(2^(n−3)/2,H)
6	−	M(2^(n−2)/2,H)	7	−	M(2^(n−1)/2,C)

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).

8

7

6

5

4

3

2

1

0

−1

−2

−3

−4

−5

−6

−7

−8

0

R

1

R^2

C

2

M2(R)

M2(R)

H

3

M2(C)

M2^2(R)

M2(C)

H^2

4

M2(H)

M4(R)

M4(R)

M2(H)

M2(H)

5

M2^2(H)

M4(C)

M4^2(R)

M4(C)

M2^2(H)

M4(C)

6

M4(H)

M4(H)

M8(R)

M8(R)

M4(H)

M4(H)

M8(R)

7

M8(C)

M4^2(H)

M8(C)

M8^2(R)

M8(C)

M4^2(H)

M8(C)

M8^2(R)

8

M16(R)

M8(H)

M8(H)

M16(R)

M16(R)

M8(H)

M8(H)

M16(R)

M16(R)

ω^2

+

−

−

+

+

−

−

+

+

−

−

+

+

−

−

+

+

Symmetries

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

$C\ell_{p+1,q+1}(\mathbf{R}) = \mathrm{M}_2(C\ell_{p,q}(\mathbf{R}))$

$C\ell_{p+4,q}(\mathbf{R}) = C\ell_{p,q+4}(\mathbf{R})$

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

From these Bott periodicity follows:

$C\ell_{p+8,q}(\mathbf{R}) = C\ell_{p+4,q+4}(\mathbf{R}) = M_{2^4}(C\ell_{p,q}(\mathbf{R})) .$

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

$C\ell_{p+k,q}(\mathbf{R}) = C\ell_{p,q+k}(\mathbf{R}) .$

(The table is symmetric about columns with signature 1, 5, −3, −7, and so forth.) Thus if the signature satisfies p − q ≡ 1 (mod 4),

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

Bibliography

• Paolo Budinich and Andrzej Trautman, The Spinorial Chessboard, Springer Verlag, Berlin-New York 1988, ISBN 9783540190783.

See also

• Dirac algebra Cℓ_1,3(C)
• Pauli algebra Cℓ_3,0(C)
• Spacetime algebra Cℓ_1,3(R)
• Clifford module
• Spin representation