Classification of Clifford algebras

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In mathematics, in particular in the theory of nondegenerate quadratic forms on real and complex vector spaces, the finite-dimensional Clifford algebras have been completely classified. In each case, the Clifford algebra is isomorphic to a matrix algebra over R, C, or H (the quaternions), or to a direct sum of two such algebras, though not in a canonical way.

Notation and conventions. In this article we will use 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 K(n). The direct sum of algebras will be denoted by K²(n) = K(n) ⊕ K(n).

[edit] 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

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

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:

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

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	Cℓ_n(C)
2m	C(2^m)
2m+1	C(2^m) ⊕ C(2^m)

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

[edit] Real case

The real case is slightly more complicated, exhibiting a periodicity of 8 rather than 2. 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. The algebra Cℓ_p,q(R) will therefore have p vectors which square to +1 and q vectors which square to −1. The unit pseudoscalar in Cℓ_p,q(R) is defined as

$\omega = e_1e_2\cdots e_{n}.$

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.

The classification goes as follows: if n is even (equivalently, if 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 (or 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 then, just as in the complex case, the algebra Cℓ_p,q(R) decomposes into a direct sum of isomorphic algebras

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

each of which is central simple and so isomorphic to matrix algebra over R or H. If n is odd and ω² = −1 then the center of Cℓ_p,q(R) is isomorphic to C and can be consider 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):

n is even/odd,
ω² = ±1,
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	ω²	Cℓ_p,q(R) (p+q = 2m)	p−q mod 8	ω²	Cℓ_p,q(R) (p+q = 2m + 1)
0	+	R(2^m)	1	+	R(2^m)⊕R(2^m)
2	−	R(2^m)	3	−	C(2^m)
4	+	H(2^m−1)	5	+	H(2^m−1)⊕H(2^m−1)
6	−	H(2^m−1)	7	−	C(2^m)

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) ≅ H(2) is found in row 4, column −2).

−1

−2

−3

−4

−5

−6

−7

−8

R²

R(2)

C(2)

R²(2)

C(2)

H²

H(2)

R(4)

H(2)

H²(2)

C(4)

R²(4)

C(4)

H²(2)

C(4)

H(4)

R(8)

H(4)

R(8)

C(8)

H²(4)

C(8)

R²(8)

C(8)

H²(4)

C(8)

R²(8)

R(16)

H(8)

R(16)

H(8)

R(16)

ω²

−

There is a tangled web of symmetries and relationships in the above table. For example, the entire table is symmetric about the column 1 (as well as the columns 5, −3, and −7). Going over 4 spots in any row yields an identical algebra.