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).

1 Complex case
2 Real case
3 See also

[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 significantly more complicated, exhibiting a periodicity of 8 rather than 2, and there is a 2-parameter family of Clifford algebras.

[edit] Classification of quadratic form

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. The algebra Cℓ_p,q(R) will therefore have p vectors which square to +1 and q vectors which square to −1.

[edit] Unit pseudoscalar

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

$\omega = e_1e_2\cdots e_n.$

This is a longest element, and it corresponds to a volume form (in the exterior algebra) or a longest element in the Bruhat order in a Coxeter 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.

[edit] Center

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 (if and only if $p-q \equiv 1 \mod{4}$ ) then, just as in the complex case, the algebra Cℓ_p,q(R) decomposes into a direct sum of isomorphic algebras

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

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

If n is odd and ω² = −1 (if and only if $p-q \equiv -1 \mod{4}$ ) 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.

[edit] 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 $\mathbb{R} \oplus \mathbb{R}$ or $\mathbb{C}$
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	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

−

[edit] Symmetries

There is a tangled web of symmetries and relationships in the above table. Write $A[n] := A \otimes M_n(\mathbf{R})$ for $n \times n$ matrices with coefficients in $A$ , and $C l (p + 1, q + 1)$ for the real Clifford algebra.