Classification of Clifford algebras
{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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
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_{n}^{2}(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
where n = dim V, so there is essentially only one Clifford algebra in each dimension. We will denote the Clifford algebra on C^{n} 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 ω^{2} = 1. Define the operators
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
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):
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 2^{n}. What we have then is the following table:
n | Cℓ_{n}(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 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:
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
{{#invoke:see also|seealso}} The unit pseudoscalar in Cℓ_{p,q}(R) is defined as
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 , one can either reverse the order of the second group, yielding , or apply a perfect shuffle, yielding . These both have sign , which is 4-periodic (proof), and combined with , this shows that the square of ω is given by
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 ω^{2} = +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
each of which is central simple and so isomorphic to matrix algebra over R or H.
If n is odd and ω^{2} = −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: ω^{2} = ±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} | Cℓ_{p,q}(R) (n = p+q) |
p−q mod 8 | ω^{2} | Cℓ_{p,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, Cℓ_{2}(C) and Cℓ_{3,0}(R) are both determined to be M_{2}(C). It is important to note that there is a difference in the classifying isomorphisms used. Since the Cℓ_{2}(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, Cℓ_{2}(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) ≅ M_{2}(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 | M_{2}(R) | M_{2}(R) | H | ||||||||||||||
3 | M_{2}(C) | M_{2}^{2}(R) | M_{2}(C) | H^{2} | |||||||||||||
4 | M_{2}(H) | M_{4}(R) | M_{4}(R) | M_{2}(H) | M_{2}(H) | ||||||||||||
5 | M_{2}^{2}(H) | M_{4}(C) | M_{4}^{2}(R) | M_{4}(C) | M_{2}^{2}(H) | M_{4}(C) | |||||||||||
6 | M_{4}(H) | M_{4}(H) | M_{8}(R) | M_{8}(R) | M_{4}(H) | M_{4}(H) | M_{8}(R) | ||||||||||
7 | M_{8}(C) | M_{4}^{2}(H) | M_{8}(C) | M_{8}^{2}(R) | M_{8}(C) | M_{4}^{2}(H) | M_{8}(C) | M_{8}^{2}(R) | |||||||||
8 | M_{16}(R) | M_{8}(H) | M_{8}(H) | M_{16}(R) | M_{16}(R) | M_{8}(H) | M_{8}(H) | M_{16}(R) | M_{16}(R) | ||||||||
ω^{2} | + | − | − | + | + | − | − | + | + | − | − | + | + | − | − | + | + |
Symmetries
There is a tangled web of symmetries and relationships in the above table.
Going over 4 spots in any row yields an identical algebra.
From these Bott periodicity follows:
If the signature satisfies p − q ≡ 1 (mod 4) then
(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),
See also
- Dirac algebra Cℓ_{1,3}(C)
- Pauli algebra Cℓ_{3,0}(C)
- Spacetime algebra Cℓ_{1,3}(R)
- Clifford module
- Spin representation