Classification of Clifford algebras

In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified as rings. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H, or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cl_1,1 and Cl_2,0, which are both isomorphic as rings 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, and other structure, such as the distinguished subspace of generators V, are not used here. This article uses the sign convention for Clifford multiplication so that
for all vectors v in the vector space of generators V, where Q is the quadratic form on the vector space V. We will denote the algebra of matrices with entries in the division algebra K by M_n or End. The direct sum of two such identical algebras will be denoted by, which is isomorphic to.

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: their 2-fold/8-fold periodic embeddings of the classical groups in each other, 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, so there is essentially only one Clifford algebra for each dimension. This is because the complex numbers include i by which and so positive or negative terms are equivalent. We will denote the Clifford algebra on Cⁿ with the standard quadratic form by Cl_n.
There are two separate cases to consider, according to whether n is even or odd. When n is even, the algebra Cl_n 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 as well. We can always find a normalized pseudoscalar ω such that. 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 into a direct sum of two algebras
where
The algebras Cl_n^± 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 :
and therefore isomorphic. 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 is 2ⁿ. What we have then is the following table:

n	Cl_n	Cl	N
even	End	End ⊕ End	2^n/2
odd	End ⊕ End	End	2^/2

The even subalgebra Cl of Cl_n is isomorphic to Cl_n−1. When n is even, the even subalgebra can be identified with the block diagonal matrices. When n is odd, the even subalgebra consists of those elements of for which the two pieces are identical. Picking either piece then gives an isomorphism with.

Complex spinors in even dimension

The classification allows Dirac spinors and Weyl spinors to be defined in even dimension.
In even dimension n, the Clifford algebra Cl_n is isomorphic to End, which has its fundamental representation on. A complex Dirac spinor is an element of Δ_n. The term complex signifies that it is the element of a representation space of a complex Clifford algebra, rather than that is an element of a complex vector space.
The even subalgebra Cl_n⁰ is isomorphic to and therefore decomposes to the direct sum of two irreducible representation spaces, each isomorphic to C^N/2. A left-handed complex Weyl spinor is an element of Δ.

Proof of the structure theorem for complex Clifford algebras

The structure theorem is simple to prove inductively. For base cases, Cl₀ is simply, while Cl₁ is given by the algebra by defining the only gamma matrix as.
We will also need. The Pauli matrices can be used to generate the Clifford algebra by setting,. The span of the generated algebra is End.
The proof is completed by constructing an isomorphism. Let γ_a generate Cl_n, and generate Cl₂. Let be the chirality element satisfying and. These can be used to construct gamma matrices for Cl_n+2 by setting for and for. These can be shown to satisfy the required Clifford algebra and by the universal property of Clifford algebras, there is an isomorphism.
Finally, in the even case this means by the induction hypothesis. The odd case follows similarly as the tensor product distributes over direct sums.

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 an isotropic quadratic form:
where is the dimension of the vector space. The pair of integers 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 Rp'',q is denoted Cl_p,''q.
A standard orthonormal basis for Rp'',q consists of mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1.

Unit pseudoscalar

Given a standard basis as defined in the previous subsection, the unit pseudoscalar in Cl_p,''q is defined as
This is both a Coxeter element of sorts and a longest element of a Coxeter group in the Bruhat order; this is an analogy. It corresponds to and generalizes a volume form, and lifts reflection through the origin.
To compute the square, one can either reverse the order of the second group, yielding sgne''₁e₂⋅⋅⋅e_ne_n⋅⋅⋅e₂e₁, or apply a perfect shuffle, yielding sgne₁e₁e₂e₂⋅⋅⋅e_ne_n. These both have sign, which is 4-periodic, and combined with, this shows that the square of ω is given by
Note that, unlike the complex case, it is not in general possible to find a pseudoscalar that squares to +1.

Center

If n is even, the algebra Cl_p,''q is central simple and so isomorphic to a matrix algebra over R or H by the Artin–Wedderburn theorem.
If n'' 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 then, just as in the complex case, the algebra Cl_p,''q decomposes into a direct sum of isomorphic algebras
each of which is central simple and so isomorphic to a matrix algebra over R or H.
If n'' is odd and then the center of Cl_p,''q 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:

signature mod 2: n'' is even/odd: central simple or not
signature mod 4: : if not central simple, center is or C
signature mod 8: the Brauer class of the algebra or even subalgebra is R or H

Each of these properties depends only on the signature modulo 8. The complete classification table is given below. The size of the matrices is determined by the requirement that Cl_p,''q have dimension 2p''+q.

p − q mod 8	ω²	Cl_p,''q	p'' − q mod 8	ω²	Cl_p,''q
0	+	M_N''	1	+	M_N ⊕ M_N
2	−	M_N	3	−	M_N
4	+	M_N/2	5	+	M_N/2 ⊕ M_N/2
6	−	M_N/2	7	−	M_N

It may be seen that of all matrix ring types mentioned, there is only one type shared by complex and real algebras: the type M_2^m. For example, Cl₂ and Cl_3,0 are both determined to be M₂. It is important to note that there is a difference in the classifying isomorphisms used. Since the Cl₂ is algebra isomorphic via a C-linear map, and Cl_3,0 is algebra isomorphic via an R-linear map, Cl₂ and Cl_3,0 are R-algebra isomorphic.
A table of this classification for follows. Here runs vertically and runs horizontally.

Symmetries

There is a tangled web of symmetries and relationships in the above table. Most importantly, we have
In terms of the table, the first rule says that going down one step from the Clifford algebra we obtain a Clifford algebra that consists of matrices in.
The other two rules imply that
which says that going right 4 steps in any row yields an identical algebra. From all these, Bott periodicity follows:
Furthermore, if the signature satisfies then
This says that the table is symmetric about columns where ..., −7, −3, 1, 5, 9,....