Classification of Clifford algebras

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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 for a nondegenerate quadratic form have been completely classified. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over , , or (the quaternions), 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 and , which are both isomorphic as rings to the ring of two-by-two matrices over the real numbers.

Notation and conventions[edit]

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 in the vector space of generators , where is the quadratic form on the vector space . We will denote the algebra of matrices with entries in the division algebra K by or . The direct sum of two such identical algebras will be denoted by , which is isomorphic to .

Bott periodicity[edit]

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 (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[edit]

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 by which and so positive or negative terms are equivalent. We will denote the Clifford algebra on with the standard quadratic form by or .

There are two separate cases to consider, according to whether is even or odd. When is even, the algebra is central simple and so by the Artin–Wedderburn theorem is isomorphic to a matrix algebra over .

When is odd, the center includes not only the scalars but the pseudoscalars (degree elements) as well. We can always find a normalized pseudoscalar 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 into a direct sum of two algebras


The algebras are just the positive and negative eigenspaces of and the are just the projection operators. Since is odd, these algebras are mixed by (the linear map on defined by ):

and therefore isomorphic (since is an automorphism). These two isomorphic algebras are each central simple and so, again, isomorphic to a matrix algebra over . The sizes of the matrices can be determined from the fact that the dimension of is . What we have then is the following table:

Classification of complex Clifford algebras

The even subalgebra of is (non-canonically) isomorphic to . When is even, the even subalgebra can be identified with the block diagonal matrices (when partitioned into 2×2 block matrix). When is odd, the even subalgebra are those elements of for which the two factors are identical. Picking either piece then gives an isomorphism with .

Complex spinors in even dimension[edit]

The classification allows Dirac spinors and Weyl spinors to be defined in even dimension.[1] In even dimension , the Clifford algebra is isomorphic to which has its fundamental representation on . A complex Dirac spinor is an element of The complex refers to the fact it is the element of a representation space of a complex Clifford algebra, rather than the fact it is an element of a complex vector space.

The even subalgebra is isomorphic to and therefore decomposes to the direct sum of two irreducible representation spaces , each isomorphic to . A left-handed (respectively right-handed) complex Weyl spinor is an element of (respectively,).

Proof of the structure theorem for complex Clifford algebras[edit]

The theorem is simple to prove inductively. For base cases, is vacuously , while 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 .

The proof is completed by constructing an isomorphism . Let generate , and generate . Let be the chirality element satisfying and . These can be used to construct gamma matrices for 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 and we're done. The odd case follows similarly as the tensor product distributes over direct sums.

Real case[edit]

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[edit]

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 Rp,q. The Clifford algebra on Rp,q is denoted Clp,q(R).

A standard orthonormal basis {ei} for Rp,q consists of n = p + q mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1.

Unit pseudoscalar[edit]

The unit pseudoscalar in Clp,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 in general possible to find a pseudoscalar that squares to +1.


If n (equivalently, pq) is even, the algebra Clp,q(R) is central simple and so isomorphic to a matrix algebra over R or H by the Artin–Wedderburn theorem.

If n (equivalently, pq) 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 pq ≡ 1 (mod 4)) then, just as in the complex case, the algebra Clp,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 ω2 = −1 (equivalently, if pq ≡ −1 (mod 4)) then the center of Clp,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.


All told there are three properties which determine the class of the algebra Clp,q(R):

  • signature mod 2: n is even/odd: central simple or not
  • signature mod 4: ω2 = ±1: if not central simple, center is RR 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 pq modulo 8. The complete classification table is given below. The size of the matrices is determined by the requirement that Clp,q(R) have dimension 2p+q.

pq mod 8 ω2 Clp,q(R)
(n = p + q)
pq mod 8 ω2 Clp,q(R)
(n = p + q)
0 + M(2n/2,R) 1 + M(2(n−1)/2,R) ⊕ M(2(n−1)/2,R)
2 M(2n/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(2m,C). For example, Cl2(C) and Cl3,0(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 Cl3,0(R) is algebra isomorphic via an R-linear map, Cl2(C) and Cl3,0(R) are R-algebra isomorphic.

A table of this classification for p + q ≤ 8 follows. Here p + q runs vertically and pq runs horizontally (e.g. the algebra Cl1,3(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 R2 C
2 M2(R) M2(R) H
3 M2(C) M22(R) M2(C) H2
4 M2(H) M4(R) M4(R) M2(H) M2(H)
5 M22(H) M4(C) M42(R) M4(C) M22(H) M4(C)
6 M4(H) M4(H) M8(R) M8(R) M4(H) M4(H) M8(R)
7 M8(C) M42(H) M8(C) M82(R) M8(C) M42(H) M8(C) M82(R)
8 M16(R) M8(H) M8(H) M16(R) M16(R) M8(H) M8(H) M16(R) M16(R)
ω2 + + + + + + + + +


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 pq ≡ 1 (mod 4) then

(The table is symmetric about columns with signature ..., −7, −3, 1, 5, ...)

Thus if the signature satisfies pq ≡ 1 (mod 4),

See also[edit]


  1. ^ Hamilton, Mark J. D. (2017). Mathematical gauge theory : with applications to the standard model of particle physics. Cham, Switzerland. pp. 346–347. ISBN 9783319684383.