In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the finite-dimensional Clifford algebras for a nondegenerate quadratic form are completely classified as rings. In general, the Clifford algebra is either a central simple algebra or a direct sum of two copies of such an algebra. For Clifford algebras over real or complex field, this means that the Clifford algebra is isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two such algebras that are (non-canonically) isomorphic. The dimensions of the matrix algebra, and what division ring (R, C, H) can be determined by the dimension of the vector space and invariants of the quadratic form (its signature, over the reals).
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<sub>n</sub>(K) or End(K<sup>n</sup>). The direct sum of two such identical algebras will be denoted by , which is isomorphic to .
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 over the complex numbers one may multiply a basis vector by i, so positive and negative squares are equivalent. We will denote the Clifford algebra on C<sup>n</sup> with the standard quadratic form by Cl<sub>n</sub>(C).
There are two separate cases to consider, according to whether n is even or odd. When n is even, the algebra Cl<sub>n</sub>(C) is central simple and so by the ArtinâÂÂWedderburn 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. After rescaling the volume element by a nonzero complex scalar if necessary, one may choose a normalized pseudoscalar ÃÂ 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<sub>n</sub>(C) into a direct sum of two algebras
where
The algebras Cl<sub>n</sub><sup>±</sup>(C) are just the positive and negative eigenspaces of ÃÂ, and the P<sub>±</sub> are the corresponding projection operators. Since àis odd, these algebras are exchanged by the involution ñ induced by on the generating space:
and are therefore isomorphic. Each of these two summands is central simple and hence isomorphic to a matrix algebra over C. The sizes of the matrices are determined from the fact that the dimension of Cl<sub>n</sub>(C) is 2<sup>n</sup>. What one obtains is the following table:
The even subalgebra Cl(C) is (non-canonically) isomorphic to Cl<sub>nâÂÂ1</sub>(C). When n is even, the even subalgebra can be identified with the block diagonal matrices (after writing elements in block form). When n is odd, the even subalgebra consists of those elements of for which the two components are equal. Projection onto either factor then gives an isomorphism with .
The classification allows Dirac spinors and Weyl spinors to be defined in even dimension.
In even dimension n, the Clifford algebra Cl<sub>n</sub>(C) is isomorphic to End(C<sup>N</sup>), which has its fundamental representation on . A complex Dirac spinor is an element of ÃÂ<sub>n</sub>. The word complex indicates that this is a module for a complex Clifford algebra, not merely that the underlying vector space is complex.
The even subalgebra Cl<sub>n</sub><sup>0</sup>(C) is isomorphic to and therefore its spinor module decomposes as the direct sum of two irreducible representation spaces , each isomorphic to C<sup>N/2</sup>. A left-handed (respectively right-handed) complex Weyl spinor is an element of ÃÂ (respectively, ÃÂ).
The structure theorem may be proved inductively. For the base cases, Cl<sub>0</sub>(C) is simply , while Cl<sub>1</sub>(C) is the algebra , obtained by taking the unique generator to be .
One also needs . The Pauli matrices give a concrete realization: if one sets and , then these generate a copy of Cl<sub>2</sub>(C) whose span is all of End(C<sup>2</sup>).
The inductive step is the standard 2-periodicity isomorphism
To construct it, let ó<sub>a</sub> generate Cl<sub>n</sub>(C), and let generate Cl<sub>2</sub>(C). Let be the chirality element in Cl<sub>2</sub>(C), so that and each anticommutes with ÃÂ. Then one obtains generators for Cl<sub>n+2</sub>(C) by setting
These satisfy the Clifford relations, so by the universal property of Clifford algebras they induce an isomorphism .
Finally, if n is even and , then
Since , this gives the even-dimensional case in dimension . The odd-dimensional case follows similarly, using that tensor product distributes over direct sums.
A standard proof proceeds from three ingredients: the low-dimensional base cases, the 2-periodicity isomorphism
and the identification of the even subalgebra
See, for example, or .
For the base cases, one has
and
The first is immediate. For the second, if is the generator with , then
are central orthogonal idempotents with , so the algebra splits as the direct sum of the two one-dimensional ideals and .
Next, one needs the two-dimensional case
A concrete realization is obtained from the Pauli matrices:
These satisfy , so by the universal property they define a homomorphism . Since the image contains , it has dimension 4 and hence is all of .
The key step is the 2-periodicity isomorphism. Let generate , let generate , and set
Then and anticommutes with both and . Define elements of by
Because and anticommutes with the generators of , the elements satisfy the Clifford relations for the standard quadratic form on . Therefore the universal property gives a homomorphism
Both algebras have dimension , so this homomorphism is an isomorphism.
It follows by induction on that
Indeed, the case is , and each application of 2-periodicity tensors with , doubling the matrix size.
For odd dimension, let . The volume element is central because is odd, and over it may be rescaled so that . Hence
are central orthogonal idempotents, giving a decomposition
On the other hand, the even subalgebra is isomorphic to , and projection onto either summand identifies each simple factor with that even subalgebra. Since
one obtains
This proves the classification:
Equivalently, the complex Clifford algebras are 2-periodic, and the even subalgebra of is isomorphic to .
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.
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 a diagonal form
where 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<sup>p,q</sup>. The Clifford algebra on R<sup>p,q</sup> is denoted Cl<sub>p,q</sub>(R).
A standard orthonormal basis {e<sub>i</sub>} for R<sup>p,q</sup> consists of mutually orthogonal vectors, p of which have norm +1 and q of which have norm âÂÂ1.
Given a standard basis as defined in the previous subsection, the unit pseudoscalar in Cl<sub>p,q</sub>(R) is defined as
It is the Clifford-algebra analogue of the volume element.
To compute the square , one may reverse the order of the second factor and then commute equal basis vectors together. This introduces the sign , and since for and for the remaining basis vectors, one obtains
Note that, unlike the complex case, it is not in general possible to find a pseudoscalar that squares to +1.
If n (equivalently, ) is even, the algebra Cl<sub>p,q</sub>(R) is central simple and so isomorphic to a matrix algebra over R or H by the ArtinâÂÂWedderburn theorem.
If n is odd then the algebra is no longer central simple: its center contains the pseudoscalar as well as the scalars. If n is odd and (equivalently, if ) then, just as in the complex case, the algebra Cl<sub>p,q</sub>(R) 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 (equivalently, if ) then the center of Cl<sub>p,q</sub>(R) is isomorphic to C, and the algebra may be regarded as a complex central simple algebra; hence it is isomorphic to a matrix algebra over C.
All told there are three properties which determine the class of the algebra Cl<sub>p,q</sub>(R):
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<sub>p,q</sub>(R) have dimension 2<sup>p+q</sup>.
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<sub>2<sup>m</sup></sub>(C). For example, Cl<sub>2</sub>(C) and Cl<sub>3,0</sub>(R) are both isomorphic to M<sub>2</sub>(C). It is important to distinguish the categories in which these isomorphisms are taken: Cl<sub>2</sub>(C) is classified as a C-algebra, whereas Cl<sub>3,0</sub>(R) is classified as an R-algebra. Thus the two are R-algebra isomorphic, but not canonically as complex algebras.
A table of this classification for follows. Here runs vertically and runs horizontally (e.g. the algebra is found in row 4, column âÂÂ2).
There is a tangled web of symmetries and relationships in the above table. Most importantly, one has the standard real-periodicity isomorphisms
In terms of the table, the first rule says that going down one step from the Clifford algebra yields , which consists of matrices over . The other two rules imply that
and from these one obtains Bott periodicity in the form
Furthermore, if the signature satisfies then
This says that the table is symmetric about columns where ..., âÂÂ7, âÂÂ3, 1, 5, 9,....
The 8-fold periodicity over the real numbers is part of Bott periodicity, the corresponding periodicity for the homotopy groups of the stable orthogonal group; similarly, over the complex numbers one has 2-fold periodicity for the stable unitary group. In Bott's geometric description, the relevant loop spaces are modeled by successive quotients of the classical groups, which are compact symmetric spaces. In stable group theory, loop spaces enter because Bott periodicity identifies the stable classical groups, up to homotopy, with iterated loop spaces of the corresponding classifying spaces. The matching 2-fold and 8-fold algebraic periodicities of complex and real Clifford algebras are part of the same picture.
Note that in the real classification, in general,
In the sign convention used in this article, exchanging p and q replaces the quadratic form by its negative, so it sends the signature difference to . Since the isomorphism class of the real Clifford algebra is determined by , one should compare the entries in the classification table for residues d and modulo 8.
These entries agree only when , that is, only when or . In all other congruence classes, the algebras are of different types. For example,
while
So the failure of symmetry appears already in the first few low-dimensional cases.
There are two different mechanisms behind this asymmetry. In odd dimension, the distinction is visible in the center. If , then and the algebra splits as a direct sum of two simple ideals, so its center is . If instead , then and the center is . Thus swapping p and q can change the center from split real to complex.
In even dimension, both algebras are central simple, so the distinction is instead in their Brauer classes. For example, when the algebra is a split matrix algebra over , while when âÂÂequivalently âÂÂthe algebra is a matrix algebra over . So swapping p and q can also change a split algebra into a quaternionic one.
Equivalently, one has
if and only if or . This is simply the fixed-point condition for the involution on the real classification table.
This is one reason sign conventions matter in the literature: authors using the opposite convention for Clifford multiplication often write for what this article denotes by . The non-symmetry under is a property of real Clifford algebras, not just a notational artefact.
This asymmetry belongs to the full Clifford algebra, not to the spin group. Let be a real quadratic space. The spin group is defined inside the even Clifford algebra by
where is generated by the unit vectors with . Under the standard twisted-adjoint action, such a vector acts on by reflection in the hyperplane orthogonal to , so products of an even number of unit vectors act by orientation-preserving orthogonal transformations.
Now replacing by does not change the orthogonal group: the same linear maps preserve and , so and hence . This is why one has and correspondingly . The point is that although the Clifford algebras need not be the same, the spin group is built from even products of the same reflections, inside the even Clifford algebra.
The lowest-dimensional examples already show the distinction. In the sign convention used in this article,
so the full algebras are different, but in both cases the even subalgebra is just . Hence
A more instructive example is
Here the full algebras, and therefore their irreducible real modules, are of different types: in the first case the irreducible module is real 2-dimensional, whereas in the second it is quaternionic 1-dimensional. But the spin group only sees the even subalgebra. In both signatures the even subalgebra is generated by 1 and the bivector , and
Therefore
and in either case the spin group is the circle group
So the full Clifford algebra can distinguish real and quaternionic module types even when the associated spin group cannot: after passing to the even subalgebra, both cases are governed by the same complex structure.
The same phenomenon persists in higher dimensions. For example, although and are different entries in the real classification table, the associated spin groups are both the double cover of the Lorentz group; in particular
and hence also .
Let F be a field of characteristic not 2, and let be a nondegenerate quadratic form on a finite-dimensional F-vector space . Over such a field, the classification of Clifford algebras is naturally expressed in terms of the center and a Brauer class rather than by a periodic matrix table.
If is even, then the full Clifford algebra is a central simple algebra over . Its Brauer class
is called the Clifford invariant of . The center of the even Clifford algebra is the quadratic étale -algebra
where is the signed discriminant of . Thus is either a separable quadratic extension field of or the split algebra .
If is odd, then the even Clifford algebra is central simple over . In this case the relevant Clifford invariant is
while the full Clifford algebra has center and satisfies
Thus, in odd dimension, the isomorphism class of the full Clifford algebra is determined by the quadratic étale center together with the Brauer class .
An explicit computation of may be made after diagonalizing
The associated Hasse invariant is the 2-torsion Brauer class
where denotes the class of the quaternion algebra generated by with , , and . The Clifford invariant is obtained from the Hasse invariant by a universal correction depending only on :
Here is the determinant of a Gram matrix, viewed in . In this sense, the Brauer class of the relevant Clifford algebra is the standard Clifford invariant of the quadratic form.
Over , this recovers the usual real classification table above. The Brauer group has two elements, represented by the split class and the class of the quaternion algebra . For a diagonal form of signature , the Hasse invariant is
since over the quaternion class is nontrivial exactly when both and are negative. The formula above therefore determines abstractly whether the relevant central simple algebra is split or quaternionic. In even dimension this yields matrix algebras over or ; in odd dimension one combines the same Brauer-class computation for with the center , which is either or . When , the full Clifford algebra is a complex matrix algebra, because
The same viewpoint extends to nonarchimedean local fields. If is a local field of characteristic not 2, then quadratic spaces over are classified up to isometry by dimension, determinant, and Clifford invariant; equivalently, one may use dimension, determinant, and Hasse invariant. The Brauer group has exactly two elements of order dividing 2, namely the split class and the class of the unique quaternion division algebra over . Accordingly, the Brauer-class part of the Clifford-algebra classification over is especially simple. If has even dimension , then is isomorphic either to or to , where is the quaternion division algebra over . If has odd dimension , then is isomorphic either to or to ; the full Clifford algebra is then obtained from by adjoining its quadratic étale center. In practice one diagonalizes , computes the Hilbert-symbol product , and then obtains from the same formula relating Hasse and Clifford invariants.
The preceding discussion assumed that the ground field has characteristic different from 2. In characteristic 2, the polar form of a quadratic form is alternating, so a nonsingular quadratic space must have even dimension. Odd-dimensional forms are still important, but they are treated using the theory of regular (or âÂÂ1/2-regularâÂÂ) quadratic forms rather than the nonsingular theory.
For this reason, the characteristic-2 theory is usually formulated not only in terms of quadratic forms, but in terms of quadratic pairs on central simple algebras. In that setting the discriminant and the even Clifford algebra are defined for quadratic pairs and play the role of the corresponding invariants in characteristic different from 2. Accordingly, there is no direct analogue of the real-signature classification table in characteristic 2 without first reformulating the theory in this language.