Classification of Clifford algebras

Template:Short description 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).

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 $$v^2=Q(v)1$$ 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 Template:Nowrap matrices with entries in the division algebra K by M_n(K) or End(Kⁿ). The direct sum of two such identical algebras will be denoted by Template:Nowrap, which is isomorphic to Template:Nowrap.

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 Template:Nowrap, 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 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. After rescaling the volume element by a nonzero complex scalar if necessary, one may choose a normalized pseudoscalar ω such that Template:Nowrap. 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

${\mathrm{C}\mathrm{l}}_n(𝐂)={\mathrm{C}\mathrm{l}}_n^{+}(𝐂)\oplus {\mathrm{C}\mathrm{l}}_n^{-}(𝐂),$

where

${\mathrm{C}\mathrm{l}}_n^{\pm }(𝐂)=P_{\pm }{\mathrm{C}\mathrm{l}}_n(𝐂).$

The algebras Cl_n^±(C) are just the positive and negative eigenspaces of ω, and the P_± are the corresponding projection operators. Since ω is odd, these algebras are exchanged by the involution α induced by Template:Nowrap on the generating space:

$\alpha ({\mathrm{C}\mathrm{l}}_n^{\pm }(𝐂))={\mathrm{C}\mathrm{l}}_n^{\mp }(𝐂),$

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_n(C) is 2ⁿ. What one obtains is the following table:

Classification of complex Clifford algebras

n	Cln(C)	ClTemplate:Su(C)	N
even	End(CN)	End(CN/2) ⊕ End(CN/2)	2n/2
odd	End(CN) ⊕ End(CN)	End(CN)	2(n−1)/2

The even subalgebra ClTemplate:Su(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 (after writing elements in Template:Nowrap block form). When n is odd, the even subalgebra consists of those elements of Template:Nowrap for which the two components are equal. Projection onto either factor then gives an isomorphism with Template:Nowrap.

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(C) is isomorphic to End(C^N), which has its fundamental representation on Template:Nowrap. A complex Dirac spinor is an element of Δ_n. 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_n⁰(C) is isomorphic to Template:Nowrap and therefore its spinor module decomposes as the direct sum of two irreducible representation spaces Template:Nowrap, each isomorphic to C^N/2. A left-handed (respectively right-handed) complex Weyl spinor is an element of ΔTemplate:Su (respectively, ΔTemplate:Su).

Proof of the structure theorem for complex Clifford algebras

The structure theorem may be proved inductively. For the base cases, Cl₀(C) is simply Template:Nowrap, while Cl₁(C) is the algebra Template:Nowrap, obtained by taking the unique generator to be Template:Nowrap.

One also needs Template:Nowrap. The Pauli matrices give a concrete realization: if one sets Template:Nowrap and Template:Nowrap, then these generate a copy of Cl₂(C) whose span is all of End(C²).

The inductive step is the standard 2-periodicity isomorphism

${\mathrm{C}\mathrm{l}}_{n+2}(𝐂)\cong {\mathrm{C}\mathrm{l}}_n(𝐂)\otimes {\mathrm{C}\mathrm{l}}_2(𝐂).$

To construct it, let γ_a generate Cl_n(C), and let ${\tilde{\gamma }}_1,{\tilde{\gamma }}_2$ generate Cl₂(C). Let Template:Nowrap be the chirality element in Cl₂(C), so that Template:Nowrap and each ${\tilde{\gamma }}_a$ anticommutes with ω. Then one obtains generators for Cl_n+2(C) by setting

${\mathrm{\Gamma }}_a={\gamma }_a\otimes \omega (1\le a\le n),$

${\mathrm{\Gamma }}_{n+1}=1\otimes {\tilde{\gamma }}_1,{\mathrm{\Gamma }}_{n+2}=1\otimes {\tilde{\gamma }}_2.$

These satisfy the Clifford relations, so by the universal property of Clifford algebras they induce an isomorphism Template:Nowrap.

Finally, if n is even and Template:Nowrap, then

${\mathrm{C}\mathrm{l}}_{n+2}(𝐂)\cong \mathrm{End}({𝐂}^N)\otimes \mathrm{End}({𝐂}^2)\cong \mathrm{End}({𝐂}^{2N}).$

Since Template:Nowrap, this gives the even-dimensional case in dimension Template:Nowrap. The odd-dimensional case follows similarly, using that tensor product distributes over direct sums.

Proof of the structure theorem for complex Clifford algebras

A standard proof proceeds from three ingredients: the low-dimensional base cases, the 2-periodicity isomorphism

${\mathrm{C}\mathrm{l}}_{n+2}(𝐂)\cong {\mathrm{C}\mathrm{l}}_n(𝐂)\otimes {\mathrm{C}\mathrm{l}}_2(𝐂),$

and the identification of the even subalgebra

${\mathrm{C}\mathrm{l}}_{n+1}(𝐂{)}^0\cong {\mathrm{C}\mathrm{l}}_n(𝐂).$

See, for example, Template:Harvtxt or Template:Harvtxt.

For the base cases, one has

${\mathrm{C}\mathrm{l}}_0(𝐂)\cong 𝐂$

and

${\mathrm{C}\mathrm{l}}_1(𝐂)\cong 𝐂\oplus 𝐂.$

The first is immediate. For the second, if $e$ is the generator with $e^2=1$, then

$P_{\pm }=\frac{1}{2}(1\pm e)$

are central orthogonal idempotents with $P_{+}+P_{-}=1$, so the algebra splits as the direct sum of the two one-dimensional ideals $𝐂P_{+}$ and $𝐂P_{-}$.

Next, one needs the two-dimensional case

${\mathrm{C}\mathrm{l}}_2(𝐂)\cong M_2(𝐂).$

A concrete realization is obtained from the Pauli matrices:

${\gamma }_1={\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\gamma }_2={\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right).$

These satisfy ${\gamma }_i{\gamma }_j+{\gamma }_j{\gamma }_i=2{\delta }_{ij}$, so by the universal property they define a homomorphism ${\mathrm{C}\mathrm{l}}_2(𝐂)\to M_2(𝐂)$. Since the image contains $1,{\gamma }_1,{\gamma }_2,{\gamma }_1{\gamma }_2$, it has dimension 4 and hence is all of $M_2(𝐂)$.

The key step is the 2-periodicity isomorphism. Let ${\gamma }_1,\dots ,{\gamma }_n$ generate ${\mathrm{C}\mathrm{l}}_n(𝐂)$, let ${\tilde{\gamma }}_1,{\tilde{\gamma }}_2$ generate ${\mathrm{C}\mathrm{l}}_2(𝐂)$, and set

$\omega =i{\tilde{\gamma }}_1{\tilde{\gamma }}_2.$

Then ${\omega }^2=1$ and $\omega$ anticommutes with both ${\tilde{\gamma }}_1$ and ${\tilde{\gamma }}_2$. Define elements of ${\mathrm{C}\mathrm{l}}_n(𝐂)\otimes {\mathrm{C}\mathrm{l}}_2(𝐂)$ by

${\mathrm{\Gamma }}_a={\gamma }_a\otimes \omega (1\le a\le n),$

${\mathrm{\Gamma }}_{n+1}=1\otimes {\tilde{\gamma }}_1,{\mathrm{\Gamma }}_{n+2}=1\otimes {\tilde{\gamma }}_2.$

Because ${\omega }^2=1$ and $\omega$ anticommutes with the generators of ${\mathrm{C}\mathrm{l}}_2(𝐂)$, the elements ${\mathrm{\Gamma }}_1,\dots ,{\mathrm{\Gamma }}_{n+2}$ satisfy the Clifford relations for the standard quadratic form on ${𝐂}^{n+2}$. Therefore the universal property gives a homomorphism

${\mathrm{C}\mathrm{l}}_{n+2}(𝐂)\to {\mathrm{C}\mathrm{l}}_n(𝐂)\otimes {\mathrm{C}\mathrm{l}}_2(𝐂).$

Both algebras have dimension $2^{n+2}$, so this homomorphism is an isomorphism.

It follows by induction on $m$ that

${\mathrm{C}\mathrm{l}}_{2m}(𝐂)\cong {\mathrm{C}\mathrm{l}}_0(𝐂)\otimes {\mathrm{C}\mathrm{l}}_2(𝐂{)}^{\otimes m}\cong M_{2^m}(𝐂).$

Indeed, the case $m=0$ is ${\mathrm{C}\mathrm{l}}_0(𝐂)\cong 𝐂$, and each application of 2-periodicity tensors with ${\mathrm{C}\mathrm{l}}_2(𝐂)\cong M_2(𝐂)$, doubling the matrix size.

For odd dimension, let $n=2m+1$. The volume element $\omega =e_1{e}_2\cdots e_n$ is central because $n$ is odd, and over $𝐂$ it may be rescaled so that ${\omega }^2=1$. Hence

$P_{\pm }=\frac{1}{2}(1\pm \omega )$

are central orthogonal idempotents, giving a decomposition

${\mathrm{C}\mathrm{l}}_{2m+1}(𝐂)={\mathrm{C}\mathrm{l}}_{2m+1}^{+}(𝐂)\oplus {\mathrm{C}\mathrm{l}}_{2m+1}^{-}(𝐂).$

On the other hand, the even subalgebra is isomorphic to ${\mathrm{C}\mathrm{l}}_{2m}(𝐂)$, and projection onto either summand identifies each simple factor with that even subalgebra. Since

${\mathrm{C}\mathrm{l}}_{2m}(𝐂)\cong M_{2^m}(𝐂),$

one obtains

${\mathrm{C}\mathrm{l}}_{2m+1}(𝐂)\cong M_{2^m}(𝐂)\oplus M_{2^m}(𝐂).$

This proves the classification:

${\mathrm{C}\mathrm{l}}_{2m}(𝐂)\cong M_{2^m}(𝐂),{\mathrm{C}\mathrm{l}}_{2m+1}(𝐂)\cong M_{2^m}(𝐂)\oplus M_{2^m}(𝐂).$

Equivalently, the complex Clifford algebras are 2-periodic, and the even subalgebra of ${\mathrm{C}\mathrm{l}}_{n+1}(𝐂)$ is isomorphic to ${\mathrm{C}\mathrm{l}}_n(𝐂)$.

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 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 a diagonal form

$Q(u)=u_1^2+\cdots +u_p^2-u_{p+1}^2-\cdots -u_{p+q}^2$

where Template:Nowrap 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 Template:Nowrap mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1.

Unit pseudoscalar

Template:See also Given a standard basis Template:Mset as defined in the previous subsection, the unit pseudoscalar in Cl_p,q(R) is defined as

$\omega =e_1{e}_2\cdots e_n.$

It is the Clifford-algebra analogue of the volume element.

To compute the square ${\omega }^2=(e_1{e}_2\cdots e_n)(e_1{e}_2\cdots e_n)$, one may reverse the order of the second factor and then commute equal basis vectors together. This introduces the sign Template:Nowrap, and since Template:Nowrap for Template:Nowrap and Template:Nowrap for the remaining Template:Nowrap basis vectors, one obtains

${\omega }^2=(-1{)}^{\frac{n(n-1)}{2}}(-1{)}^q=(-1{)}^{\frac{(p-q)(p-q-1)}{2}}=\{\begin{array}{cc}+1& p-q\equiv 0,1(\mathrm{mod}4)\\ -1& p-q\equiv 2,3(\mathrm{mod}4).\end{array}$

Note that, unlike the complex case, it is not in general possible to find a pseudoscalar that squares to +1.

Center

If n (equivalently, Template:Nowrap) 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 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 Template:Nowrap (equivalently, if Template:Nowrap) then, just as in the complex case, the algebra Cl_p,q(R) decomposes into a direct sum of isomorphic algebras

${\mathrm{Cl}}_{p,q}(𝐑)={\mathrm{Cl}}_{p,q}^{+}(𝐑)\oplus {\mathrm{Cl}}_{p,q}^{-}(𝐑),$

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

If n is odd and Template:Nowrap (equivalently, if Template:Nowrap) then the center of Cl_p,q(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.

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, determining whether the algebra is central simple or not;
• signature mod 4: Template:Nowrap, determining in the odd-dimensional case whether the center is Template:Nowrap or C;
• signature mod 8: the Brauer class of the algebra (n even) or of the even subalgebra (n odd), determining whether the central simple factor is split or quaternionic.

Each of these properties depends only on the signature Template:Nowrap 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	Clp,q(R) (N = 2(p+q)/2)		p − q mod 8	ω2	Clp,q(R) (N = 2(p+q−1)/2)
0	+	MN(R)		1	+	MN(R) ⊕ MN(R)
2	−	MN(R)		3	−	MN(C)
4	+	MN/2(H)		5	+	MN/2(H) ⊕ MN/2(H)
6	−	MN/2(H)		7	−	MN(C)

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(C). For example, Cl₂(C) and Cl_3,0(R) are both isomorphic to M₂(C). It is important to distinguish the categories in which these isomorphisms are taken: Cl₂(C) is classified as a C-algebra, whereas Cl_3,0(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 Template:Nowrap follows. Here Template:Nowrap runs vertically and Template:Nowrap runs horizontally (e.g. the algebra Template:Nowrap 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)

MTemplate:Nowrap(R)

M2(C)

H2

4

M2(H)

M4(R)

M4(R)

M2(H)

M2(H)

5

MTemplate:Nowrap(H)

M4(C)

MTemplate:Nowrap(R)

M4(C)

MTemplate:Nowrap(H)

M4(C)

6

M4(H)

M4(H)

M8(R)

M8(R)

M4(H)

M4(H)

M8(R)

7

M8(C)

MTemplate:Nowrap(H)

M8(C)

MTemplate:Nowrap(R)

M8(C)

MTemplate:Nowrap(H)

M8(C)

MTemplate:Nowrap(R)

8

M16(R)

M8(H)

M8(H)

M16(R)

M16(R)

M8(H)

M8(H)

M16(R)

M16(R)

ω2

+

−

−

+

+

−

−

+

+

−

−

+

+

−

−

+

+

Symmetries

There is a tangled web of symmetries and relationships in the above table. Most importantly, one has the standard real-periodicity isomorphisms

$\begin{array}{cc}{\mathrm{Cl}}_{p+1,q+1}(𝐑)& \cong {\mathrm{Cl}}_{p,q}(𝐑)\otimes {\mathrm{M}}_2(𝐑),\\ {\mathrm{Cl}}_{q,p+2}(𝐑)& \cong {\mathrm{Cl}}_{p,q}(𝐑)\otimes 𝐇,\\ {\mathrm{Cl}}_{q+2,p}(𝐑)& \cong {\mathrm{Cl}}_{p,q}(𝐑)\otimes {\mathrm{M}}_2(𝐑).\end{array}$

In terms of the table, the first rule says that going down one step from the Clifford algebra ${\mathrm{Cl}}_{p,q}(𝐑)$ yields ${\mathrm{Cl}}_{p+1,q+1}(𝐑)$, which consists of $2\times 2$ matrices over ${\mathrm{Cl}}_{p,q}(𝐑)$. The other two rules imply that

${\mathrm{Cl}}_{p+4,q}(𝐑)\cong {\mathrm{Cl}}_{p,q+4}(𝐑)$

and from these one obtains Bott periodicity in the form

${\mathrm{Cl}}_{p+8,q}(𝐑)\cong {\mathrm{Cl}}_{p+4,q+4}(𝐑)\cong {\mathrm{Cl}}_{p,q+8}(𝐑)\cong {\mathrm{M}}_{16}({\mathrm{Cl}}_{p,q}(𝐑)).$

Furthermore, if the signature satisfies Template:Nowrap then

${\mathrm{Cl}}_{p+k,q}(𝐑)\cong {\mathrm{Cl}}_{p,q+k}(𝐑).$

This says that the table is symmetric about columns where $p-q=$ ..., −7, −3, 1, 5, 9,....

Bott periodicity

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.

Failure of symmetry under swapping p and q

Note that in the real classification, in general,

${\mathrm{Cl}}_{p,q}(𝐑)\cong ̸{\mathrm{Cl}}_{q,p}(𝐑).$

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 Template:Nowrap to Template:Nowrap. Since the isomorphism class of the real Clifford algebra is determined by Template:Nowrap, one should compare the entries in the classification table for residues d and Template:Nowrap modulo 8.

These entries agree only when Template:Nowrap, that is, only when Template:Nowrap or Template:Nowrap. In all other congruence classes, the algebras are of different types. For example,

${\mathrm{Cl}}_{1,0}(𝐑)\cong 𝐑\oplus 𝐑,{\mathrm{Cl}}_{0,1}(𝐑)\cong 𝐂,$

while

${\mathrm{Cl}}_{2,0}(𝐑)\cong M_2(𝐑),{\mathrm{Cl}}_{0,2}(𝐑)\cong 𝐇.$

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 Template:Nowrap, then Template:Nowrap and the algebra splits as a direct sum of two simple ideals, so its center is Template:Nowrap. If instead Template:Nowrap, then Template:Nowrap and the center is Template:Math. 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 Template:Nowrap the algebra is a split matrix algebra over Template:Math, while when Template:Nowrap—equivalently Template:Nowrap—the algebra is a matrix algebra over Template:Math. So swapping p and q can also change a split algebra into a quaternionic one.

Equivalently, one has

${\mathrm{Cl}}_{p,q}(𝐑)\cong {\mathrm{Cl}}_{q,p}(𝐑)$

if and only if Template:Nowrap or Template:Nowrap. This is simply the fixed-point condition for the involution $d\mapsto -d$ 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 ${\mathrm{Cl}}_{p,q}$ for what this article denotes by ${\mathrm{Cl}}_{q,p}$. The non-symmetry under $(p,q)\mapsto (q,p)$ 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 Template:Nowrap be a real quadratic space. The spin group is defined inside the even Clifford algebra by

$\mathrm{Spin}(V,q)=\mathrm{Pin}(V,q)\cap {\mathrm{Cl}}^0(V,q),$

where Template:Nowrap is generated by the unit vectors Template:Nowrap with Template:Nowrap. Under the standard twisted-adjoint action, such a vector acts on Template:Nowrap by reflection in the hyperplane orthogonal to Template:Nowrap, so products of an even number of unit vectors act by orientation-preserving orthogonal transformations.

Now replacing Template:Nowrap by Template:Nowrap does not change the orthogonal group: the same linear maps preserve Template:Nowrap and Template:Nowrap, so Template:Nowrap and hence Template:Nowrap. This is why one has Template:Nowrap and correspondingly Template:Nowrap. 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,

${\mathrm{Cl}}_{1,0}(𝐑)\cong 𝐑\oplus 𝐑,{\mathrm{Cl}}_{0,1}(𝐑)\cong 𝐂,$

so the full algebras are different, but in both cases the even subalgebra is just Template:Math. Hence

$\mathrm{Spin}(1,0)\cong \mathrm{Spin}(0,1)\cong \{\pm 1\}.$

A more instructive example is

${\mathrm{Cl}}_{2,0}(𝐑)\cong M_2(𝐑),{\mathrm{Cl}}_{0,2}(𝐑)\cong 𝐇.$

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 Template:Nowrap, and

$(e_1{e}_2{)}^2=-e_1^2{e}_2^2=-1.$

Therefore

${\mathrm{Cl}}_{2,0}^0(𝐑)\cong {\mathrm{Cl}}_{0,2}^0(𝐑)\cong 𝐂,$

and in either case the spin group is the circle group

$\{\cos \theta +\sin \theta e_1{e}_2:\theta \in 𝐑\}\cong U(1).$

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 ${\mathrm{Cl}}_{1,3}(𝐑)$ and ${\mathrm{Cl}}_{3,1}(𝐑)$ are different entries in the real classification table, the associated spin groups are both the double cover of the Lorentz group; in particular

$\mathrm{Spin}(1,3)\cong {\mathrm{SL}}_2(𝐂),$

and hence also $\mathrm{Spin}(3,1)\cong {\mathrm{SL}}_2(𝐂)$.

General fields

Let F be a field of characteristic not 2, and let $q$ be a nondegenerate quadratic form on a finite-dimensional F-vector space $V$. 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 $\dim V=2m$ is even, then the full Clifford algebra $\mathrm{Cl}(V,q)$ is a central simple algebra over $F$. Its Brauer class

$c(q):=[\mathrm{Cl}(V,q)]\in \mathrm{Br}(F)$

is called the Clifford invariant of $q$. The center of the even Clifford algebra ${\mathrm{Cl}}^0(V,q)$ is the quadratic étale $F$-algebra

$Z(q)=F[x]/(x^2-\delta (q))$,

where $\delta (q)=(-1{)}^{m}det(q)$ is the signed discriminant of $q$. Thus $Z(q)$ is either a separable quadratic extension field of $F$ or the split algebra $F\oplus F$.

If $\dim V=2m+1$ is odd, then the even Clifford algebra ${\mathrm{Cl}}^0(V,q)$ is central simple over $F$. In this case the relevant Clifford invariant is

$c(q):=[{\mathrm{Cl}}^0(V,q)]\in \mathrm{Br}(F),$

while the full Clifford algebra has center $Z(q)$ and satisfies

$\mathrm{Cl}(V,q)\cong {\mathrm{Cl}}^0(V,q){\otimes }_{F}Z(q).$

Thus, in odd dimension, the isomorphism class of the full Clifford algebra is determined by the quadratic étale center $Z(q)$ together with the Brauer class $c(q)$.

An explicit computation of $c(q)$ may be made after diagonalizing

$q\cong ⟨a_1,\dots ,a_n⟩.$

The associated Hasse invariant is the 2-torsion Brauer class

$s(q)=\prod _{1\le i<j\le n}(a_i,a_j)\in \mathrm{Br}(F)[2],$

where $(a_i,a_j)$ denotes the class of the quaternion algebra generated by $i,j$ with $i^2=a_i$, $j^2=a_j$, and $ij=-ji$. The Clifford invariant is obtained from the Hasse invariant by a universal correction depending only on $nmod8$:

$c(q)=s(q)\cdot \{\begin{array}{cc}1,& n\equiv 1,2(\mathrm{mod}8),\\ (-1,-detq),& n\equiv 3,4(\mathrm{mod}8),\\ (-1,-1),& n\equiv 5,6(\mathrm{mod}8),\\ (-1,detq),& n\equiv 0,7(\mathrm{mod}8).\end{array}$

Here $detq$ is the determinant of a Gram matrix, viewed in $F^{\times }/F^{\times 2}$. In this sense, the Brauer class of the relevant Clifford algebra is the standard Clifford invariant of the quadratic form.

Over $𝐑$, this recovers the usual real classification table above. The Brauer group $\mathrm{Br}(𝐑)$ has two elements, represented by the split class and the class of the quaternion algebra $𝐇$. For a diagonal form of signature $(p,q)$, the Hasse invariant is

$s(q)=[𝐇{]}^{(\genfrac{}{}{0ex}{}{q}{2})},$

since over $𝐑$ the quaternion class $(a,b)$ is nontrivial exactly when both $a$ and $b$ 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 $𝐑$ or $𝐇$; in odd dimension one combines the same Brauer-class computation for ${\mathrm{Cl}}^0(V,q)$ with the center $Z(q)$, which is either $𝐑\times 𝐑$ or $𝐂$. When $Z(q)\cong 𝐂$, the full Clifford algebra is a complex matrix algebra, because

$𝐂{\otimes }_{𝐑}𝐇\cong M_2(𝐂).$

The same viewpoint extends to nonarchimedean local fields. If $K$ is a local field of characteristic not 2, then quadratic spaces over $K$ are classified up to isometry by dimension, determinant, and Clifford invariant; equivalently, one may use dimension, determinant, and Hasse invariant. The Brauer group $\mathrm{Br}(K)$ has exactly two elements of order dividing 2, namely the split class and the class of the unique quaternion division algebra over $K$. Accordingly, the Brauer-class part of the Clifford-algebra classification over $K$ is especially simple. If $q$ has even dimension $2m$, then $\mathrm{Cl}(q)$ is isomorphic either to $M_{2^m}(K)$ or to $M_{2^{m-1}}(D)$, where $D$ is the quaternion division algebra over $K$. If $q$ has odd dimension $2m+1$, then ${\mathrm{Cl}}^0(q)$ is isomorphic either to $M_{2^m}(K)$ or to $M_{2^{m-1}}(D)$; the full Clifford algebra is then obtained from ${\mathrm{Cl}}^0(q)$ by adjoining its quadratic étale center. In practice one diagonalizes $q$, computes the Hilbert-symbol product $s(q)=\prod _{i<j}(a_i,a_j)$, and then obtains $c(q)$ from the same formula relating Hasse and Clifford invariants.

Characteristic two

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 “1/2-regular”) 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.

See also

• Clifford algebra
• Dirac algebra Cl_1,3(C)
• Pauli algebra Cl_3,0(R)
• Spacetime algebra Cl_1,3(R)
• Clifford module
• Spin representation

Sources

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