Classification of Clifford algebras

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 $\mathbb {R}$, $\mathbb {C}$, or $\mathbb {H}$ (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 $\mathrm {Cl} _{2,0}(\mathbb {R} )$ and $\mathrm {Cl} _{2,0}(\mathbb {R} )$, 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, are not used here. This article uses the (+) sign convention for Clifford multiplication so that

$$v^{2}=Q(v)$$

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 $n\times n$ matrices with entries in the division algebra K by $M_{n}(K),M(n,K)$ or $\mathrm {End} (K^{n})$. The direct sum of two such identical algebras will be denoted by $M_{n}(K)\oplus M_{n}(K)=:M_{n}^{2}(K)$, which is isomorphic to $M_{n}(K\oplus K)$.

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: 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

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 $n=\mathrm {dim} (V)$, so there is essentially only one Clifford algebra for each dimension. This is because the complex numbers include $i$ by which $-u_{k}^{2}=+(iu_{k})^{2}$ and so positive or negative terms are equivalent. We will denote the Clifford algebra on $\mathbb {C} ^{n}$ with the standard quadratic form by $\mathrm {Cl} _{n}(\mathbb {C} )$ or $\mathbb {C} l(n)$.

There are two separate cases to consider, according to whether $n$ is even or odd. When $n$ is even, the algebra $\mathbb {C} l(n)$ is central simple and so by the Artin–Wedderburn theorem is isomorphic to a matrix algebra over $\mathbb {C}$.

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 $\omega$ such that $\omega ^{2}=1$. 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 $\mathbb {C} l(n)$ into a direct sum of two algebras

$\mathbb {C} l(n)=\mathbb {C} l^{+}(n)\oplus \mathbb {C} l^{-}(n),$

where

$\mathbb {C} l^{\pm }(n)=P_{\pm }\mathbb {C} l(n).$

The algebras $\mathbb {C} l^{\pm }(n)$ are just the positive and negative eigenspaces of $\omega$ and the $P_{\pm }$ are just the projection operators. Since $\omega$ is odd, these algebras are mixed by $\alpha$ (the linear map on $V$ defined by $v\mapsto -v$):

$\alpha \left(\mathbb {C} l^{\pm }(n)\right)=\mathbb {C} l^{\mp }(n),$

and therefore isomorphic (since $\alpha$ is an automorphism). These two isomorphic algebras are each central simple and so, again, isomorphic to a matrix algebra over $\mathbb {C}$. The sizes of the matrices can be determined from the fact that the dimension of $\mathbb {C} l(n)$ is $2^{n}$. What we have then is the following table:

Classification of complex Clifford algebras

$n$	$\mathbb {C} l(n)$	$\mathbb {C} l^{0}(n)$	$N$
Even	$\mathrm {End} (\mathbb {C} ^{N})$	$\mathrm {End} (\mathbb {C} ^{N/2})\oplus \mathrm {End} (\mathbb {C} ^{N/2})$	$2^{n/2}$
Odd	$\mathrm {End} (\mathbb {C} ^{N})\oplus \mathrm {End} (\mathbb {C} ^{N})$	$\mathrm {End} (\mathbb {C} ^{N})$	$2^{(n-1)/2}$

The even subalgebra $\mathbb {C} l^{0}(n)$ of $\mathbb {C} l(n)$ is (non-canonically) isomorphic to $\mathbb {C} l(n-1)$. 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 $\mathrm {End} (\mathbb {C} ^{N})\oplus \mathrm {End} (\mathbb {C} ^{N})$ for which the two factors are identical. Picking either piece then gives an isomorphism with $\mathbb {C} l(n)\cong \mathrm {End} (\mathbb {C} ^{N})$.

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 $\mathbb {C} l(n)$ is isomorphic to $\mathrm {End} (\mathbb {C} ^{N})$ which has its fundamental representation on $\Delta _{n}:=\mathbb {C} ^{N}$. A complex Dirac spinor is an element of $\Delta _{n}.$ 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 $\mathbb {C} l^{0}(n)$ is isomorphic to $\mathrm {End} (\mathbb {C} ^{N/2})\oplus \mathrm {End} (\mathbb {C} ^{N/2})$ and therefore decomposes to the direct sum of two irreducible representation spaces $\Delta _{n}^{+}\oplus \Delta _{n}^{-}$, each isomorphic to $\mathbb {C} ^{N/2}$. A left-handed (respectively right-handed) complex Weyl spinor is an element of $\Delta _{n}^{+}$ (respectively,$\Delta _{n}^{-}$).

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 the standard diagonal form:

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

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

See also: Pseudoscalar

The unit pseudoscalar in Cl_p,q(R) is defined as

$\omega =e_{1}e_{2}\cdots e_{n}.$

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 $\omega ^{2}=(e_{1}e_{2}\cdots e_{n})(e_{1}e_{2}\cdots e_{n})$, one can either reverse the order of the second group, yielding ${\mbox{sgn}}(\sigma )e_{1}e_{2}\cdots e_{n}e_{n}\cdots e_{2}e_{1}$, or apply a perfect shuffle, yielding ${\mbox{sgn}}(\sigma )(e_{1}e_{1}e_{2}e_{2}\cdots e_{n}e_{n})$. These both have sign $(-1)^{\lfloor n/2\rfloor }=(-1)^{n(n-1)/2}$, which is 4-periodic (proof), and combined with $e_{i}e_{i}=\pm 1$, this shows that the square of ω is given by

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

Note that, unlike the complex case, it is not in general possible to find a pseudoscalar that 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 ω² = +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

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

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 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: ω² = ±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	ω2	Clp,q(R) (n = p + q)		p − q 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(2^m,C). For example, Cl₂(C) and Cl_3,0(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 Cl_3,0(R) is algebra isomorphic via an R-linear map, Cl₂(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) ≅ 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

+

−

−

+

+

−

−

+

+

−

−

+

+

−

−

+

+

Symmetries

There is a tangled web of symmetries and relationships in the above table.

${\begin{aligned}\operatorname {Cl} _{p+1,q+1}(\mathbf {R} )&=\mathrm {M} _{2}(\operatorname {Cl} _{p,q}(\mathbf {R} ))\\\operatorname {Cl} _{p+4,q}(\mathbf {R} )&=\operatorname {Cl} _{p,q+4}(\mathbf {R} )\end{aligned}}$

Going over 4 spots in any row yields an identical algebra.

From these Bott periodicity follows:

$\operatorname {Cl} _{p+8,q}(\mathbf {R} )=\operatorname {Cl} _{p+4,q+4}(\mathbf {R} )=M_{2^{4}}(\operatorname {Cl} _{p,q}(\mathbf {R} )).$

If the signature satisfies p − q ≡ 1 (mod 4) then

$\operatorname {Cl} _{p+k,q}(\mathbf {R} )=\operatorname {Cl} _{p,q+k}(\mathbf {R} ).$

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

Thus if the signature satisfies p − q ≡ 1 (mod 4),

$\operatorname {Cl} _{p+k,q}(\mathbf {R} )=\operatorname {Cl} _{p,q+k}(\mathbf {R} )=\operatorname {Cl} _{p-k+k,q+k}(\mathbf {R} )=\mathrm {M} _{2^{k}}(\operatorname {Cl} _{p-k,q}(\mathbf {R} ))=\mathrm {M} _{2^{k}}(\operatorname {Cl} _{p,q-k}(\mathbf {R} )).$

See also

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

References

• Budinich, Paolo; Trautman, Andrzej (1988). The Spinorial Chessboard. Springer Verlag. ISBN 978-3-540-19078-3.
• Lawson, H. Blaine; Michelsohn, Marie-Louise (2016). Spin Geometry. Princeton Mathematical Series. Vol. 38. Princeton University Press. ISBN 978-1-4008-8391-2.
• Porteous, Ian R. (1995). Clifford Algebras and the Classical Groups. Cambridge Studies in Advanced Mathematics. Vol. 50. Cambridge University Press. ISBN 978-0-521-55177-9.