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 R, C, or 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 Cl_1,1(R) and Cl_2,0(R), which are both isomorphic as rings to the ring of two-by-two matrices over the real numbers.

The significance of this result is that the additional structure on a Clifford algebra relative to the "underlying" associative algebra — namely, the structure given by the grade involution automorphism and reversal anti-automorphism (and their composition, the Clifford conjugation) — is in general an essential part of its definition, not a procedural artifact of its construction as the quotient of a tensor algebra by an ideal. The category of Clifford algebras is not just a selection from the category of matrix rings, picking out those in which the ring product can be constructed as the Clifford product for some vector space and quadratic form. With few exceptions, "forgetting" the additional structure (in the category theory sense of a forgetful functor) is not reversible.

Continuing the example above: Cl_1,1(R) and Cl_2,0(R) share the same associative algebra structure, isomorphic to (and commonly denoted as) the matrix algebra M₂(R). But they are distinguished by different choices of grade involution — of which two-dimensional subring, closed under the ring product, to designate as the even subring — and therefore of which of the various anti-automorphisms of M₂(R) can accurately represent the reversal anti-automorphism of the Clifford algebra. These distinguished (anti-)automorphisms are structures on the tensor algebra which are preserved by the "quotient by ideal" construction of the Clifford algebra. The matrix algebra representation admits them, but only the Clifford algebra distinguishes them from other elements of the matrix algebra's (anti-)automorphism group.

Of the four real degrees of freedom in M₂(R)), only one, generated by $-i\sigma _{2}=\left[{\begin{array}{rr}0&-1\\1&0\end{array}}\right]$ , is reversed by the obvious anti-automorphism of M₂(R), the matrix transpose. (The designation $-i\sigma _{2}$ refers to the Pauli matrices, which are here just a conventional way of naming the degrees of freedom in a two by two matrix.) In the case of Cl_2,0(R), where both degrees of freedom in the odd part have positive norm, the obvious assignment of the even subalgebra to the matrices spanned by 1 and $-i\sigma _{2}$ (and the odd part to the other two non-identity symmetric degrees of freedom) is consistent with selecting the matrix transpose as the reversal anti-automorphism. In this representation, the grade involution can be expressed as the inner automorphism $A\rightarrow (-i\sigma _{2})A{(-i\sigma _{2})}^{-1}$ .

But in the case of Cl_1,1(R), the degree of freedom with negative norm lies in the odd part, and the reversal anti-automorphism must instead reverse one of the non-identity degrees of freedom that has positive norm. The general anti-automorphism of a full matrix algebra (over a field other than $\mathbb {Z} _{2}$ ) is easily proven to be an anti-similarity transformation $A\rightarrow TA^{t}T^{-1}$ for some invertible matrix T. (These anti-automorphisms form the odd part of a Z₂-graded matrix transformation group of which the inner automorphisms are the even subgroup; together, they form the group of Jordan automorphisms. It's not quite this simple in the case of a matrix group over a general commutative ring or a C^*-algebra.) If we choose to represent the degree of freedom with positive norm in the odd part of the Clifford algebra using one of the two Pauli matrices with real entries, $\sigma _{1}=\left[{\begin{array}{rr}0&1\\1&0\end{array}}\right]$ , then the non-identity degree of freedom in the even subalgebra corresponds to the remaining Pauli matrix, $\sigma _{3}=\left[{\begin{array}{rr}1&0\\0&-1\end{array}}\right]$ . The anti-similarity transformation that reverses only this last degree of freedom, which is $A\rightarrow \sigma _{1}A^{t}\sigma _{1}^{-1}$ , is then an appropriate representation of the reversal anti-automorphism on Cl_1,1(R). In this representation, the grade involution can be expressed as the inner automorphism $A\rightarrow \sigma _{3}A\sigma _{3}^{-1}$ .

Looking back at the case of Cl_2,0(R), we may observe that "the degree of freedom with negative norm" need not be generated by $\pm i\sigma _{2}$ ; one could just as easily choose $\pm \left[{\begin{array}{rr}{(1-k^{2})}^{\frac {1}{2}}&-k\\k&-{(1-k^{2})}^{\frac {1}{2}}\end{array}}\right]$ for any $\left|k\right|>=1$ , which is a similarity transformation away from $\pm i\sigma _{2}$ . What matters is not the relationship of matrix degrees of freedom to the matrix transpose but their relationship to the particular anti-similarity transformation that represents the reversal anti-automorphism of the Clifford algebra. (And, of course, to the automorphism chosen as the grade involution, which must be selected from those which have the right signature relative to the chosen reversal anti-automorphism. This is a somewhat subtle point; these two operations on the representation need not commute on the entire matrix algebra in which the representation is embedded! But for the representation to be faithful, they do need to commute on the representation itself, and that's a property that has to be verified separately from their individual properties.)

Looking at the other two Clifford algebras with four real degrees of freedom, Cl_0,2(R) and Cl₁(C), we find that they are distinguished from the above two and from one another by their structure as associative algebras. Both may be represented as subalgebras of the matrix algebra M₄(R), and in a sense as subalgebras of M₂(C). Setting Cl₁(C) aside, one may observe that the same assignment of the even subalgebra to the matrices spanned by 1 and $-i\sigma _{2}$ works for Cl_0,2(R) as for Cl_2,0(R), with the odd part spanned by $-i\sigma _{1}$ and $-i\sigma _{3}$ — complex, not real, 2 by 2 matrices. This is consistent with selecting the complex matrix transpose — not Hermitian conjugation — as the reversal anti-automorphism. In this representation, the grade involution can again be expressed as the inner automorphism $A\rightarrow (-i\sigma _{2})A{(-i\sigma _{2})}^{-1}$ ; it can also be expressed as complex conjugation of the matrix entries (still not Hermitian conjugation). Calling this associative algebra H (Hamilton's quaternions) risks emphasizing the composition of these two operations (the Clifford conjugation, which does indeed coincide here with Hermitian conjugation and thus with the quaternion conjugation) over the others (which, as they invert only one or two of the non-identity degrees of freedom, necessarily involve "breaking the symmetry" of the imaginary quaternions). Perhaps it is arguable whether the reversal anti-automorphism or the Clifford conjugation is "more fundamental", but there can be more to the grade involution than just the composition of the two; a general associative algebra can have outer automorphisms, too.

When extending this analysis to other Clifford algebras, it is important to remember that the identification of general anti-automorphisms with anti-similarity transformations only holds for a full matrix algebra over a (non- $\mathbb {Z} _{2}$ ) field. The quaternions H are a division algebra but not a field, and a matrix with quaternion entries is equivalent to a sparse matrix over a complex (or real) field; the general anti-automorphism of such a matrix algebra is still an anti-similarity transformation parameterized by some matrix T in the full matrix algebra, but that T may lie outside the subring representing the matrices with quaternion entries. Similarly, a direct sum of two copies of a matrix algebra — of which the simplest case is the associative algebra underlying Cl_1,0(R), D = R² — may also be represented as a sparse matrix algebra (as when identifying (a, x) with $a\cdot 1+x\cdot \sigma _{3}$ ); but the matrix parameter of a general anti-similarity transformation may not lie in this same algebra. (In this example, the reversal anti-automorphism is simply the identity on Cl_1,0(R), but the Clifford conjugation corresponds to $A\rightarrow TA^{t}T^{-1}$ with a T lying outside D, such as $\sigma _{1}$ or $-i\sigma _{2}$ .)

Discussion of (anti-)automorphism groups of an associative algebra over C — and therefore of a Clifford algebra over field C — has to account for two different ways to be "anti-": order reversal of the algebra product and complex antilinearity relative to the scalar product. Complex conjugation of the entries in a particular complex matrix representation of a Clifford algebra, much like the matrix transpose, is not as special as it first seems; what matters is not the relationship of matrix degrees of freedom to the complex conjugation but their relationship to the complex antilinear automorphism that represents "charge conjugation" of all components in the Clifford algebra. (The set of all such complex antilinear automorphisms presumably forms the odd part of a different Z₂-graded matrix transformation group of which the even subgroup is, or perhaps includes, the inner automorphisms; adjoining the anti-similarity transformations as well as the complex antilinear anti-automorphisms, one might obtain a Z₂×Z₂-graded matrix transformation group

The relevant antilinear automorphism is, in any case, restricted by the complex Clifford algebra construction as a quotient of the (covariant) tensor algebra over the field C by the complex bilinear extension of a symmetric real-valued quadratic form on the underlying real (co)vector space to a complexified version of that space. (See section 4.1 of Vaz and da Rocha; the point is that the two-operand quadratic form is by construction not only complex bilinear — not linear in one operand and antilinear in the other — but consistent with a real quadratic form on the R^n subspace associated with the original real axes.) So a statement like "there is essentially only one complex Clifford algebra for each value of the dimension n" is true in the qualified sense that, given a complex Clifford algebra constructed in this way, one can extract "real" subalgebras for any signature p+q=n by choosing different antilinear automorphisms to represent complex conjugation on the complexified (co)vector space. But it is not automatically true that a particular antilinear automorphism has a "natural" representation relative to the otherwise simplest representation as a complex matrix algebra. The complex Clifford algebras for different quadratic form signatures share an associative algebra structure, but may have inequivalent antilinear "charge conjugation" automorphisms as well as reversal anti-automorphisms and grade involution automorphisms.

Returning to Cl₁(C), we find that it is consistent as an associative algebra with C⊕C (the complexification of the representation of Cl_1,0(R) by R⊕R), but that we can represent "charge conjugation" simply as complex conjugation in at most one of the two choices of signature for the real quadratic form. In the representation identifying (a, x) with $a\cdot 1+x\cdot \sigma _{3}$ , the other signature is obtained by $A\rightarrow TA^{*}T^{-1}$ with $T=\sigma _{1}$ (or $-i\sigma _{2}$ , etc.). Since there is only one odd (complex) degree of freedom, these two antilinear automorphisms are related by the grade involution automorphism itself, and the reversal anti-automorphism is the identity; the general case is more complicated, of course.

A representation of a real Clifford algebra using a matrix with complex entries — such as Cl_3,0(R), which has the same structure as an associative algebra over the reals as M₂(C) — has anti-automorphisms as a real algebra that are not represented by anti-similarity transforms on the complex matrix. (In the simplest such case, Cl_0,1(R) represented by C, the reversal anti-automorphism is again simply the identity but the Clifford conjugation corresponds to complex conjugation — in a way that has nothing to do with "charge conjugation".) And although similarity transformations (inner automorphisms) form the entirety of the even subalgebra of the automorphisms of M_n(R) or M_n(C), a general associative algebra may also have outer automorphisms that cannot be represented as similarity transformations, and it is possible a priori for the grade involution of a Clifford algebra representable as M_n(K) to be representable only as an outer, rather than an inner, automorphism of M_n(K) (for non-field K). The reversal anti-automorphism and Clifford conjugation anti-automorphism are related by the grade involution automorphism, but one cannot necessarily recover the correct grade involution automorphism given representations of these two anti-automorphisms on an otherwise adequate matrix algebra.

In short (too late!), one must be cautious with statements like "Cl₂(C) and Cl_3,0(R) are both determined to be M₂(C)," or even "Cl_4,2(R) and Cl_6,0(R) are both represented by M₈(R)." That's really only true at the level of the associative algebra structure, and even then it's an abuse of notation to denote a subalgebra of M₄(R) as M₂(C). Really classifying the Clifford algebras would require assigning each a representative embedding as a subalgebra of a matrix ring over the correct field — e. g., not "M₄(H)" but some appropriately constrained 64-real-dimensional subalgebra of (probably) M₁₆(R) — with a specific choice of grade involution automorphism (especially if it's an outer automorphism), reversal anti-automorphism, and (if the field is C) antilinear charge conjugation automorphism. These are of course only unique up to some group of automorphisms of the full matrix ring.

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)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 n × n 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 M_n(K) ⊕ M_n(K), which is isomorphic to M_n(K ⊕ 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 = dim(V), so there is essentially only one Clifford algebra for each dimension. This is because the complex numbers include i by which −u_k² = +(iu_k)² and so positive or negative terms 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. We can always find a normalized pseudoscalar ω such that ω² = 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 Cl_n(C) into a direct sum of two algebras

\mathrm {Cl} _{n}(\mathbf {C} )=\mathrm {Cl} _{n}^{+}(\mathbf {C} )\oplus \mathrm {Cl} _{n}^{-}(\mathbf {C} ),

where

\mathrm {Cl} _{n}^{\pm }(\mathbf {C} )=P_{\pm }\mathrm {Cl} _{n}(\mathbf {C} ).

The algebras Cl_n^±(C) 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 (the linear map on V defined by v ↦ −v):

\alpha \left(\mathrm {Cl} _{n}^{\pm }(\mathbf {C} )\right)=\mathrm {Cl} _{n}^{\mp }(\mathbf {C} ),

and therefore isomorphic (since α is an automorphism). 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(C) is 2ⁿ. What we have then is the following table:

Classification of complex Clifford algebras
n	Cl_n(C)	Cl^[0] _n(C)	N
even	End(C^N)	End(C^N/2) ⊕ End(C^N/2)	2^n/2
odd	End(C^N) ⊕ End(C^N)	End(C^N)	2^(n−1)/2

The even subalgebra Cl^[0]
_n(C) of Cl_n(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 (when partitioned into 2 × 2 block matrices). When n is odd, the even subalgebra consists of those elements of End(C^N) ⊕ End(C^N) for which the two pieces are identical. Picking either piece then gives an isomorphism with Cl_n(C) ≅ End(C^N).

Complex spinors in even dimension

The classification allows Dirac spinors and Weyl spinors to be defined in even dimension.^[1]

In even dimension n, the Clifford algebra Cl_n(C) is isomorphic to End(C^N), which has its fundamental representation on Δ_n := C^N. 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(C) is isomorphic to End(C^N/2) ⊕ End(C^N/2) and therefore decomposes to the direct sum of two irreducible representation spaces Δ⁺
_n ⊕ Δ⁻
_n, each isomorphic to C^N/2. A left-handed (respectively right-handed) complex Weyl spinor is an element of Δ⁺
_n (respectively, Δ⁻
_n).

Proof of the structure theorem for complex Clifford algebras

The structure theorem is simple to prove inductively. For base cases, Cl₀(C) is simply C ≅ End(C), while Cl₁(C) is given by the algebra C ⊕ C ≅ End(C) ⊕ End(C) by defining the only gamma matrix as γ₁ = (1, −1).

We will also need Cl₂(C) ≅ End(C²). The Pauli matrices can be used to generate the Clifford algebra by setting γ₁ = σ₁, γ₂ = σ₂. The span of the generated algebra is End(C²).

The proof is completed by constructing an isomorphism Cl_n+2(C) ≅ Cl_n(C) ⊗ Cl₂(C). Let γ_a generate Cl_n(C), and ${\tilde {\gamma }}_{a}$ generate Cl₂(C). Let ω = i ${\tilde {\gamma }}_{1}{\tilde {\gamma }}_{2}$ be the chirality element satisfying ω² = 1 and ${\tilde {\gamma }}_{a}$ ω + ω ${\tilde {\gamma }}_{a}$ = 0. These can be used to construct gamma matrices for Cl_n+2(C) by setting Γ_a = γ_a ⊗ ω for 1 ≤ a ≤ n and Γ_a = 1 ⊗ ${\tilde {\gamma }}_{a-n}$ for a = n + 1, n + 2. These can be shown to satisfy the required Clifford algebra and by the universal property of Clifford algebras, there is an isomorphism Cl_n(C) ⊗ Cl₂(C) → Cl_n+2(C).

Finally, in the even case this means by the induction hypothesis Cl_n+2(C) ≅ End(C^N) ⊗ End(C²) ≅ End(C^N+1). 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 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

Given a standard basis {e_i} 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}.

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 ω² = (e₁e₂⋅⋅⋅e_n)(e₁e₂⋅⋅⋅e_n), one can either reverse the order of the second group, yielding sgn(σ)e₁e₂⋅⋅⋅e_ne_n⋅⋅⋅e₂e₁, or apply a perfect shuffle, yielding sgn(σ)e₁e₁e₂e₂⋅⋅⋅e_ne_n. These both have sign (−1)^⌊n/2⌋ = (−1)^n(n−1)/2, which is 4-periodic (proof), and combined with e_ie_i = ±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	Cl_p,q(R) (N = 2^(p+q)/2)	p − q mod 8	ω²	Cl_p,q(R) (N = 2^(p+q−1)/2)
0	+	M_N(R)	1	+	M_N(R) ⊕ M_N(R)
2	−	M_N(R)	3	−	M_N(C)
4	+	M_N/2(H)	5	+	M_N/2(H) ⊕ M_N/2(H)
6	−	M_N/2(H)	7	−	M_N(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 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)

H

3

M₂(C)

M₂²(R)

M₂(C)

H²

4

M₂(H)

M₄(R)

M₂(H)

5

M₂²(H)

M₄(C)

M₄²(R)

M₄(C)

M₂²(H)

M₄(C)

6

M₄(H)

M₈(R)

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)

M₁₆(R)

M₈(H)

M₁₆(R)

ω²

+

−

+

−

+

−

+

−

+

Each real Clifford algebra contains a finite group of signed monomials, constructed by beginning with the sign group {+1, –1} in the scalar subalgebra and repeatedly multiplying by elements of the orthonormal basis. This group is always of order 2^p+q+1, and it has a center either of order 2 for even p + q (the original sign group) or of order 4 for odd p + q (the Klein four-group V₄ = Z₂×Z₂ for p – q = 1 mod 4, the cyclic group Z₄ for p – q = –1 mod 4). Choosing one element from each orbit of the sign group yields a basis for the real vector space underlying the Clifford algebra; this does not generally produce a subgroup of the signed monomial group. Instead, one may quotient out the sign group to obtain an elementary abelian group of unsigned monomial axes.

The signed monomial group does, however, have two additional distinguished normal subgroups: the scalar/pseudoscalar subgroup (which is the center for odd p + q) and the subgroup of even grade (which contains the scalar/pseudoscalar subgroup for even p + q). For all p + q ≥ 1, the full group is an extension of the even subgroup by a group of order 2. For odd p + q, the pseudoscalar is of odd grade and commutes with the basis monomials, and so the full group may be constructed as a central extension of the even subgroup, extending the sign subgroup to the scalar/pseudoscalar subgroup. This case is simply denoted using the ∗ symbol to represent the central product of groups with respect to the Z₂ sign group, even when the center of one side is larger; so if G is of order 2^j and H is of order 2^k, G∗H is of order 2^j+k–1. For example, when p=0 and q=3, the pseudoscalar squares to +1 and the even subgroup is the quaternion group Q₈ (also known as the extraspecial group of order 2³ and type –, 2_–¹⁺²); if we adopt a convention of writing the full signed monomial group in the odd case as (scalar/pseudoscalar subgroup)∗(even subgroup), this is V₄∗Q₈. Two such expressions may have the same product group but represent different splits into central scalar/pseudoscalar subgroup and even subgroup, as in Z₄∗Q₈ (p=3,q=0) and Z₄∗D₈ (p=1,q=2); here D₈ indicates the dihedral group with 8 elements, which some authors denote D₄ (too confusing in this context). Both of these are equivalent to the Pauli group G₁, whose center is of course Z₄ either way; but the different choice of grade involution automorphism results in a different fixed-point subgroup of elements of even grade.

For even p + q, the pseudoscalar is of even grade and anti-commutes with the basis monomials, and therefore lies in the center of the even subgroup (but not in the center of the full group). The even subgroup may be constructed (not necessarily uniquely) as a central extension of a smaller subgroup by this center. In this case, the full group can be obtained by an extension of the order-4 scalar/pseudoscalar subgroup to a non-abelian subgroup of order 8; but the choice of extension depends on the choice of normal subgroup within the even subgroup in the previous central extension construction. (For example, when p=0 and q=4, the even subgroup is the Pauli group G₁, which may be expressed as either D₈∗Z₄ or Q₈∗Z₄; the full group 2₊¹⁺⁴ is obtained by the extension of Z₄ to D₈ in the first case and to Q₈ in the second case.) In order to avoid a misleading implication that one choice or the other is correct, one may denote this nested set of groups as (scalar/pseudoscalar group)⊲(subgroup of even grade)⊲(full group) — in this case, Z₄⊲G₁⊲2₊¹⁺⁴. (The scalar/pseudoscalar group is always the center of the even subgroup, but is called out in this convention as a reminder of its significance to the Clifford algebra structure.) For p + q ≥ 6, p – q = 2 mod 4, the subgroup of even grade is similarly expressible as either 2₊^1+2k∗Z₄ or 2_–^1+2k∗Z₄ for the appropriate value of k; in the absence of a name like G₁, this group is denoted 2_±^1+2k∗Z₄ to emphasize that neither is particularly preferred.

A table of the signed monomial groups for p + q ≤ 8 follows, expressed using these conventions (and truncated to one cycle of p – q mod 8). It contains strictly more information than the matrix classification above, but still omits the reversal anti-automorphism. This anti-automorphism further distinguishes among Clifford algebra structures, as can be easily seen by comparing its effect on the odd parts of Cl_4,0(R) and Cl_0,4(R); it fixes the grade 1 monomials and reverses the sign on the grade 3 monomials, which are of order 2 and 4 respectively in Cl_4,0(R) but of order 4 and 2 respectively in Cl_0,4(R).

	4	3	2	1	0	−1	−2	−3	-4
0					Z₂
1				V₄∗Z₂		Z₄∗Z₂
2			Z₄⊲Z₄⊲D₈		V₄⊲V₄⊲D₈		Z₄⊲Z₄⊲Q₈
3		Z₄∗Q₈		V₄∗D₈		Z₄∗D₈		V₄∗Q₈
4	V₄⊲(Q₈∗V₄)⊲2_–¹⁺⁴		Z₄⊲G₁⊲2₊¹⁺⁴		V₄⊲(D₈∗V₄)⊲2₊¹⁺⁴		Z₄⊲G₁⊲2_–¹⁺⁴		V₄⊲(Q₈∗V₄)⊲2_–¹⁺⁴
5		Z₄∗2_–¹⁺⁴		V₄∗2₊¹⁺⁴		Z₄∗2₊¹⁺⁴		V₄∗2_–¹⁺⁴
6	V₄⊲(2_–¹⁺⁴∗V₄)⊲2_–¹⁺⁶		Z₄⊲(2_±¹⁺⁴∗Z₄)⊲2₊¹⁺⁶		V₄⊲(2₊¹⁺⁴∗V₄)⊲2₊¹⁺⁶		Z₄⊲(2_±¹⁺⁴∗Z₄)⊲2_–¹⁺⁶		V₄⊲(2_–¹⁺⁴∗V₄)⊲2_–¹⁺⁶
7		Z₄∗2_–¹⁺⁶		V₄∗2₊¹⁺⁶		Z₄∗2₊¹⁺⁶		V₄∗2_–¹⁺⁶
8	V₄⊲(2_–¹⁺⁶∗V₄)⊲2_–¹⁺⁸		Z₄⊲(2_±¹⁺⁶∗Z₄)⊲2₊¹⁺⁸		V₄⊲(2₊¹⁺⁶∗V₄)⊲2₊¹⁺⁸		Z₄⊲(2_±¹⁺⁶∗Z₄)⊲2_–¹⁺⁸		V₄⊲(2_–¹⁺⁶∗V₄)⊲2_–¹⁺⁸

One of the facts visible in this table (and easily confirmed from the Clifford algebra construction) is that the signed monomial group for (p,q) is isomorphic to the even subgroup of the signed monomial group for (p,q+1). The grade involution automorphism is inner for even p+q, represented by conjugation with either of the signed pseudoscalars. It is outer for odd p+q, but is represented in the group extension from (p,q) to (p,q+1) by conjugation with the added basis monomial (with either sign).

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} )).

References

↑ Hamilton, Mark J. D. (2017). Mathematical gauge theory : with applications to the standard model of particle physics. Cham, Switzerland. pp. 346–347. ISBN 9783319684383.{{cite book}}: CS1 maint: location missing publisher (link)

Sources

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.

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[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.{{cite book}}: CS1 maint: location missing publisher (link)

[1]

Notation and conventions

Bott periodicity

Complex case

Complex spinors in even dimension

Proof of the structure theorem for complex Clifford algebras

Real case

Classification of quadratic forms

Unit pseudoscalar

Center

Classification

Symmetries

See also

References

Sources