Classification of Clifford algebras
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In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified as rings. 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.
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, 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 n × n matrices with entries in the division algebra K by M_{n}(K) or End(K^{n}). 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
[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 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_{k}^{2} = +(iu_{k})^{2} and so positive or negative terms are equivalent. We will denote the Clifford algebra on C^{n} 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 ω^{2} = 1. Define the operators
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
where
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):
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 2^{n}. What we have then is the following table:
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}^{[0]}(C) ≅ End(C^{N}).
Complex spinors in even dimension
[edit]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_{n}^{0}(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
[edit]The structure theorem is simple to prove inductively. For base cases, Cl_{0}(C) is simply C ≅ End(C), while Cl_{1}(C) is given by the algebra C ⊕ C ≅ End(C) ⊕ End(C) by defining the only gamma matrix as γ_{1} = (1, −1).
We will also need Cl_{2}(C) ≅ End(C^{2}). The Pauli matrices can be used to generate the Clifford algebra by setting γ_{1} = σ_{1}, γ_{2} = σ_{2}. The span of the generated algebra is End(C^{2}).
The proof is completed by constructing an isomorphism Cl_{n+2}(C) ≅ Cl_{n}(C) ⊗ Cl_{2}(C). Let γ_{a} generate Cl_{n}(C), and generate Cl_{2}(C). Let ω = i be the chirality element satisfying ω^{2} = 1 and ω + ω = 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 ⊗ 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) ⊗ Cl_{2}(C) → Cl_{n+2}(C).
Finally, in the even case this means by the induction hypothesis Cl_{n+2}(C) ≅ End(C^{N}) ⊗ End(C^{2}) ≅ End(C^{N+1}). 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 an isotropic quadratic 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 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
[edit]Given a standard basis {e_{i}} as defined in the previous subsection, the unit pseudoscalar in Cl_{p,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 ω^{2} = (e_{1}e_{2}⋅⋅⋅e_{n})(e_{1}e_{2}⋅⋅⋅e_{n}), one can either reverse the order of the second group, yielding sgn(σ)e_{1}e_{2}⋅⋅⋅e_{n}e_{n}⋅⋅⋅e_{2}e_{1}, or apply a perfect shuffle, yielding sgn(σ)e_{1}e_{1}e_{2}e_{2}⋅⋅⋅e_{n}e_{n}. These both have sign (−1)^{⌊n/2⌋} = (−1)^{n(n−1)/2}, which is 4-periodic (proof), and combined with e_{i}e_{i} = ±1, 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.
Center
[edit]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 ω^{2} = +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
each of which is central simple and so isomorphic to matrix algebra over R or H.
If n is odd and ω^{2} = −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
[edit]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: ω^{2} = ±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 | ω^{2} | Cl_{p,q}(R) (N = 2^{(p+q)/2}) |
p − q mod 8 | ω^{2} | 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_{2m}(C). For example, Cl_{2}(C) and Cl_{3,0}(R) are both determined to be M_{2}(C). It is important to note that there is a difference in the classifying isomorphisms used. Since the Cl_{2}(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, Cl_{2}(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) ≅ M_{2}(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 | R^{2} | C | |||||||||||||||
2 | M_{2}(R) | M_{2}(R) | H | ||||||||||||||
3 | M_{2}(C) | M^{2} _{2}(R) |
M_{2}(C) | H^{2} | |||||||||||||
4 | M_{2}(H) | M_{4}(R) | M_{4}(R) | M_{2}(H) | M_{2}(H) | ||||||||||||
5 | M^{2} _{2}(H) |
M_{4}(C) | M^{2} _{4}(R) |
M_{4}(C) | M^{2} _{2}(H) |
M_{4}(C) | |||||||||||
6 | M_{4}(H) | M_{4}(H) | M_{8}(R) | M_{8}(R) | M_{4}(H) | M_{4}(H) | M_{8}(R) | ||||||||||
7 | M_{8}(C) | M^{2} _{4}(H) |
M_{8}(C) | M^{2} _{8}(R) |
M_{8}(C) | M^{2} _{4}(H) |
M_{8}(C) | M^{2} _{8}(R) |
|||||||||
8 | M_{16}(R) | M_{8}(H) | M_{8}(H) | M_{16}(R) | M_{16}(R) | M_{8}(H) | M_{8}(H) | M_{16}(R) | M_{16}(R) | ||||||||
ω^{2} | + | − | − | + | + | − | − | + | + | − | − | + | + | − | − | + | + |
Symmetries
[edit]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 p − q ≡ 1 (mod 4) then
(The table is symmetric about columns with signature ..., −7, −3, 1, 5, ...)
Thus if the signature satisfies p − q ≡ 1 (mod 4),
See also
[edit]- Dirac algebra Cl_{1,3}(C)
- Pauli algebra Cl_{3,0}(R)
- Spacetime algebra Cl_{1,3}(R)
- Clifford module
- Spin representation
References
[edit]- ^ Hamilton, Mark J. D. (2017). Mathematical gauge theory : with applications to the standard model of particle physics. Cham, Switzerland. pp. 346–347. ISBN 9783319684383.
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Sources
[edit]- 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.