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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Classification of Clifford algebras | 1/7 | https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras | reference | science, encyclopedia | 2026-05-05T09:08:16.280956+00:00 | kb-cron |
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
{\displaystyle 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 Mn(K) or End(Kn). The direct sum of two such identical algebras will be denoted by Mn(K) ⊕ Mn(K), which is isomorphic to Mn(K ⊕ K).
== 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
+
⋯
+
u
n
2
,
{\displaystyle 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 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 Cln(C). There are two separate cases to consider, according to whether n is even or odd. When n is even, the algebra Cln(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 ω2 = 1. Define the operators
P
±
=
1
2
(
1
±
ω
)
.
{\displaystyle 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 Cln(C) into a direct sum of two algebras
C
l
n
(
C
)
=
C
l
n
+
(
C
)
⊕
C
l
n
−
(
C
)
,
{\displaystyle \mathrm {Cl} _{n}(\mathbf {C} )=\mathrm {Cl} _{n}^{+}(\mathbf {C} )\oplus \mathrm {Cl} _{n}^{-}(\mathbf {C} ),}
where
C
l
n
±
(
C
)
=
P
±
C
l
n
(
C
)
.
{\displaystyle \mathrm {Cl} _{n}^{\pm }(\mathbf {C} )=P_{\pm }\mathrm {Cl} _{n}(\mathbf {C} ).}
The algebras Cln±(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 v ↦ −v on the generating space:
α
(
C
l
n
±
(
C
)
)
=
C
l
n
∓
(
C
)
,
{\displaystyle \alpha \left(\mathrm {Cl} _{n}^{\pm }(\mathbf {C} )\right)=\mathrm {Cl} _{n}^{\mp }(\mathbf {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 Cln(C) is 2n. What one obtains is the following table:
The even subalgebra Cl[0]n(C) is (non-canonically) isomorphic to Cln−1(C). When n is even, the even subalgebra can be identified with the block diagonal matrices (after writing elements in 2 × 2 block form). When n is odd, the even subalgebra consists of those elements of End(CN) ⊕ End(CN) for which the two components are equal. Projection onto either factor then gives an isomorphism with Cln0 ≅ End(CN).
=== 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 Cln(C) is isomorphic to End(CN), which has its fundamental representation on Δn := CN. A complex Dirac spinor is an element of Δn. 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 Cln0(C) is isomorphic to End(CN/2) ⊕ End(CN/2) and therefore its spinor module decomposes as the direct sum of two irreducible representation spaces Δ+n ⊕ Δ−n, each isomorphic to CN/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 may be proved inductively. For the base cases, Cl0(C) is simply C ≅ End(C), while Cl1(C) is the algebra C ⊕ C ≅ End(C) ⊕ End(C), obtained by taking the unique generator to be γ1 = (1, −1). One also needs Cl2(C) ≅ End(C2). The Pauli matrices give a concrete realization: if one sets γ1 = σ1 and γ2 = σ2, then these generate a copy of Cl2(C) whose span is all of End(C2). The inductive step is the standard 2-periodicity isomorphism