--- title: "Classification of Clifford algebras" chunk: 2/7 source: "https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras" category: "reference" tags: "science, encyclopedia" date_saved: "2026-05-05T09:08:16.280956+00:00" instance: "kb-cron" --- C l n + 2 ( C ) ≅ C l n ( C ) ⊗ C l 2 ( C ) . {\displaystyle \mathrm {Cl} _{n+2}(\mathbf {C} )\cong \mathrm {Cl} _{n}(\mathbf {C} )\otimes \mathrm {Cl} _{2}(\mathbf {C} ).} To construct it, let γa generate Cln(C), and let γ ~ 1 , γ ~ 2 {\displaystyle {\tilde {\gamma }}_{1},{\tilde {\gamma }}_{2}} generate Cl2(C). Let ω = i\tilde\gamma_1 \tilde\gamma_2 be the chirality element in Cl2(C), so that ω2 = 1 and each γ ~ a {\displaystyle {\tilde {\gamma }}_{a}} anticommutes with ω. Then one obtains generators for Cln+2(C) by setting Γ a = γ a ⊗ ω ( 1 ≤ a ≤ n ) , {\displaystyle \Gamma _{a}=\gamma _{a}\otimes \omega \qquad (1\leq a\leq n),} Γ n + 1 = 1 ⊗ γ ~ 1 , Γ n + 2 = 1 ⊗ γ ~ 2 . {\displaystyle \Gamma _{n+1}=1\otimes {\tilde {\gamma }}_{1},\qquad \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 Cln(C) ⊗ Cl2(C) \to Cln+2(C). Finally, if n is even and Cln(C) ≅ End(CN), then C l n + 2 ( C ) ≅ End ⁡ ( C N ) ⊗ End ⁡ ( C 2 ) ≅ End ⁡ ( C 2 N ) . {\displaystyle \mathrm {Cl} _{n+2}(\mathbf {C} )\cong \operatorname {End} (\mathbf {C} ^{N})\otimes \operatorname {End} (\mathbf {C} ^{2})\cong \operatorname {End} (\mathbf {C} ^{2N}).} Since 2N = 2(n+2)/2, this gives the even-dimensional case in dimension n+2. 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 C l n + 2 ( C ) ≅ C l n ( C ) ⊗ C l 2 ( C ) , {\displaystyle \mathrm {Cl} _{n+2}(\mathbf {C} )\cong \mathrm {Cl} _{n}(\mathbf {C} )\otimes \mathrm {Cl} _{2}(\mathbf {C} ),} and the identification of the even subalgebra C l n + 1 ( C ) 0 ≅ C l n ( C ) . {\displaystyle \mathrm {Cl} _{n+1}(\mathbf {C} )^{0}\cong \mathrm {Cl} _{n}(\mathbf {C} ).} See, for example, Porteous (1995) or Lawson & Michelsohn (2016). For the base cases, one has C l 0 ( C ) ≅ C {\displaystyle \mathrm {Cl} _{0}(\mathbf {C} )\cong \mathbf {C} } and C l 1 ( C ) ≅ C ⊕ C . {\displaystyle \mathrm {Cl} _{1}(\mathbf {C} )\cong \mathbf {C} \oplus \mathbf {C} .} The first is immediate. For the second, if e {\displaystyle e} is the generator with e 2 = 1 {\displaystyle e^{2}=1} , then P ± = 1 2 ( 1 ± e ) {\displaystyle P_{\pm }={\frac {1}{2}}(1\pm e)} are central orthogonal idempotents with P + + P − = 1 {\displaystyle P_{+}+P_{-}=1} , so the algebra splits as the direct sum of the two one-dimensional ideals C P + {\displaystyle \mathbf {C} P_{+}} and C P − {\displaystyle \mathbf {C} P_{-}} . Next, one needs the two-dimensional case C l 2 ( C ) ≅ M 2 ( C ) . {\displaystyle \mathrm {Cl} _{2}(\mathbf {C} )\cong M_{2}(\mathbf {C} ).} A concrete realization is obtained from the Pauli matrices: γ 1 = σ 1 = ( 0 1 1 0 ) , γ 2 = σ 2 = ( 0 − i i 0 ) . {\displaystyle \gamma _{1}=\sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\qquad \gamma _{2}=\sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}.} These satisfy γ i γ j + γ j γ i = 2 δ i j {\displaystyle \gamma _{i}\gamma _{j}+\gamma _{j}\gamma _{i}=2\delta _{ij}} , so by the universal property they define a homomorphism C l 2 ( C ) → M 2 ( C ) {\displaystyle \mathrm {Cl} _{2}(\mathbf {C} )\to M_{2}(\mathbf {C} )} . Since the image contains 1 , γ 1 , γ 2 , γ 1 γ 2 {\displaystyle 1,\gamma _{1},\gamma _{2},\gamma _{1}\gamma _{2}} , it has dimension 4 and hence is all of M 2 ( C ) {\displaystyle M_{2}(\mathbf {C} )} . The key step is the 2-periodicity isomorphism. Let γ 1 , … , γ n {\displaystyle \gamma _{1},\dots ,\gamma _{n}} generate C l n ( C ) {\displaystyle \mathrm {Cl} _{n}(\mathbf {C} )} , let γ ~ 1 , γ ~ 2 {\displaystyle {\tilde {\gamma }}_{1},{\tilde {\gamma }}_{2}} generate C l 2 ( C ) {\displaystyle \mathrm {Cl} _{2}(\mathbf {C} )} , and set ω = i γ ~ 1 γ ~ 2 . {\displaystyle \omega =i{\tilde {\gamma }}_{1}{\tilde {\gamma }}_{2}.} Then ω 2 = 1 {\displaystyle \omega ^{2}=1} and ω {\displaystyle \omega } anticommutes with both γ ~ 1 {\displaystyle {\tilde {\gamma }}_{1}} and γ ~ 2 {\displaystyle {\tilde {\gamma }}_{2}} . Define elements of C l n ( C ) ⊗ C l 2 ( C ) {\displaystyle \mathrm {Cl} _{n}(\mathbf {C} )\otimes \mathrm {Cl} _{2}(\mathbf {C} )} by Γ a = γ a ⊗ ω ( 1 ≤ a ≤ n ) , {\displaystyle \Gamma _{a}=\gamma _{a}\otimes \omega \qquad (1\leq a\leq n),} Γ n + 1 = 1 ⊗ γ ~ 1 , Γ n + 2 = 1 ⊗ γ ~ 2 . {\displaystyle \Gamma _{n+1}=1\otimes {\tilde {\gamma }}_{1},\qquad \Gamma _{n+2}=1\otimes {\tilde {\gamma }}_{2}.} Because ω 2 = 1 {\displaystyle \omega ^{2}=1} and ω {\displaystyle \omega } anticommutes with the generators of C l 2 ( C ) {\displaystyle \mathrm {Cl} _{2}(\mathbf {C} )} , the elements Γ 1 , … , Γ n + 2 {\displaystyle \Gamma _{1},\dots ,\Gamma _{n+2}} satisfy the Clifford relations for the standard quadratic form on C n + 2 {\displaystyle \mathbf {C} ^{n+2}} . Therefore the universal property gives a homomorphism C l n + 2 ( C ) → C l n ( C ) ⊗ C l 2 ( C ) . {\displaystyle \mathrm {Cl} _{n+2}(\mathbf {C} )\to \mathrm {Cl} _{n}(\mathbf {C} )\otimes \mathrm {Cl} _{2}(\mathbf {C} ).}