17 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Classification of Clifford algebras | 2/7 | https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras | reference | science, encyclopedia | 2026-05-05T09:08:16.280956+00:00 | 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} ).}