14 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Classification of Clifford algebras | 3/7 | https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras | reference | science, encyclopedia | 2026-05-05T09:08:16.280956+00:00 | kb-cron |
Both algebras have dimension
2
n
+
2
{\displaystyle 2^{n+2}}
, so this homomorphism is an isomorphism. It follows by induction on
m
{\displaystyle m}
that
C
l
2
m
(
C
)
≅
C
l
0
(
C
)
⊗
C
l
2
(
C
)
⊗
m
≅
M
2
m
(
C
)
.
{\displaystyle \mathrm {Cl} _{2m}(\mathbf {C} )\cong \mathrm {Cl} _{0}(\mathbf {C} )\otimes \mathrm {Cl} _{2}(\mathbf {C} )^{\otimes m}\cong M_{2^{m}}(\mathbf {C} ).}
Indeed, the case
m
=
0
{\displaystyle m=0}
is
C
l
0
(
C
)
≅
C
{\displaystyle \mathrm {Cl} _{0}(\mathbf {C} )\cong \mathbf {C} }
, and each application of 2-periodicity tensors with
C
l
2
(
C
)
≅
M
2
(
C
)
{\displaystyle \mathrm {Cl} _{2}(\mathbf {C} )\cong M_{2}(\mathbf {C} )}
, doubling the matrix size. For odd dimension, let
n
=
2
m
+
1
{\displaystyle n=2m+1}
. The volume element
ω
=
e
1
e
2
⋯
e
n
{\displaystyle \omega =e_{1}e_{2}\cdots e_{n}}
is central because
n
{\displaystyle n}
is odd, and over
C
{\displaystyle \mathbf {C} }
it may be rescaled so that
ω
2
=
1
{\displaystyle \omega ^{2}=1}
. Hence
P
±
=
1
2
(
1
±
ω
)
{\displaystyle P_{\pm }={\frac {1}{2}}(1\pm \omega )}
are central orthogonal idempotents, giving a decomposition
C
l
2
m
+
1
(
C
)
=
C
l
2
m
+
1
+
(
C
)
⊕
C
l
2
m
+
1
−
(
C
)
.
{\displaystyle \mathrm {Cl} _{2m+1}(\mathbf {C} )=\mathrm {Cl} _{2m+1}^{+}(\mathbf {C} )\oplus \mathrm {Cl} _{2m+1}^{-}(\mathbf {C} ).}
On the other hand, the even subalgebra is isomorphic to
C
l
2
m
(
C
)
{\displaystyle \mathrm {Cl} _{2m}(\mathbf {C} )}
, and projection onto either summand identifies each simple factor with that even subalgebra. Since
C
l
2
m
(
C
)
≅
M
2
m
(
C
)
,
{\displaystyle \mathrm {Cl} _{2m}(\mathbf {C} )\cong M_{2^{m}}(\mathbf {C} ),}
one obtains
C
l
2
m
+
1
(
C
)
≅
M
2
m
(
C
)
⊕
M
2
m
(
C
)
.
{\displaystyle \mathrm {Cl} _{2m+1}(\mathbf {C} )\cong M_{2^{m}}(\mathbf {C} )\oplus M_{2^{m}}(\mathbf {C} ).}
This proves the classification:
C
l
2
m
(
C
)
≅
M
2
m
(
C
)
,
C
l
2
m
+
1
(
C
)
≅
M
2
m
(
C
)
⊕
M
2
m
(
C
)
.
{\displaystyle \mathrm {Cl} _{2m}(\mathbf {C} )\cong M_{2^{m}}(\mathbf {C} ),\qquad \mathrm {Cl} _{2m+1}(\mathbf {C} )\cong M_{2^{m}}(\mathbf {C} )\oplus M_{2^{m}}(\mathbf {C} ).}
Equivalently, the complex Clifford algebras are 2-periodic, and the even subalgebra of
C
l
n
+
1
(
C
)
{\displaystyle \mathrm {Cl} _{n+1}(\mathbf {C} )}
is isomorphic to
C
l
n
(
C
)
{\displaystyle \mathrm {Cl} _{n}(\mathbf {C} )}
.
== 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 a diagonal form
Q
(
u
)
=
u
1
2
+
⋯
+
u
p
2
−
u
p
+
1
2
−
⋯
−
u
p
+
q
2
{\displaystyle 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 Rp,q. The Clifford algebra on Rp,q is denoted Clp,q(R). A standard orthonormal basis {ei} for Rp,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 {ei} as defined in the previous subsection, the unit pseudoscalar in Clp,q(R) is defined as
ω
=
e
1
e
2
⋯
e
n
.
{\displaystyle \omega =e_{1}e_{2}\cdots e_{n}.}
It is the Clifford-algebra analogue of the volume element. To compute the square ω2 = (e1e2\cdots en)(e1e2\cdots en), one may reverse the order of the second factor and then commute equal basis vectors together. This introduces the sign (−1)n(n−1)/2, and since ei2 = +1 for i \le p and ei2 = -1 for the remaining q basis vectors, one obtains
ω
2
=
(
−
1
)
n
(
n
−
1
)
2
(
−
1
)
q
=
(
−
1
)
(
p
−
q
)
(
p
−
q
−
1
)
2
=
{
+
1
p
−
q
≡
0
,
1
(
mod
4
)
−
1
p
−
q
≡
2
,
3
(
mod
4
)
.
{\displaystyle \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{\pmod {4}}\\-1&p-q\equiv 2,3{\pmod {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 Clp,q(R) is central simple and so isomorphic to a matrix algebra over R or H by the Artin–Wedderburn theorem. If n is odd then the algebra is no longer central simple: its center contains the pseudoscalar 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 Clp,q(R) decomposes into a direct sum of isomorphic algebras
Cl
p
,
q
(
R
)
=
Cl
p
,
q
+
(
R
)
⊕
Cl
p
,
q
−
(
R
)
,
{\displaystyle \operatorname {Cl} _{p,q}(\mathbf {R} )=\operatorname {Cl} _{p,q}^{+}(\mathbf {R} )\oplus \operatorname {Cl} _{p,q}^{-}(\mathbf {R} ),}