13 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Classification of Clifford algebras | 6/7 | https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras | reference | science, encyclopedia | 2026-05-05T09:08:16.280956+00:00 | kb-cron |
and hence also
Spin
(
3
,
1
)
≅
SL
2
(
C
)
{\displaystyle \operatorname {Spin} (3,1)\cong \operatorname {SL} _{2}(\mathbf {C} )}
.
== General fields == Let F be a field of characteristic not 2, and let
q
{\displaystyle q}
be a nondegenerate quadratic form on a finite-dimensional F-vector space
V
{\displaystyle V}
. Over such a field, the classification of Clifford algebras is naturally expressed in terms of the center and a Brauer class rather than by a periodic matrix table. If
dim
V
=
2
m
{\displaystyle \dim V=2m}
is even, then the full Clifford algebra
Cl
(
V
,
q
)
{\displaystyle \operatorname {Cl} (V,q)}
is a central simple algebra over
F
{\displaystyle F}
. Its Brauer class
c
(
q
)
:=
[
Cl
(
V
,
q
)
]
∈
Br
(
F
)
{\displaystyle c(q):=[\operatorname {Cl} (V,q)]\in \operatorname {Br} (F)}
is called the Clifford invariant of
q
{\displaystyle q}
. The center of the even Clifford algebra
Cl
0
(
V
,
q
)
{\displaystyle \operatorname {Cl} ^{0}(V,q)}
is the quadratic étale
F
{\displaystyle F}
-algebra
Z
(
q
)
=
F
[
x
]
/
(
x
2
−
δ
(
q
)
)
{\displaystyle Z(q)=F[x]/(x^{2}-\delta (q))}
, where
δ
(
q
)
=
(
−
1
)
m
det
(
q
)
{\displaystyle \delta (q)=(-1)^{m}\det(q)}
is the signed discriminant of
q
{\displaystyle q}
. Thus
Z
(
q
)
{\displaystyle Z(q)}
is either a separable quadratic extension field of
F
{\displaystyle F}
or the split algebra
F
⊕
F
{\displaystyle F\oplus F}
. If
dim
V
=
2
m
+
1
{\displaystyle \dim V=2m+1}
is odd, then the even Clifford algebra
Cl
0
(
V
,
q
)
{\displaystyle \operatorname {Cl} ^{0}(V,q)}
is central simple over
F
{\displaystyle F}
. In this case the relevant Clifford invariant is
c
(
q
)
:=
[
Cl
0
(
V
,
q
)
]
∈
Br
(
F
)
,
{\displaystyle c(q):=[\operatorname {Cl} ^{0}(V,q)]\in \operatorname {Br} (F),}
while the full Clifford algebra has center
Z
(
q
)
{\displaystyle Z(q)}
and satisfies
Cl
(
V
,
q
)
≅
Cl
0
(
V
,
q
)
⊗
F
Z
(
q
)
.
{\displaystyle \operatorname {Cl} (V,q)\cong \operatorname {Cl} ^{0}(V,q)\otimes _{F}Z(q).}
Thus, in odd dimension, the isomorphism class of the full Clifford algebra is determined by the quadratic étale center
Z
(
q
)
{\displaystyle Z(q)}
together with the Brauer class
c
(
q
)
{\displaystyle c(q)}
. An explicit computation of
c
(
q
)
{\displaystyle c(q)}
may be made after diagonalizing
q
≅
⟨
a
1
,
…
,
a
n
⟩
.
{\displaystyle q\cong \langle a_{1},\dots ,a_{n}\rangle .}
The associated Hasse invariant is the 2-torsion Brauer class
s
(
q
)
=
∏
1
≤
i
<
j
≤
n
(
a
i
,
a
j
)
∈
Br
(
F
)
[
2
]
,
{\displaystyle s(q)=\prod _{1\leq i<j\leq n}(a_{i},a_{j})\in \operatorname {Br} (F)[2],}
where
(
a
i
,
a
j
)
{\displaystyle (a_{i},a_{j})}
denotes the class of the quaternion algebra generated by
i
,
j
{\displaystyle i,j}
with
i
2
=
a
i
{\displaystyle i^{2}=a_{i}}
,
j
2
=
a
j
{\displaystyle j^{2}=a_{j}}
, and
i
j
=
−
j
i
{\displaystyle ij=-ji}
. The Clifford invariant is obtained from the Hasse invariant by a universal correction depending only on
n
mod
8
{\displaystyle n{\bmod {8}}}
:
c
(
q
)
=
s
(
q
)
⋅
{
1
,
n
≡
1
,
2
(
mod
8
)
,
(
−
1
,
−
det
q
)
,
n
≡
3
,
4
(
mod
8
)
,
(
−
1
,
−
1
)
,
n
≡
5
,
6
(
mod
8
)
,
(
−
1
,
det
q
)
,
n
≡
0
,
7
(
mod
8
)
.
{\displaystyle c(q)=s(q)\cdot {\begin{cases}1,&n\equiv 1,2{\pmod {8}},\\(-1,-\det q),&n\equiv 3,4{\pmod {8}},\\(-1,-1),&n\equiv 5,6{\pmod {8}},\\(-1,\det q),&n\equiv 0,7{\pmod {8}}.\end{cases}}}
Here
det
q
{\displaystyle \det q}
is the determinant of a Gram matrix, viewed in
F
×
/
F
×
2
{\displaystyle F^{\times }/F^{\times 2}}
. In this sense, the Brauer class of the relevant Clifford algebra is the standard Clifford invariant of the quadratic form. Over
R
{\displaystyle \mathbf {R} }
, this recovers the usual real classification table above. The Brauer group
Br
(
R
)
{\displaystyle \operatorname {Br} (\mathbf {R} )}
has two elements, represented by the split class and the class of the quaternion algebra
H
{\displaystyle \mathbf {H} }
. For a diagonal form of signature
(
p
,
q
)
{\displaystyle (p,q)}
, the Hasse invariant is
s
(
q
)
=
[
H
]
(
q
2
)
,
{\displaystyle s(q)=[\mathbf {H} ]^{\binom {q}{2}},}
since over
R
{\displaystyle \mathbf {R} }
the quaternion class
(
a
,
b
)
{\displaystyle (a,b)}
is nontrivial exactly when both
a
{\displaystyle a}
and
b
{\displaystyle b}
are negative. The formula above therefore determines abstractly whether the relevant central simple algebra is split or quaternionic. In even dimension this yields matrix algebras over
R
{\displaystyle \mathbf {R} }
or
H
{\displaystyle \mathbf {H} }
; in odd dimension one combines the same Brauer-class computation for
Cl
0
(
V
,
q
)
{\displaystyle \operatorname {Cl} ^{0}(V,q)}
with the center
Z
(
q
)
{\displaystyle Z(q)}
, which is either
R
×
R
{\displaystyle \mathbf {R} \times \mathbf {R} }
or
C
{\displaystyle \mathbf {C} }
. When
Z
(
q
)
≅
C
{\displaystyle Z(q)\cong \mathbf {C} }
, the full Clifford algebra is a complex matrix algebra, because
C
⊗
R
H
≅
M
2
(
C
)
.
{\displaystyle \mathbf {C} \otimes _{\mathbf {R} }\mathbf {H} \cong M_{2}(\mathbf {C} ).}