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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Classification of Clifford algebras | 4/7 | https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras | reference | science, encyclopedia | 2026-05-05T09:08:16.280956+00:00 | kb-cron |
each of which is central simple and so isomorphic to a matrix algebra over R or H. If n is odd and ω2 = −1 (equivalently, if p − q ≡ −1 (mod 4)) then the center of Clp,q(R) is isomorphic to C, and the algebra may be regarded as a complex central simple algebra; hence it is isomorphic to a matrix algebra over C.
=== Classification === All told there are three properties which determine the class of the algebra Clp,q(R):
signature mod 2: n is even/odd, determining whether the algebra is central simple or not; signature mod 4: ω2 = ±1, determining in the odd-dimensional case whether the center is R ⊕ R or C; signature mod 8: the Brauer class of the algebra (n even) or of the even subalgebra (n odd), determining whether the central simple factor is split or quaternionic. Each of these properties depends only on the signature p − q modulo 8. The complete classification table is given below. The size of the matrices is determined by the requirement that Clp,q(R) have dimension 2p+q.
It may be seen that of all matrix-ring types mentioned, there is only one type shared by complex and real algebras: the type M2m(C). For example, Cl2(C) and Cl3,0(R) are both isomorphic to M2(C). It is important to distinguish the categories in which these isomorphisms are taken: Cl2(C) is classified as a C-algebra, whereas Cl3,0(R) is classified as an R-algebra. Thus the two are R-algebra isomorphic, but not canonically as complex algebras. A table of this classification for p + q ≤ 8 follows. Here p + q runs vertically and p − q runs horizontally (e.g. the algebra Cl1,3(R) ≅ M2(H) is found in row 4, column −2).
=== Symmetries === There is a tangled web of symmetries and relationships in the above table. Most importantly, one has the standard real-periodicity isomorphisms
Cl
p
+
1
,
q
+
1
(
R
)
≅
Cl
p
,
q
(
R
)
⊗
M
2
(
R
)
,
Cl
q
,
p
+
2
(
R
)
≅
Cl
p
,
q
(
R
)
⊗
H
,
Cl
q
+
2
,
p
(
R
)
≅
Cl
p
,
q
(
R
)
⊗
M
2
(
R
)
.
{\displaystyle {\begin{aligned}\operatorname {Cl} _{p+1,q+1}(\mathbf {R} )&\cong \operatorname {Cl} _{p,q}(\mathbf {R} )\otimes \operatorname {M} _{2}(\mathbf {R} ),\\\operatorname {Cl} _{q,p+2}(\mathbf {R} )&\cong \operatorname {Cl} _{p,q}(\mathbf {R} )\otimes \mathbf {H} ,\\\operatorname {Cl} _{q+2,p}(\mathbf {R} )&\cong \operatorname {Cl} _{p,q}(\mathbf {R} )\otimes \operatorname {M} _{2}(\mathbf {R} ).\end{aligned}}}
In terms of the table, the first rule says that going down one step from the Clifford algebra
Cl
p
,
q
(
R
)
{\displaystyle \operatorname {Cl} _{p,q}(\mathbf {R} )}
yields
Cl
p
+
1
,
q
+
1
(
R
)
{\displaystyle \operatorname {Cl} _{p+1,q+1}(\mathbf {R} )}
, which consists of
2
×
2
{\displaystyle 2\times 2}
matrices over
Cl
p
,
q
(
R
)
{\displaystyle \operatorname {Cl} _{p,q}(\mathbf {R} )}
. The other two rules imply that
Cl
p
+
4
,
q
(
R
)
≅
Cl
p
,
q
+
4
(
R
)
{\displaystyle \operatorname {Cl} _{p+4,q}(\mathbf {R} )\cong \operatorname {Cl} _{p,q+4}(\mathbf {R} )}
and from these one obtains Bott periodicity in the form
Cl
p
+
8
,
q
(
R
)
≅
Cl
p
+
4
,
q
+
4
(
R
)
≅
Cl
p
,
q
+
8
(
R
)
≅
M
16
(
Cl
p
,
q
(
R
)
)
.
{\displaystyle \operatorname {Cl} _{p+8,q}(\mathbf {R} )\cong \operatorname {Cl} _{p+4,q+4}(\mathbf {R} )\cong \operatorname {Cl} _{p,q+8}(\mathbf {R} )\cong \operatorname {M} _{16}(\operatorname {Cl} _{p,q}(\mathbf {R} )).}
Furthermore, if the signature satisfies p − q ≡ 1 (mod 4) then
Cl
p
+
k
,
q
(
R
)
≅
Cl
p
,
q
+
k
(
R
)
.
{\displaystyle \operatorname {Cl} _{p+k,q}(\mathbf {R} )\cong \operatorname {Cl} _{p,q+k}(\mathbf {R} ).}
This says that the table is symmetric about columns where
p
−
q
=
{\displaystyle p-q=}
..., −7, −3, 1, 5, 9,....
=== Bott periodicity === The 8-fold periodicity over the real numbers is part of Bott periodicity, the corresponding periodicity for the homotopy groups of the stable orthogonal group; similarly, over the complex numbers one has 2-fold periodicity for the stable unitary group. In Bott's geometric description, the relevant loop spaces are modeled by successive quotients of the classical groups, which are compact symmetric spaces. In stable group theory, loop spaces enter because Bott periodicity identifies the stable classical groups, up to homotopy, with iterated loop spaces of the corresponding classifying spaces. The matching 2-fold and 8-fold algebraic periodicities of complex and real Clifford algebras are part of the same picture.
=== Failure of symmetry under swapping p and q === Note that in the real classification, in general,
Cl
p
,
q
(
R
)
≇
Cl
q
,
p
(
R
)
.
{\displaystyle \operatorname {Cl} _{p,q}(\mathbf {R} )\not \cong \operatorname {Cl} _{q,p}(\mathbf {R} ).}
In the sign convention used in this article, exchanging p and q replaces the quadratic form by its negative, so it sends the signature difference p − q to q − p = −(p − q). Since the isomorphism class of the real Clifford algebra is determined by p − q (mod 8), one should compare the entries in the classification table for residues d and −d modulo 8. These entries agree only when d ≡ −d (mod 8), that is, only when d ≡ 0 or 4 (mod 8). In all other congruence classes, the algebras are of different types. For example,