5.9 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Classification of Clifford algebras | 7/7 | https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras | reference | science, encyclopedia | 2026-05-05T09:08:16.280956+00:00 | kb-cron |
The same viewpoint extends to nonarchimedean local fields. If
K
{\displaystyle K}
is a local field of characteristic not 2, then quadratic spaces over
K
{\displaystyle K}
are classified up to isometry by dimension, determinant, and Clifford invariant; equivalently, one may use dimension, determinant, and Hasse invariant. The Brauer group
Br
(
K
)
{\displaystyle \operatorname {Br} (K)}
has exactly two elements of order dividing 2, namely the split class and the class of the unique quaternion division algebra over
K
{\displaystyle K}
. Accordingly, the Brauer-class part of the Clifford-algebra classification over
K
{\displaystyle K}
is especially simple. If
q
{\displaystyle q}
has even dimension
2
m
{\displaystyle 2m}
, then
Cl
(
q
)
{\displaystyle \operatorname {Cl} (q)}
is isomorphic either to
M
2
m
(
K
)
{\displaystyle M_{2^{m}}(K)}
or to
M
2
m
−
1
(
D
)
{\displaystyle M_{2^{m-1}}(D)}
, where
D
{\displaystyle D}
is the quaternion division algebra over
K
{\displaystyle K}
. If
q
{\displaystyle q}
has odd dimension
2
m
+
1
{\displaystyle 2m+1}
, then
Cl
0
(
q
)
{\displaystyle \operatorname {Cl} ^{0}(q)}
is isomorphic either to
M
2
m
(
K
)
{\displaystyle M_{2^{m}}(K)}
or to
M
2
m
−
1
(
D
)
{\displaystyle M_{2^{m-1}}(D)}
; the full Clifford algebra is then obtained from
Cl
0
(
q
)
{\displaystyle \operatorname {Cl} ^{0}(q)}
by adjoining its quadratic étale center. In practice one diagonalizes
q
{\displaystyle q}
, computes the Hilbert-symbol product
s
(
q
)
=
∏
i
<
j
(
a
i
,
a
j
)
{\displaystyle s(q)=\prod _{i<j}(a_{i},a_{j})}
, and then obtains
c
(
q
)
{\displaystyle c(q)}
from the same formula relating Hasse and Clifford invariants.
=== Characteristic two === The preceding discussion assumed that the ground field has characteristic different from 2. In characteristic 2, the polar form of a quadratic form is alternating, so a nonsingular quadratic space must have even dimension. Odd-dimensional forms are still important, but they are treated using the theory of regular (or “1/2-regular”) quadratic forms rather than the nonsingular theory. For this reason, the characteristic-2 theory is usually formulated not only in terms of quadratic forms, but in terms of quadratic pairs on central simple algebras. In that setting the discriminant and the even Clifford algebra are defined for quadratic pairs and play the role of the corresponding invariants in characteristic different from 2. Accordingly, there is no direct analogue of the real-signature classification table in characteristic 2 without first reformulating the theory in this language.
== See also == Clifford algebra Dirac algebra Cl1,3(C) Pauli algebra Cl3,0(R) Spacetime algebra Cl1,3(R) Clifford module Spin representation
== References ==
== Sources == Atiyah, Michael F.; Bott, Raoul; Shapiro, Arnold (1964). "Clifford modules". Topology. 3 (Suppl. 1): 3–38. doi:10.1016/0040-9383(64)90003-5. Bott, Raoul (1970). "The periodicity theorem for the classical groups and some of its applications". Advances in Mathematics. 4 (3): 353–411. doi:10.1016/0001-8708(70)90030-7. Budinich, Paolo; Trautman, Andrzej (1988). The Spinorial Chessboard. Springer Verlag. ISBN 978-3-540-19078-3. Lam, T. Y. (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. Lawson, H. Blaine; Michelsohn, Marie-Louise (2016). Spin Geometry. Princeton Mathematical Series. Vol. 38. Princeton University Press. ISBN 978-1-4008-8391-2. Porteous, Ian R. (1995). Clifford Algebras and the Classical Groups. Cambridge Studies in Advanced Mathematics. Vol. 50. Cambridge University Press. ISBN 978-0-521-55177-9.