Clifford algebra
A finite-dimensional associative algebra over a commutative ring; it was first investigated by W. Clifford in 1876. Let be a commutative ring with an identity, let
be a free
-module and let
be a quadratic form on
. By the Clifford algebra of the quadratic form
(or of the pair
) one means the quotient algebra
of the tensor algebra
of the
-module
by the two-sided ideal generated by the elements of the form
, where
. Elements of
are identified with their corresponding cosets in
. For any
one has
, where
is the symmetric bilinear form associated with
.
For the case of the null quadratic form ,
is the same as the exterior algebra
of
. If
, the field of real numbers, and
is a non-degenerate quadratic form on the
-dimensional vector space
over
, then
is the algebra
of alternions, where
is the number of positive squares in the canonical form of
(cf. Alternion).
Let be a basis of the
-module
. Then the elements
form a basis of the
-module
. In particular,
is a free
-module of rank
. If in addition the
are orthogonal with respect to
, then
can be presented as a
-algebra with generators
and relations
and
. The submodule of
generated by products of an even number of elements of
forms a subalgebra of
, denoted by
.
Suppose that is a field and that the quadratic form
is non-degenerate. For even
,
is a central simple algebra over
of dimension
, the subalgebra
is separable, and its centre
has dimension 2 over
. If
is algebraically closed, then when
is even
is a matrix algebra and
is a product of two matrix algebras. (If, on the other hand,
is odd, then
is a matrix algebra and
is a product of two matrix algebras.)
The invertible elements of
(or of
) for which
form the Clifford group
(or the special Clifford group
) of the quadratic form
. The restriction of the transformation
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to the subspace defines a homomorphism
, where
is the orthogonal group of the quadratic form
. The kernel
consists of the invertible elements of the algebra
and
. If
is even, then
and
is a subgroup of index 2 in
, which in the case when
is not of characteristic 2, is the same as the special orthogonal group
. If
is odd, then
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Let be the anti-automorphism of
induced by the anti-automorphism
![]() |
of the tensor algebra . The group
![]() |
is called the spinor group of the quadratic form (or of the Clifford algebra
).
The homomorphism has kernel
. If
or
and
is positive definite, then
and
coincides with the classical spinor group.
References
[1] | N. Bourbaki, "Elements of mathematics" , Addison-Wesley (1966–1977) (Translated from French) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929 MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210 Zbl 05948094 Zbl 1145.17002 Zbl 1145.17001 Zbl 1116.28002 Zbl 1108.26003 Zbl 1106.46005 Zbl 1107.01001 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1107.54001 Zbl 1123.22005 Zbl 1120.17002 Zbl 1120.17001 Zbl 1179.58001 Zbl 1106.46004 Zbl 1105.18001 Zbl 1106.46003 Zbl 1107.13002 Zbl 1107.13001 |
[2] | J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) Zbl 0221.20056 |
[3] | A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) MR0412321 Zbl 0342.22001 |
[4] | E. Cartan, "Leçons sur la théorie des spineurs" , Hermann (1938) Zbl 0022.17101 Zbl 0019.36301 Zbl 64.1382.04 |
Comments
The algebra generated by products of an even number of elements of the free
-module
is also called the even Clifford algebra of the quadratic form
. See also the articles Exterior algebra (or Grassmann algebra), and Cartan method of exterior forms for more details in the case
.
References
[a1] | C. Chevalley, "The algebraic theory of spinors" , Columbia Univ. Press (1954) MR0060497 Zbl 0057.25901 |
[a2] | O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) Zbl 0259.10018 |
[a3] | C. Chevalley, "The construction and study of certain important algebras" , Math. Soc. Japan (1955) pp. Chapt. III MR0072867 |
Clifford algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clifford_algebra&oldid=24051