Spinor group
of a non-generate quadratic form on an
-dimensional vector space
(
) over a field
A connected linear algebraic group which is the simply-connected covering of the irreducible component of the identity of the orthogonal group
of the form
. If
, then
coincides with the special orthogonal group
. The spinor group is constructed in the following way. Let
be the Clifford algebra of the pair
, let
(
) be the subspace of
generated by products of an even (odd) number of elements of
, and let
be the canonical anti-automorphism of
defined by the formula
![]() |
The inclusion enables one to define the Clifford group
![]() |
and the even (or special) Clifford group
![]() |
The spinor group is defined by
![]() |
The spinor group is a quasi-simple (when
), connected, simply-connected, linear algebraic group, of type
when
and of type
when
; if
it is
and if
it is
. The following isomorphisms hold:
![]() |
![]() |
There is a linear representation of
in
defined by
![]() |
If ,
![]() |
The group has a faithful linear representation in
(see Spinor representation).
If is the field of real numbers and
is positive (or negative) definite, then the group
of real points of the algebraic group
is sometimes also called a spinor group. This is a connected simply-connected compact Lie group which is a two-sheeted covering of the special orthogonal group
. The following isomorphisms hold:
![]() |
![]() |
(see Symplectic group),
![]() |
References
[1] | H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) |
[2] | J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) |
[3] | E. Cartan, "Leçons sur la théorie des spineurs" , 2 , Hermann (1938) |
[4] | M.M. Postnikov, "Lie groups and Lie algebras" , Moscow (1982) (In Russian) |
[5] | C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) |
Comments
See also Quadratic form.
is the so-called even Clifford algebra of
.
References
[a1] | N. Bourbaki, "Algèbre. Formes sesquilineares et formes quadratiques" , Eléments de mathématiques , Hermann (1959) pp. Chapt. 9 |
[a2] | C. Chevalley, "The algebraic theory of spinors" , Columbia Univ. Press (1954) |
[a3] | Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) |
Spinor group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spinor_group&oldid=18620