Spinor representation
spin representation
The simplest faithful linear representation (cf. Faithful representation; Linear representation) of the spinor group , or the linear representation of the corresponding even Clifford algebra
(see Spinor group;
is a quadratic form). If the ground field
is algebraically closed, then the algebra
is isomorphic to the complete matrix algebra
(where
) or to the algebra
(where
). Therefore there is defined a linear representation
of the algebra
on the space of dimension
over
; this representation is called a spinor representation. The restriction of
to
is called the spinor representation of
. For odd
, the spinor representation is irreducible, and for even
it splits into the direct sum of two non-equivalent irreducible representations
and
, which are called half-spin (or) representations. The elements of the space of the spinor representation are called spinors, and those of the space of the half-spinor representation half-spinors. The spinor representation of the spinor group
is self-dual for any
, whereas the half-spinor representations
and
of the spinor group
are self-dual for even
and dual to one another for odd
. The spinor representation of
is faithful for all
, while the half-spinor representations of
are faithful for odd
, but have a kernel of order two when
is even.
For a quadratic form on a space
over some subfield
, the spinor representation is not always defined over
. However, if the Witt index of
is maximal, that is, equal to
(in particular, if
is algebraically closed), then the spinor and half-spinor representations are defined over
. In this case these representations can be described in the following way if
(see [1]). Let
and
be
-subspaces of the
-space
that are maximal totally isotropic (with respect to the symmetric bilinear form on
associated with
) and let
. Let
be the subalgebra of the Clifford algebra
generated by the subspace
, and let
be the product of
vectors forming a
-basis of
. If
is even,
, then the spinor representation is realized in the left ideal
and acts there by left translation:
(
,
). Furthermore, the mapping
defines an isomorphism of vector spaces
that enables one to realize the spinor representation in
, which is naturally isomorphic to the exterior algebra over
. The half-spinor representations
and
are realized in the
-dimensional subspaces
and
.
If is odd, then
can be imbedded in the
-dimensional vector space
over
. One defines a quadratic form
on
by putting
for all
and
.
is a non-degenerate quadratic form of maximal Witt index defined over
on the even-dimensional vector space
. The spinor representation of
(or of
) is obtained by restricting any of the half-spinor representations of
(or of
) to the subalgebra
(or
, respectively).
The problem of classifying spinors has been solved when and
is an algebraically closed field of characteristic 0 (see [4], [8], [9]). The problem consists of the following: 1) describe the orbits of
in the spinor space by giving a representative of each orbit; 2) calculate the stabilizers in
of each of these representatives; and 3) describe the algebra of invariants of the linear group
.
The existence of spinor and half-spinor representations of the Lie algebra of
was discovered by E. Cartan in 1913, when he classified the finite-dimensional representations of simple Lie algebras [6]. In 1935, R. Brauer and H. Weyl described spinor and half-spinor representations in terms of Clifford algebras [5]. P. Dirac [3] showed how spinors could be used in quantum mechanics to describe the rotation of an electron.
References
[1] | N. Bourbaki, "Algèbre" , Eléments de mathématiques , Hermann (1970) pp. Chapt. II. Algèbre linéaire |
[2] | H. Weyl, "Classical groups, their invariants and representations" , Princeton Univ. Press (1946) (Translated from German) |
[3] | P.A.M. Dirac, "Principles of quantum mechanics" , Clarendon Press (1958) |
[4] | V.L. Popov, "Classification of spinors of dimension fourteen" Trans. Moscow Math. Soc. , 1 (1980) pp. 181–232 Trudy Moskov. Mat. Obshch. , 37 (1978) pp. 173–217 |
[5] | R. Brauer, H. Weyl, "Spinors in ![]() |
[6] | E. Cartan, "Les groupes projectifs qui ne laissant invariante aucune multiplicité plane" Bull. Soc. Math. France , 41 (1913) pp. 53–96 |
[7] | C. Chevalley, "The algebraic theory of spinors" , Columbia Univ. Press (1954) |
[8] | V. Gatti, E. Viniberghi, "Spinors in ![]() |
[9] | J.I. Igusa, "A classification of spinors up to dimension twelve" Amer. J. Math. , 92 : 4 (1970) pp. 997–1028 |
Spinor representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spinor_representation&oldid=24167