Weyl group
The Weyl group of symmetries of a root system. Depending on the actual realization of the root system, different Weyl groups are considered: Weyl groups of a semi-simple splittable Lie algebra, of a symmetric space, of an algebraic group, etc.
Let be a connected affine algebraic group defined over an algebraically closed field
. The Weyl group of
with respect to a torus
is the quotient group
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considered as a group of automorphisms of induced by the conjugations of
by elements of
. Here
is the normalizer (cf. Normalizer of a subset) and
is the centralizer of
in
. The group
is finite. If
is a maximal torus,
is said to be the Weyl group
of the algebraic group
. This definition does not depend on the choice of a maximal torus
(up to isomorphism). The action by conjugation of
on the set
of Borel subgroups (cf. Borel subgroup) in
containing
induces a simply transitive action of
on
. The action by conjugation of
on
induces an adjoint action of
on the Lie algebra
of
. Let
be the set of non-zero weights of the weight decomposition of
with respect to this action, which means that
is the root system of
with respect to
(cf. Weight of a representation of a Lie algebra).
is a subset of the group
of rational characters of the torus
, and
is invariant with respect to the action of
on
.
Let be a reductive group, let
be the connected component of the identity of its centre and let
be a maximal torus of
. The vector space
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is canonically identified with a subspace of the vector space
![]() |
As a subset of , the set
is a reduced root system in
, and the natural action of
on
defines an isomorphism between
and the Weyl group of the root system
. Thus,
displays all the properties of a Weyl group of a reduced root system; e.g. it is generated by reflections (cf. Reflection).
The Weyl group of a Tits system is a generalization of this situation (for its exact definition see Tits system).
The Weyl group of a finite-dimensional reductive Lie algebra
over an algebraically closed field of characteristic zero is defined as the Weyl group of its adjoint group. The adjoint action of
in the Cartan subalgebra
of
is a faithful representation of
. The group
is often identified with the image of this representation, being regarded as the corresponding linear group in
generated by the reflections. The concept of a "Weyl group" was first used by H. Weyl
in the special case of finite-dimensional semi-simple Lie algebras over the field of complex numbers. A Weyl group may also be defined for an arbitrary splittable semi-simple finite-dimensional Lie algebra, as the Weyl group of its root system. A relative Weyl group may be defined for an affine algebraic group defined over an algebraically non-closed field. If
is a maximal
-split torus of
, then the quotient group
(the normalizer of
over its centralizer in
), regarded as the group of automorphisms of
induced by the conjugations of
by elements of
, is said to be the relative Weyl group of
.
For the Weyl group of a symmetric space, see Symmetric space. The Weyl group of a real connected non-compact semi-simple algebraic group is identical with the Weyl group of the corresponding symmetric space. For the affine Weyl group see Root system.
References
[1a] | H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch linearen Transformationen I" Math. Z. , 23 (1925) pp. 271–309 |
[1b] | H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch linearen Transformationen II" Math. Z. , 24 (1925) pp. 328–395 |
[2] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |
[3] | N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) |
[4] | N. Bourbaki, "Lie groups and Lie algebras" , Elements of mathematics , Hermann (1975) (Translated from French) |
[5a] | A. Borel, J. Tits, "Groupes réductifs" Publ. Math. I.H.E.S. , 27 (1965) pp. 55–150 |
[5b] | A. Borel, J. Tits, "Complément à l'article "Groupes réductifs" " Publ. Math. I.H.E.S. , 41 (1972) pp. 253–276 |
[6] | F. Bruhat, J. Tits, "Groupes algébriques simples sur un corps local" T.A. Springer (ed.) et al. (ed.) , Proc. Conf. local fields (Driebergen, 1966) , Springer (1967) pp. 23–36 |
[7] | S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) |
Comments
The affine Weyl group is the Weyl group of an affine Kac–Moody algebra. One may define a Weyl group for an arbitrary Kac–Moody algebra.
The Weyl group as an abstract group is a Coxeter group.
Weyl groups play an important role in representation theory (see Character formula).
References
[a1] | J. Tits, "Reductive groups over local fields" A. Borel (ed.) W. Casselman (ed.) , Automorphic forms, representations and ![]() |
[a2] | J.E. Humphreys, "Reflection groups and Coxeter groups" , Cambridge Univ. Press (1991) |
The Weyl group of a connected compact Lie group is the quotient group
, where
is the normalizer in
of a maximal torus
of
. This Weyl group is isomorphic to a finite group of linear transformations of the Lie algebra
of
(the isomorphism is realized by the adjoint representation of
in
), and may be characterized with the aid of the root system
of the Lie algebra
of
(with respect to
), as follows: If
is a system of simple roots of the algebra, which are linear forms on the real vector space
, the Weyl group is generated by the reflections in the hyperplanes
. Thus,
is the Weyl group of the system
(as a linear group in
).
has a simple transitive action on the set of all chambers (cf. Chamber) of
(which, in this case, are referred to as Weyl chambers). It should be noted that, in general,
is not the semi-direct product of
and
; all the cases in which it is have been studied. The Weyl group of
is isomorphic to the Weyl group of the corresponding complex semi-simple algebraic group
(cf. Complexification of a Lie group).
A.S. Fedenko
Weyl group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_group&oldid=21960