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

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

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 MR1544744
[1b]	H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch linearen Transformationen II" Math. Z. , 24 (1925) pp. 328–395 MR1544744
[2]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[3]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[4]	N. Bourbaki, "Lie groups and Lie algebras" , Elements of mathematics , Hermann (1975) (Translated from French) MR2109105 MR1890629 MR1728312 MR0979493 MR0682756 MR0524568 Zbl 0319.17002
[5a]	A. Borel, J. Tits, "Groupes réductifs" Publ. Math. I.H.E.S. , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[5b]	A. Borel, J. Tits, "Complément à l'article "Groupes réductifs" " Publ. Math. I.H.E.S. , 41 (1972) pp. 253–276 MR0315007
[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 MR0230838 Zbl 0263.14016
[7]	S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) MR0145455 Zbl 0111.18101

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 -functions , Proc. Symp. Pure Math. , 33:1 , Amer. Math. Soc. (1979) pp. 29–69 MR0546588 Zbl 0415.20035
[a2]	J.E. Humphreys, "Reflection groups and Coxeter groups" , Cambridge Univ. Press (1991) MR1066460 Zbl 0768.20016 Zbl 0725.20028

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