Namespaces
Variants
Actions

Weyl group

From Encyclopedia of Mathematics
Revision as of 16:55, 17 December 2019 by Ulf Rehmann (talk | contribs) (tex done)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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 $ G $ be a connected affine algebraic group defined over an algebraically closed field $ k $ . The Weyl group of $ G $ with respect to a torus $ T \subset G $ is the quotient group $$ W(T,\ G) = N _{G} (T) / Z _{G} (T), $$ considered as a group of automorphisms of $ T $ induced by the conjugations of $ T $ by elements of $ N _{G} (T) $ . Here $ N _{G} (T) $ is the normalizer (cf. Normalizer of a subset) and $ Z _{G} (T) $ is the centralizer of $ T $ in $ G $ . The group $ W(T,\ G) $ is finite. If $ T _{0} $ is a maximal torus, $ W( T _{0} ,\ G) $ is said to be the Weyl group $ W(G) $ of the algebraic group $ G $ . This definition does not depend on the choice of a maximal torus $ T _{0} $ ( up to isomorphism). The action by conjugation of $ N _{G} ( T _{0} ) $ on the set $ B ^ {T _{0}} $ of Borel subgroups (cf. Borel subgroup) in $ G $ containing $ T _{0} $ induces a simply transitive action of $ W( T _{0} ,\ G) $ on $ B ^ {T _{0}} $ . The action by conjugation of $ T $ on $ G $ induces an adjoint action of $ T $ on the Lie algebra $ \mathfrak g $ of $ G $ . Let $ \Phi (T,\ G) $ be the set of non-zero weights of the weight decomposition of $ \mathfrak g $ with respect to this action, which means that $ \Phi (T,\ G) $ is the root system of $ \mathfrak g $ with respect to $ T $ ( cf. Weight of a representation of a Lie algebra). $ \Phi (T,\ G) $ is a subset of the group $ X(T) $ of rational characters of the torus $ T $ , and $ \Phi (T,\ G) $ is invariant with respect to the action of $ W(T,\ G) $ on $ X(T) $ .


Let $ G $ be a reductive group, let $ Z(G) ^{0} $ be the connected component of the identity of its centre and let $ T _{0} $ be a maximal torus of $ G $ . The vector space $$ X(T _{0} /Z(G) ^{0} ) _ {\mathbf Q} = X(T _{0} /Z(G) ^{0} ) \otimes _ {\mathbf Z} \mathbf Q $$ is canonically identified with a subspace of the vector space $$ X(T _{0} ) _ {\mathbf Q} = X(T _{0} ) \otimes _ {\mathbf Z} \mathbf Q . $$ As a subset of $ X {( T _{0} )} _ {\mathbf Q} $ , the set $ \Phi ( T _{0} ,\ G) $ is a reduced root system in $ X( T _{0} /Z(G) ^{0} ) _ {\mathbf Q} $ , and the natural action of $ W( T _{0} ,\ G) $ on $ {X( T _{0} )} _ {\mathbf Q} $ defines an isomorphism between $ W( T _{0} ,\ G) $ and the Weyl group of the root system $ \Phi (T _{0} ,\ G) $ . Thus, $ W(T _{0} ,\ G) $ 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 $ W $ of a finite-dimensional reductive Lie algebra $ \mathfrak g $ over an algebraically closed field of characteristic zero is defined as the Weyl group of its adjoint group. The adjoint action of $ W $ in the Cartan subalgebra $ \mathfrak p $ of $ \mathfrak g $ is a faithful representation of $ W $ . The group $ W $ is often identified with the image of this representation, being regarded as the corresponding linear group in $ \mathfrak p $ 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 $ G $ defined over an algebraically non-closed field. If $ T $ is a maximal $ k $ - split torus of $ G $ , then the quotient group $ N _{G} (T)/ Z _{G} (T) $ ( the normalizer of $ T $ over its centralizer in $ G $ ), regarded as the group of automorphisms of $ T $ induced by the conjugations of $ T $ by elements of $ N _{G} (T) $ , is said to be the relative Weyl group of $ G $ .


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 $ G $ is the quotient group $ W = N/T $ , where $ N $ is the normalizer in $ G $ of a maximal torus $ T $ of $ G $ . This Weyl group is isomorphic to a finite group of linear transformations of the Lie algebra $ \mathfrak t $ of $ T $ ( the isomorphism is realized by the adjoint representation of $ N $ in $ \mathfrak t $ ), and may be characterized with the aid of the root system $ \Delta $ of the Lie algebra $ \mathfrak g $ of $ G $ ( with respect to $ \mathfrak t $ ), as follows: If $ \alpha _{1} \dots \alpha _{r} $ is a system of simple roots of the algebra, which are linear forms on the real vector space $ \mathfrak t $ , the Weyl group is generated by the reflections in the hyperplanes $ \alpha _{i} (x) = 0 $ . Thus, $ W $ is the Weyl group of the system $ \Delta $ ( as a linear group in $ \mathfrak t $ ). $ W $ has a simple transitive action on the set of all chambers (cf. Chamber) of $ \Delta $ ( which, in this case, are referred to as Weyl chambers). It should be noted that, in general, $ N $ is not the semi-direct product of $ W $ and $ T $ ; all the cases in which it is have been studied. The Weyl group of $ G $ is isomorphic to the Weyl group of the corresponding complex semi-simple algebraic group $ G _{\mathbf C} $ ( cf. Complexification of a Lie group).

A.S. Fedenko

How to Cite This Entry:
Weyl group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_group&oldid=21960
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article