# Difference between revisions of "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 $G$ 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

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://www.encyclopediaofmath.org/index.php?title=Weyl_group&oldid=44291
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article