# 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.

How to Cite This Entry:
Weyl group. Encyclopedia of Mathematics. URL: http://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