# 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 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.