Lie groups | Finite reflection groups | Lie algebras

Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. In fact it turns out that most finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra. (Wikipedia).

Weyl group
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Related pages

Length function | Lie group | Dynkin diagram | Length of a Weyl group element | Maximal torus | Bruhat order | Weight (representation theory) | Coxeter–Dynkin diagram | Index of a subgroup | Group cohomology | Bruhat decomposition | Isometry group | Borel subgroup | Grassmannian | Hyperplane | Root system | Symmetric group | Symmetric space | Linear algebraic group | Torus | Hasse diagram | Poincaré duality | Connected space | Mathematics | Semidirect product | Cartan subalgebra | Lie algebra | Subgroup | Hermann Weyl | Algebraic group | Solvable group | Field with one element | Longest element of a Coxeter group | Presentation of a group | Semisimple Lie algebra | Outer automorphism group