Weyl group

In mathematics, the Weyl group is an important tool for the study of Lie groups and Lie algebras, and more generally of root systems. It is named after Hermann Weyl, who in 1925 recognized its importance.

Weyl group of a Lie group

It is a semisimple Lie group and

Their Iwasawa decomposition. Let the normalizer of in and the centralizer of in. The Weyl group is defined as

Is a finite group, which is produced by elements of the order 2.

Weyl group of a root system

It is a root system in a vector space, then that of the reflections in the hyperplanes generated from the roots

Generated group the Weyl group of the root system.

If a semisimple Lie group with Lie algebra, then one considers the Cartan subalgebra and the corresponding root system. The Weyl group of coincides with the Weyl group of.

Example

The Weyl group of the special linear group is the symmetric group.