Weyl group

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

Weyl group of a Lie group

Let it be a semi-simple Lie group and ${\ displaystyle G}$

{\ displaystyle G = KAN}

their Iwasawa decomposition (K is a compact subgroup, A is an Abelian and N is a nilpotent). Let it be the normalizer of in and the centralizer of in . The Weyl Group is defined as ${\ displaystyle {\ mathcal {N}} _ {G} (A)}$ ${\ displaystyle A}$ ${\ displaystyle G}$ ${\ displaystyle {\ mathcal {Z}} _ {G} (A)}$ ${\ displaystyle A}$ ${\ displaystyle G}$

{\ displaystyle W = {\ mathcal {N}} _ {G} (A) / {\ mathcal {Z}} _ {G} (A)}

.

It is a finite group created by elements of order 2.

Weyl group of a root system

Let it be a root system in a vector space , then it is called that of the reflections at the hyperplanes generated by the roots ${\ displaystyle R}$ ${\ displaystyle V}$

{\ displaystyle \ left \ {s _ {\ mathrm {A}}: \ mathrm {A} \ in R \ right \}}

generated group the Weyl group of the root system. ${\ displaystyle W}$

If is a semi-simple Lie group with Lie algebra , then consider a Cartan subalgebra and the associated root system . The Weyl group of coincides with the Weyl group of . ${\ displaystyle G}$ ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle {\ mathfrak {a}} \ subset {\ mathfrak {g}}}$ ${\ displaystyle R}$ ${\ displaystyle ({\ mathfrak {a}}, R)}$ ${\ displaystyle G}$

Longest element

The longest element of the Weyl group (for a given root system) is the element of maximum length with respect to the generating system given by reflections at the hyperplanes generated by the roots.

example

The Weyl group of the special linear group is the symmetrical group . The longest element is the permutation . ${\ displaystyle SL (n, \ mathbb {R})}$ ${\ displaystyle S_ {n}}$ ${\ displaystyle (1,2, \ ldots, n-1, n) \ to (n, n-1, \ ldots, 2,1)}$

literature

Michael Davis : The Geometry and Topology of Coxeter Groups , ISBN 978-0-691-13138-2

Web links

Alexander Kirillov : An introduction to Lie groups and Lie algebras , PDF (Chapter 8)
Encyclopedia of Mathematics, AS Fedenko