root system
A root system is a key notion in the classification and the
representation theory of reflection groups and of semi-simple Lie
algebras. Let E be a Euclidean vector space with inner product
(⋅,⋅). A root system is a finite spanning set R⊂E
such that for every u∈R, the orthogonal
reflection
v↦v-2(u,v)(u,u)u,v∈E |
preserves R.
A root system is called crystallographic if 2(u,v)(u,u) is an integer for all u,v∈R.
A root system is called reduced if for all u∈R, we have ku∈R for k=±1 only.
We call a root system indecomposable if there is no proper decomposition R=R′∪R′′ such that every vector in is orthogonal to every vector in .
Title | root system |
Canonical name | RootSystem |
Date of creation | 2013-03-22 13:11:30 |
Last modified on | 2013-03-22 13:11:30 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 13 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 17B20 |
Related topic | SimpleAndSemiSimpleLieAlgebras2 |
Related topic | LieAlgebra |
Defines | reduced root system |
Defines | root |
Defines | root space |
Defines | root decomposition |
Defines | indecomposable |
Defines | reduced |
Defines | crystallographic |