root system

A root systemMathworldPlanetmath 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 RE such that for every uR, the orthogonalMathworldPlanetmathPlanetmath reflectionMathworldPlanetmath


preserves R.

A root system is called crystallographic if 2(u,v)(u,u) is an integer for all u,vR.

A root system is called reduced if for all uR, we have kuR for k=±1 only.

We call a root system indecomposable if there is no proper decomposition R=RR′′ such that every vector in R is orthogonal to every vector in R′′.

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