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单词 RootSystem
释义

root system


A root systemMathworldPlanetmath is a key notion in the classification and therepresentation theory of reflection groups and of semi-simple Liealgebras. Let E be a Euclidean vector space with inner product(,). A root system is a finite spanning set REsuch that for every uR, the orthogonalMathworldPlanetmathPlanetmath reflectionMathworldPlanetmath

vv-2(u,v)(u,u)u,vE

preserves R.

A root system is called crystallographic if2(u,v)(u,u) is an integer for all u,vR.

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

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

Titleroot system
Canonical nameRootSystem
Date of creation2013-03-22 13:11:30
Last modified on2013-03-22 13:11:30
Ownerrmilson (146)
Last modified byrmilson (146)
Numerical id13
Authorrmilson (146)
Entry typeDefinition
Classificationmsc 17B20
Related topicSimpleAndSemiSimpleLieAlgebras2
Related topicLieAlgebra
Definesreduced root system
Definesroot
Definesroot space
Definesroot decomposition
Definesindecomposable
Definesreduced
Definescrystallographic
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