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 $(\cdot ,\cdot )$. A root system is a finite spanning set $R\subset E$ such that for every $u\in R$, the orthogonal reflection

$$v\mapsto v-2\frac{(u,v)}{(u,u)}u,v\in E$$

preserves $R$.

A root system is called crystallographic if $2\frac{(u,v)}{(u,u)}$ is an integer for all $u,v\in R$.

A root system is called reduced if for all $u\in R$, we have $ku\in R$ for $k=\pm 1$ only.

We call a root system indecomposable if there is no proper decomposition $R=R^{\prime }\cup R^{\prime \prime }$ such that every vector in $R^{\prime }$ is orthogonal to every vector in $R^{\prime \prime }$.

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