# 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

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

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