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 $R=R'\cup R''$ such that every vector in $R'$ is orthogonal to every vector in $R''$ .