simple root

vector space

Let $R\subseteq E$ be a root system, with $E$ a Euclidean vector space (http://planetmath.org/VectorSpace). If $R^{+}$ is a set of positive roots, then a root is called simple if it is positive, and not the sum of any two positive roots. The simple roots form a basis of the vector space $E$, and any positive root is a positive integer linear combination of simple roots.

A set of roots which is simple with respect to some choice of a set of positive roots is called a base. The Weyl group of the root system acts simply transitively on the set of bases.

Title	simple root
Canonical name	SimpleRoot
Date of creation	2013-03-22 13:11:49
Last modified on	2013-03-22 13:11:49
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	6
Author	mathcam (2727)
Entry type	Definition
Classification	msc 17B20
Defines	base