Root System - Positive Roots and Simple Roots

Positive Roots and Simple Roots

Given a root system Φ we can always choose (in many ways) a set of positive roots. This is a subset of Φ such that

• For each root exactly one of the roots, – is contained in .
• For any two distinct such that is a root, .

If a set of positive roots is chosen, elements of are called negative roots.

An element of is called a simple root if it cannot be written as the sum of two elements of . The set of simple roots is a basis of with the property that every vector in is a linear combination of elements of with all coefficients non-negative, or all coefficients non-positive. For each choice of positive roots, the corresponding set of simple roots is the unique set of roots such that the positive roots are exactly those that can be expressed as a combination of them with non-negative coefficients, and such that these combinations are unique.