# positiveRoots -- returns the positive roots of a simple Lie algebra

## Synopsis

• Usage:
positiveRoots(g), positiveRoots("A",2)
• Inputs:
• Outputs:

## Description

Let R be an irreducible root system of rank m, and choose a base of simple roots $\Delta = \{\alpha_1,...,\alpha_m\}$. This function returns all the roots that are nonnegative linear combinations of the simple roots. The formulas implemented here are taken from the tables following Bourbaki's Lie Groups and Lie Algebras Chapter 6.

In the example below, we see that for $sl_3$, the positive roots are $\alpha_1$, $\alpha_2$, and $\alpha_1+\alpha_2$.

 i1 : sl3=simpleLieAlgebra("A",2) o1 = sl3 o1 : LieAlgebra i2 : positiveRoots(sl3) o2 = {{2, -1}, {1, 1}, {-1, 2}} o2 : List

## Ways to use positiveRoots :

• "positiveRoots(LieAlgebra)"
• "positiveRoots(String,ZZ)"

## For the programmer

The object positiveRoots is .