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LieTypes :: positiveRoots

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



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 :

For the programmer

The object positiveRoots is a method function.