# LieAlgebra -- class for Lie algebras

## Description

This class represents Lie algebras. Currently only simple Lie algebras over the complex numbers are supported. An object of type LieAlgebra is a hash table whose keys record whether the Lie algebra is simple, the rank of the Lie algebra, and the type of the root system.

 i1 : g=simpleLieAlgebra("A",1) o1 = g o1 : LieAlgebra i2 : g=simpleLieAlgebra("E",6) o2 = g o2 : LieAlgebra

## Methods that use an object of class LieAlgebra :

• "dualCoxeterNumber(LieAlgebra)" -- see dualCoxeterNumber -- returns the dual Coxeter number of a simple Lie algebra
• expression(LieAlgebra) (missing documentation)
• "highestRoot(LieAlgebra)" -- see highestRoot -- returns the highest root of a simple Lie algebra
• "irreducibleLieAlgebraModule(List,LieAlgebra)" -- see irreducibleLieAlgebraModule -- construct the irreducible Lie algebra module with given highest weight
• "KillingForm(LieAlgebra,List,List)" -- see KillingForm -- computes the scaled Killing form applied to two weights
• LieAlgebra == LieAlgebra -- tests equality of LieAlgebra
• net(LieAlgebra) (missing documentation)
• "positiveRoots(LieAlgebra)" -- see positiveRoots -- returns the positive roots of a simple Lie algebra
• "starInvolution(List,LieAlgebra)" -- see starInvolution -- computes w* for a weight w
• "weylAlcove(ZZ,LieAlgebra)" -- see weylAlcove -- the dominant integral weights of level less than or equal to l

## For the programmer

The object LieAlgebra is a type, with ancestor classes HashTable < Thing.