# LieAlgebraModule -- class for Lie algebra modules

## Description

This class represents Lie algebra modules. Currently only modules over simple Lie algebras over the complex numbers are supported. An object of type LieAlgebraModule is a hash table recording the Lie algebra and the decomposition of the module into irreducible Lie algebra modules, which are indexed by their highest weights.

 i1 : g=simpleLieAlgebra("A",2) o1 = g o1 : LieAlgebra i2 : M=irreducibleLieAlgebraModule({1,1},g) o2 = M o2 : g module

## Methods that use an object of class LieAlgebraModule :

• "casimirScalar(LieAlgebraModule)" -- see casimirScalar -- computes the scalar by which the Casimir operator acts on an irreducible Lie algebra module
• dim(LieAlgebraModule) -- computes the dimension of a Lie algebra module as a vector space over the ground field
• expression(LieAlgebraModule) (missing documentation)
• "fusionCoefficient(LieAlgebraModule,LieAlgebraModule,LieAlgebraModule,ZZ)" -- see fusionCoefficient -- computes the multiplicity of W in the fusion product of U and V
• "fusionProduct(LieAlgebraModule,LieAlgebraModule,ZZ)" -- see fusionProduct -- computes the multiplicities of irreducibles in the decomposition of the fusion product of U and V
• "isIsomorphic(LieAlgebraModule,LieAlgebraModule)" -- see isIsomorphic -- tests whether two Lie algebra modules are isomorphic
• LieAlgebraModule ** LieAlgebraModule -- tensor product of LieAlgebraModules
• LieAlgebraModule ++ LieAlgebraModule -- direct sum of LieAlgebraModules
• multiplicity(List,LieAlgebraModule) -- compute the multiplicity of a weight in a Lie algebra module
• net(LieAlgebraModule) (missing documentation)
• "tensorCoefficient(LieAlgebraModule,LieAlgebraModule,LieAlgebraModule)" -- see tensorCoefficient -- computes the multiplicity of W in U tensor V
• "weightDiagram(LieAlgebraModule)" -- see weightDiagram -- computes the weights in a Lie algebra module and their multiplicities

## For the programmer

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