LieAlgebraModule -- class for Lie algebra modules

Description

This class represents Lie algebra modules. Currently only modules over semi-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 : simple LieAlgebra

i2 : M=irreducibleLieAlgebraModule(g,{1,1})

o2 = M

o2 : irreducible LieAlgebraModule over g

Functions and methods returning an object of class LieAlgebraModule:

adams -- Computes the action of the nth Adams operator on a Lie algebra module
adjointModule -- The adjoint module of a Lie algebra
branchingRule -- A Lie algebra module viewed as a module over a Lie subalgebra
fusionProduct -- computes the multiplicities of irreducibles in the decomposition of the fusion product of U and V
irreducibleLieAlgebraModule -- construct the irreducible Lie algebra module with given highest weight
LieAlgebraModuleFromWeights -- finds a Lie algebra module based on its weights
standardModule (missing documentation)
trivialModule -- The trivial module of a Lie algebra
zeroModule -- The zero module of a Lie algebra

Methods that use an object of class LieAlgebraModule:

adams(ZZ,LieAlgebraModule) -- see adams -- Computes the action of the nth Adams operator on a Lie algebra module
branchingRule(LieAlgebraModule,LieAlgebra) -- see branchingRule -- A Lie algebra module viewed as a module over a Lie subalgebra
branchingRule(LieAlgebraModule,List) -- see branchingRule -- A Lie algebra module viewed as a module over a Lie subalgebra
branchingRule(LieAlgebraModule,Matrix) -- see branchingRule -- A Lie algebra module viewed as a module over a Lie subalgebra
branchingRule(LieAlgebraModule,String) -- see branchingRule -- A Lie algebra module viewed as a module over a Lie subalgebra
casimirScalar(LieAlgebraModule) -- see casimirScalar -- computes the scalar by which the Casimir operator acts on an irreducible Lie algebra module
casimirSpectrum(LieAlgebraModule) -- see casimirSpectrum -- computes the eigenvalues of the Casimir operator associated to a representation
character(LieAlgebraModule) -- see character -- Computes the character of a Lie algebra module
dim(LieAlgebraModule) -- computes the dimension of a Lie algebra module as a vector space over the ground field
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
GTrepresentationMatrices(LieAlgebraModule) -- see GTrepresentationMatrices -- creates a list of matrices for the action of $\mathfrak{g}$ on Gelfand-Tsetlin basis
isIrreducible(LieAlgebraModule) -- see isIrreducible -- Whether a Lie algebra module is irreducible or not
isIsomorphic(LieAlgebraModule,LieAlgebraModule) (missing documentation)
LieAlgebraModule ** LieAlgebraModule -- tensor product of LieAlgebraModules
directSum(LieAlgebraModule) -- see LieAlgebraModule ++ LieAlgebraModule -- direct sum of LieAlgebraModules
LieAlgebraModule ++ LieAlgebraModule -- direct sum of LieAlgebraModules
LieAlgebraModule @ LieAlgebraModule -- Take the tensor product of modules over different Lie algebras
LieAlgebraModule ^ ZZ (missing documentation)
LieAlgebraModule ^** ZZ -- Computes the nth tensor power of a Lie algebra module
LieAlgebraModule _ LieAlgebraModule -- see LieAlgebraModule _ ZZ -- Pick out one irreducible submodule of a Lie algebra module
LieAlgebraModule _ List -- see LieAlgebraModule _ ZZ -- Pick out one irreducible submodule of a Lie algebra module
LieAlgebraModule _ Vector -- see LieAlgebraModule _ ZZ -- Pick out one irreducible submodule of a Lie algebra module
LieAlgebraModule _ ZZ -- Pick out one irreducible submodule of a Lie algebra module
LieAlgebraModule _* -- List irreducible submodules of a Lie algebra module
lieAlgebraRepresentation(LieAlgebraModule,LieAlgebraBasis,List) -- see lieAlgebraRepresentation -- create a LieAlgebraRepresentation
qdim(LieAlgebraModule) -- see qdim -- Compute principal specialization of character or quantum dimension
qdim(LieAlgebraModule,ZZ) -- see qdim -- Compute principal specialization of character or quantum dimension
dual(LieAlgebraModule) -- see starInvolution -- computes w* for a weight w
starInvolution(LieAlgebraModule) -- see starInvolution -- computes w* for a weight w
exteriorPower(ZZ,LieAlgebraModule) -- see symmetricPower(ZZ,LieAlgebraModule) -- Computes the nth symmetric / exterior tensor power of a Lie algebra module
symmetricPower(ZZ,LieAlgebraModule) -- Computes the nth symmetric / exterior tensor power of a Lie algebra module
tensorCoefficient(LieAlgebraModule,LieAlgebraModule,LieAlgebraModule) -- see tensorCoefficient -- computes the multiplicity of W in U tensor V
toExternalString(LieAlgebraModule) (missing documentation)
toString(LieAlgebraModule) (missing documentation)
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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/LieAlgebraRepresentations/documentation.m2:382:0.