LieAlgebraRepresentation -- class for a Lie algebra representation

Description

Let $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ be a Lie algebra representation. We implement this in the LieAlgebraRepresentation class as follows.

The user should input: the character of $\rho$, a LieAlgebraBasis of $\mathfrak{g}$, and the list of images $\rho(B_i)$ for each basis element $B_i$ in the LieAlgebraBasis.

As a first example, we create the standard representation of $sl_2$ from scratch. (This can be done automatically with standardRepresentation command.) It has highest weight $\omega_1$.

i1 : sl2 = simpleLieAlgebra("A",1)

o1 = sl2

o1 : simple LieAlgebra

i2 : V = irreducibleLieAlgebraModule({1},sl2)

o2 = V

o2 : irreducible LieAlgebraModule over sl2

i3 : LAB=lieAlgebraBasis("A",1)

o3 = Enhanced basis of sl2

o3 : LieAlgebraBasis

i4 : L = {matrix {{1, 0}, {0, -1/1}}, matrix {{0, 1}, {0, 0/1}}, matrix {{0, 0}, {1, 0/1}}}

o4 = {| 1 0  |, | 0 1 |, | 0 0 |}
      | 0 -1 |  | 0 0 |  | 1 0 |

o4 : List

i5 : rho = lieAlgebraRepresentation(V,LAB,L)

o5 = LieAlgebraRepresentation{"Basis" => Enhanced basis of sl2                        }
                              "Module" => V
                              "RepresentationMatrices" => {| 1 0  |, | 0 1 |, | 0 0 |}
                                                           | 0 -1 |  | 0 0 |  | 1 0 |

o5 : LieAlgebraRepresentation

Functions and methods returning an object of class LieAlgebraRepresentation:

adjointRepresentation -- creates the adjoint representation of a Lie algebra
deGraafRepresentation -- compute the representation with the specified highest weight using de Graaf's algorithm
lieAlgebraRepresentation -- create a LieAlgebraRepresentation
standardRepresentation -- creates the standard representation of a matrix Lie algebra
trivialRepresentation -- creates the trivial representation of a Lie algebra

Methods that use an object of class LieAlgebraRepresentation:

basisWordsFromMatrixGenerators(LieAlgebraRepresentation) -- see basisWordsFromMatrixGenerators -- express each basis element of $V(\lambda)$ as a linear combination of words in the lowering operators applied to the highest weight vector
casimirOperator(LieAlgebraRepresentation) -- see casimirOperator -- computes the Casimir operator associated to a representation
casimirProjection(LieAlgebraRepresentation,QQ) -- see casimirProjection -- projection operator to a specified eigenspace of the Casimir operator
casimirProjection(LieAlgebraRepresentation,ZZ) (missing documentation)
exteriorPower(ZZ,LieAlgebraRepresentation) -- computes the explicit action on $\bigwedge^k V$ for a $\mathfrak{g}$-module $V$
isomorphismOfRepresentations(LieAlgebraRepresentation,LieAlgebraRepresentation) -- see isomorphismOfRepresentations -- compute an explicit isomorphism between two Lie algebra representations
LieAlgebraRepresentation ** LieAlgebraRepresentation -- computes the explicit action on $V \otimes W$ given $\mathfrak{g}$-representations $V$ and $W$
tensor(LieAlgebraRepresentation,LieAlgebraRepresentation) -- see LieAlgebraRepresentation ** LieAlgebraRepresentation -- computes the explicit action on $V \otimes W$ given $\mathfrak{g}$-representations $V$ and $W$
representationWeights(LieAlgebraRepresentation) -- see representationWeights -- computes the weights of the basis of a Lie algebra module from an explicit representation
reynoldsOperator(LieAlgebraRepresentation) -- see reynoldsOperator -- computes the projection to the sum of the trivial submodules in $V$
symmetricPower(ZZ,LieAlgebraRepresentation) -- computes the explicit action on $\operatorname{Sym}^d V$ for a $\mathfrak{g}$-module $V$
UInVtensorW(LieAlgebraRepresentation,LieAlgebraRepresentation,LieAlgebraRepresentation,Matrix) -- see UInVtensorW -- computes a basis of a submodule of $V \otimes W$ isomorphic to $U$ with a given highest weight vector
UInVtensorW(LieAlgebraRepresentation,LieAlgebraRepresentation,LieAlgebraRepresentation,RingElement) (missing documentation)
VInSymdW(LieAlgebraRepresentation,ZZ,LieAlgebraRepresentation,Matrix) -- see VInSymdW -- computes a basis of a submodule of $\operatorname{Sym}^d W$ isomorphic to $V$ with a given highest weight vector
VInSymdW(LieAlgebraRepresentation,ZZ,LieAlgebraRepresentation,RingElement) (missing documentation)
VInWedgekW(LieAlgebraRepresentation,ZZ,LieAlgebraRepresentation,Matrix) -- see VInWedgekW -- computes a basis of a submodule of $\bigwedge^k W$ isomorphic to $V$ with a given highest weight vector
weightMuHighestWeightVectorsInSymdW(List,ZZ,LieAlgebraRepresentation) -- see weightMuHighestWeightVectorsInSymdW -- computes the highest weight vectors of weight mu in $\operatorname{Sym}^d W$
weightMuHighestWeightVectorsInW(List,LieAlgebraRepresentation) -- see weightMuHighestWeightVectorsInW -- computes the highest weight vectors of weight mu in W
weightNuHighestWeightVectorsInVtensorW(List,LieAlgebraRepresentation,LieAlgebraRepresentation) -- see weightNuHighestWeightVectorsInVtensorW -- computes the highest weight vectors of weight nu in $V \otimes W$

For the programmer

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

The source of this document is in LieAlgebraRepresentations/documentation.m2:1555:0.