lieAlgebraRepresentation -- create a LieAlgebraRepresentation

Usage:

lieAlgebraRepresentation(V,LAB,L)
Inputs:
- V, an instance of the type LieAlgebraModule,
- LAB, an instance of the type LieAlgebraBasis,
- L, a list,

Description

Let $\{B_i\}$ be a basis of $\mathfrak{g}$, and $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ be a Lie algebra representation with character $V$.

To construct $\rho$, we require a basis LAB of $\mathfrak{g}$, and a list L of matrices that are the images $\rho(B_i) \in \mathfrak{gl}(V)$.

First, we build the standard representation for $sl_3$. The list of matrices we need is already contained in the LieAlgebraBasis.

i1 : sl3=simpleLieAlgebra("A",2);

i2 : V=irreducibleLieAlgebraModule({1,0},sl3);

i3 : LAB = lieAlgebraBasis("A",2);

i4 : LAB#"BasisElements"

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

o4 : List

i5 : rho=lieAlgebraRepresentation(V,LAB,LAB#"BasisElements")

o5 = LieAlgebraRepresentation{"Basis" => Enhanced basis of sl3                                                                                      }
                              "Module" => V
                              "RepresentationMatrices" => {| 1 0  0 |, | 0 0 0  |, | 0 1 0 |, | 0 0 0 |, | 0 0 1 |, | 0 0 0 |, | 0 0 0 |, | 0 0 0 |}
                                                           | 0 -1 0 |  | 0 1 0  |  | 0 0 0 |  | 0 0 1 |  | 0 0 0 |  | 1 0 0 |  | 0 0 0 |  | 0 0 0 |
                                                           | 0 0  0 |  | 0 0 -1 |  | 0 0 0 |  | 0 0 0 |  | 0 0 0 |  | 0 0 0 |  | 0 1 0 |  | 1 0 0 |

o5 : LieAlgebraRepresentation

Next, we make an irreducible representation with highest weight $(2,0)$. This time, we create the list of matrices using the command GTrepresentationMatrices.

i6 : V=irreducibleLieAlgebraModule({2,0},sl3);

i7 : L = GTrepresentationMatrices(V)

o7 = {| 2 0 0  0 0  0 |, | 0 0 0 0  0 0  |, | 0 2 0 0 0 0 |, | 0 0 0 0 0 0 |,
      | 0 0 0  0 0  0 |  | 0 1 0 0  0 0  |  | 0 0 2 0 0 0 |  | 0 0 0 3 0 0 | 
      | 0 0 -2 0 0  0 |  | 0 0 2 0  0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 3 0 | 
      | 0 0 0  1 0  0 |  | 0 0 0 -1 0 0  |  | 0 0 0 0 1 0 |  | 0 0 0 0 0 0 | 
      | 0 0 0  0 -1 0 |  | 0 0 0 0  0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 4 | 
      | 0 0 0  0 0  0 |  | 0 0 0 0  0 -2 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 | 
     ------------------------------------------------------------------------
     | 0 0 0 6 0 0 |, | 0 0 0 0 0 0 |, | 0 0   0   0 0   0 |, | 0   0   0 0  
     | 0 0 0 0 3 0 |  | 1 0 0 0 0 0 |  | 0 0   0   0 0   0 |  | 0   0   0 0  
     | 0 0 0 0 0 0 |  | 0 1 0 0 0 0 |  | 0 0   0   0 0   0 |  | 0   0   0 0  
     | 0 0 0 0 0 4 |  | 0 0 0 0 0 0 |  | 0 1/3 0   0 0   0 |  | 1/3 0   0 0  
     | 0 0 0 0 0 0 |  | 0 0 0 1 0 0 |  | 0 0   2/3 0 0   0 |  | 0   1/3 0 0  
     | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0   0   0 1/2 0 |  | 0   0   0 1/2
     ------------------------------------------------------------------------
     0 0 |}
     0 0 |
     0 0 |
     0 0 |
     0 0 |
     0 0 |

o7 : List

i8 : lieAlgebraRepresentation(V,LAB,L)

o8 = LieAlgebraRepresentation{"Basis" => Enhanced basis of sl3                                                                                                                                                    }
                              "Module" => V
                              "RepresentationMatrices" => {| 2 0 0  0 0  0 |, | 0 0 0 0  0 0  |, | 0 2 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 6 0 0 |, | 0 0 0 0 0 0 |, | 0 0   0   0 0   0 |, | 0   0   0 0   0 0 |}
                                                           | 0 0 0  0 0  0 |  | 0 1 0 0  0 0  |  | 0 0 2 0 0 0 |  | 0 0 0 3 0 0 |  | 0 0 0 0 3 0 |  | 1 0 0 0 0 0 |  | 0 0   0   0 0   0 |  | 0   0   0 0   0 0 |
                                                           | 0 0 -2 0 0  0 |  | 0 0 2 0  0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 3 0 |  | 0 0 0 0 0 0 |  | 0 1 0 0 0 0 |  | 0 0   0   0 0   0 |  | 0   0   0 0   0 0 |
                                                           | 0 0 0  1 0  0 |  | 0 0 0 -1 0 0  |  | 0 0 0 0 1 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 4 |  | 0 0 0 0 0 0 |  | 0 1/3 0   0 0   0 |  | 1/3 0   0 0   0 0 |
                                                           | 0 0 0  0 -1 0 |  | 0 0 0 0  0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 4 |  | 0 0 0 0 0 0 |  | 0 0 0 1 0 0 |  | 0 0   2/3 0 0   0 |  | 0   1/3 0 0   0 0 |
                                                           | 0 0 0  0 0  0 |  | 0 0 0 0  0 -2 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0   0   0 1/2 0 |  | 0   0   0 1/2 0 0 |

o8 : LieAlgebraRepresentation

Ways to use lieAlgebraRepresentation:

lieAlgebraRepresentation(LieAlgebraModule,LieAlgebraBasis,List)

For the programmer

The object lieAlgebraRepresentation is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:1591:0.