standardRepresentation -- creates the standard representation of a matrix Lie algebra

Usage:

standardRepresentation(LAB)
Inputs:
- LAB, an instance of the type LieAlgebraBasis,
Outputs:
- V, an instance of the type LieAlgebraRepresentation,

Description

For the matrix Lie algebras $sl_n$, $sp(2n)$, $so(m)$, the basis elements in the LieAlgebraBasis are matrices. Thus we may use these matrices to define a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$.

The user may either input the type and rank, or the simple Lie algebra.

i1 : standardRepresentation("A",2)

o1 = LieAlgebraRepresentation{"Basis" => Enhanced basis of 𝔞                                                                                        }
                                                            2
                              "Module" => LL   (𝔞 )
                                            1,0  2
                              "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 |

o1 : LieAlgebraRepresentation

i2 : sl4=simpleLieAlgebra("A",3)

o2 = sl4

o2 : simple LieAlgebra

i3 : standardRepresentation(sl4)

o3 = LieAlgebraRepresentation{"Basis" => Enhanced basis of sl4                                                                                                                                                                                                  }
                              "Module" => LL     (sl4)
                                            1,0,0
                              "RepresentationMatrices" => {| 1 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 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 0 |}
                                                           | 0 -1 0 0 |  | 0 1 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 1 |  | 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 |
                                                           | 0 0  0 0 |  | 0 0 -1 0 |  | 0 0 1 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 1 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 -1 |  | 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 |  | 0 0 0 0 |  | 0 0 1 0 |  | 0 0 0 0 |  | 0 1 0 0 |  | 1 0 0 0 |

o3 : LieAlgebraRepresentation

Ways to use standardRepresentation:

standardRepresentation(LieAlgebra)
standardRepresentation(String,ZZ)

For the programmer

The object standardRepresentation is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:1659:0.