Macaulay2 ยป Documentation
Packages ยป LieAlgebraRepresentations :: adjointRepresentation
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adjointRepresentation -- creates the adjoint representation of a Lie algebra

Description

Let $\mathfrak{g}$ be a Lie algebra with basis LAB. The basis records a basis $B$ of $\mathfrak{g}$, the bracket for $\mathfrak{g}$, and a way to write elements of $\mathfrak{g} in the basis $B$. With these tools, we may write the matrix for the linear transformation $\operatorname{ad}(B_i)$ with respect to the basis $B$ for each $B_i$. This is the adjoint representation.

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

i1 : adjointRepresentation("A",2)

o1 = LieAlgebraRepresentation{"Basis" => Enhanced basis of ๐”ž                                                                                                                                                                                           }
                                                            2
                              "Module" => LL   (๐”ž )
                                            1,1  2
                              "RepresentationMatrices" => {| 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 1 |, | 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 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 0  0 1 |  | 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 2 0  0 0  0 0  |  | 0 0 -1 0 0 0 0  0  |  | -2 1 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  -1 0 0 0 |  | 0 0 0 0  0  0 0 0 |
                                                           | 0 0 0 -1 0 0  0 0  |  | 0 0 0  2 0 0 0  0  |  | 0  0 0 0 0 0 0 0  |  | 1 -2 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 0  1 0  0 0  |  | 0 0 0  0 1 0 0  0  |  | 0  0 0 1 0 0 0 0  |  | 0 0  -1 0 0 0 0 0 |  | -1 -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  0 -2 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  0 0 |  | 2 -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 0 0  0 0  1 0  |  | 0 0 0  0 0 0 -2 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 |  | -1 2 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  -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 -1 0 |  | 0  0 0 0  0  1 0 0 |  | 1 1 0 0  0  0 0 0 |

o1 : LieAlgebraRepresentation
 
i2 : sl4=simpleLieAlgebra("A",3)

o2 = sl4

o2 : simple LieAlgebra
 
i3 : adjointRepresentation(sl4)

o3 = LieAlgebraRepresentation{"Basis" => Enhanced basis of sl4                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     }
                              "Module" => LL     (sl4)
                                            1,0,1
                              "RepresentationMatrices" => {| 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 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 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 1 |, | 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  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  -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  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  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 1 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 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 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 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  -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  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  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 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 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 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 -1 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 2 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  |  | -2 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 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  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 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 -1 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 -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 -2 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 0 0 -1 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 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 -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 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 0 0 0  0 -1 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  0 0 0 0 0 0 0 0 0 0  0 |  | 0 1 -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 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 -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  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 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 0 0 |
                                                           | 0 0 0 0 0  0 1 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 -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 -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 |  | -1 -1 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 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 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 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 0  |  | 0 0 0 0  0 0  0 1 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 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 0 0 0 0 0 0 0  0 0 0 0 0  |  | 1 -1 -1 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 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 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 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 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 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 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 |  | -1 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  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 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 |
                                                           | 0 0 0 0 0  0 0 0  0 -2 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 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  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 |  | 2 -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  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 -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 0 0  0 0  0 0 0 0 -2 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 -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 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  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 -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  1 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  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 -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 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 -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  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 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 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 -1 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  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 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 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 1 0 0  0 0 0 |  | 0 0  0 0 0 0  0 0  0  0 0 0 0 0 0 |  | 1 1 -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 -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  -1 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 -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 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 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 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 1 1 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 -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 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  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  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  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 -1 0 0 0 |  | 0  0 0 0 0 0  0 0  0  1 0 0 0 0 0 |  | 1 0 1 0 0 0  0 0  0  0 0 0 0 0 0 |

o3 : LieAlgebraRepresentation
 
i4 : LAB=lieAlgebraBasis("C",2)

o4 = Enhanced basis of ๐” 
                        2

o4 : LieAlgebraBasis
 
i5 : adjointRepresentation(LAB)

o5 = LieAlgebraRepresentation{"Basis" => Enhanced basis of ๐”                                                                                                                                                                                                                                                                                     }
                                                            2
                              "Module" => LL   (๐”  )
                                            2,0  2
                              "RepresentationMatrices" => {| 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  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 1 |, | 0 0  -1 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  -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  |  | 0 0  0  0 0 0 0 1 0 0 |  | 0 0  0  0 0 0 0  0 2 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  0 0 0 0 0 |  | 0 0 0 0  -2 0  0 0 0 0 |  | 0 0 0 0 0  -1 0 0 0 0 |
                                                           | 0 0 2 0  0 0 0  0 0 0  |  | 0 0 -1 0 0 0 0 0  0  0 |  | -2 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  0 0 0 0 0 0  0 1 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  -1 0 0 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  2 0 0 0 0  0  0 |  | 0  0 0 0 0 0 0 0 0  0  |  | 2 -2 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 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 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 1 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 -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 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 2 0 0 0 0  0  |  | 0 0  0  0 0 0 0 0 0 0 |  | 0 0  -2 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 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 -2 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 1 0 |  | 0 0  0  0 0 0 0  0 0 1 |  | 0  0 0 0 0 0 0  0 0 0 |  | 2 -1 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 -1 0  0 0 0 0 |
                                                           | 0 0 0 0  0 0 0  2 0 0  |  | 0 0 0  0 0 0 0 -2 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  0 0 0 |  | 0 0  0  0 0 0 0 0  0  0 |  | -2 2 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 0 0  0 0 0 0 0  -1 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 0 0 0 -1 0  0 |  | 0  0 0 0  0  0 1 0 0 0 |  | 0 1 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 -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 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  -2 0 |  | 0  0 0 0  0  0 0 0 0 0 |  | 0 0 0 0  0  0  2 0 0 0 |  | 2 0 0 0 0  0  0 0 0 0 |

o5 : LieAlgebraRepresentation

Ways to use adjointRepresentation:

  • adjointRepresentation(LieAlgebra)
  • adjointRepresentation(LieAlgebraBasis)
  • adjointRepresentation(String,ZZ)

For the programmer

The object adjointRepresentation is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:1697:0.