SolvableRepresentation - Maple Help

LieAlgebras[SolvableRepresentation] - given a representation of a solvable algebra, find a basis for the representation space in which the representation matrices are upper triangular matrices

Calling Sequences

SolvableRepresentation( ${\mathbf{ρ}}$, options)

SolvableRepresentation(Alg, options)

Parameters

$\mathrm{ρ}$       - a representation of a solvable Lie algebra $\mathrm{𝔤}$ on a vector space $V$

alg     - a string or name, the name of a initialized solvable Lie algebra

options     -  the keyword argument output = O, where O is a list  with members  "NewBasis", ChangeOfBasisMatrix", "TransformedMatrices",  "Partition"; the keyword argument fieldextension = I

Description

 • Let be a representation of a solvable Lie algebra $\mathrm{𝔤}$ on a vector space $V$. A corollary of Lie's fundamental theorem for solvable Lie algebras (see RepresentationEigenvector) implies that there always exists a basis (possibly complex) for such that the matrix representation of $\mathrm{ρ}\left(x\right)$is upper triangular for all .
 • The program SolvableRepresentation(rho) uses the program RepresentationEigenvector to construction such a basis. In the case when the RepresentationEigenvector program returns a complex eigenvector (with associated complex eigenvalue ), the matrix representation will not be upper triangular but will contain the matrix $\left[\begin{array}{rr}a& b\\ -b& a\end{array}\right]$ on the diagonal (similar to the real Jordan form of a matrix).
 • For the second calling sequence, the program SolvableRepresentation is applied to the adjoint representation of the algebra Alg.
 • The output is a 4-element sequence. The 1st element is a new basis $\mathrm{ℬ}$ for$V$ in which the representation is upper triangular, the 2nd element is the change of basis matrix, the 3rd element is the representation in the new basis. The 4th element $P$ gives the partition defining the size of the diagonal block matrices. If , then the subspaces   are $\mathrm{ρ}-$invariant subspaces. If, for example,  then all the eigenvectors calculated by RepresentationEigenvector are real. If C = then the vectors and $ℬ$$\left[3\right]$ are the real and imaginary parts of a complex eigenvector. The precise form of the output can be specified by the user with the keyword argument output = O, where O is a list with members "NewBasis", ChangeOfBasisMatrix", "TransformedMatrices", "Partition".
 • With the option fieldextension = I, a complex basis will be returned (if needed) which puts the representation into upper triangular form.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$$\mathrm{with}\left(\mathrm{Library}\right):$

Example 1.

We define a 5-dimensional representation of a 3-dimensional solvable Lie algebra.

 > $L≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{alg1},\left[3\right]\right],\left[\left[\left[1,2,2\right],1\right],\left[\left[2,3,2\right],1\right]\right]\right]\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(L\right):$
 alg1 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5}\right],\mathrm{V1}\right):$
 V1 > $M≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[8,8,0,0,0\right],\left[-1,5,6,0,0\right],\left[0,-2,2,4,0\right],\left[0,0,-3,-1,2\right],\left[0,0,0,-4,-4\right]\right],\left[\left[8,16,0,0,0\right],\left[-1,4,12,0,0\right],\left[0,-2,0,8,0\right],\left[0,0,-3,-4,4\right],\left[0,0,0,-4,-8\right]\right],\left[\left[-4,-8,0,0,0\right],\left[1,-1,-6,0,0\right],\left[0,2,2,-4,0\right],\left[0,0,3,5,-2\right],\left[0,0,0,4,8\right]\right]\right]\right):$
 V1 > $\mathrm{ρ1}≔\mathrm{Representation}\left(\mathrm{alg1},\mathrm{V1},M\right)$

We find a new basis for the representation space in which the matrices are all upper triangular.

 alg1 > $\mathrm{B1},\mathrm{P1},\mathrm{newrho},\mathrm{Part1}≔\mathrm{SolvableRepresentation}\left(\mathrm{ρ1}\right)$

To verify this result we use the ChangeRepresentationBasis command to change basis in the representation space.

 V1 > $\mathrm{ChangeRepresentationBasis}\left(\mathrm{ρ1},\mathrm{B1},\mathrm{V1}\right)$

Example 2.

We define a 6-dimensional representation of a 3-dimensional solvable Lie algebra.

 alg1 > $\mathrm{L2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg2},\left[3\right]\right],\left[\left[\left[1,3,2\right],-1\right],\left[\left[1,3,1\right],3\right],\left[\left[2,3,1\right],1\right],\left[\left[2,3,2\right],3\right]\right]\right]\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{+}{3}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{3}{}{\mathrm{e2}}\right]$ (2.2)
 alg1 > $\mathrm{DGsetup}\left(\mathrm{L2}\right):$
 Alg2 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5},\mathrm{x6}\right],\mathrm{V2}\right):$
 V2 > $M≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[0,0,0,0,0,0\right],\left[0,0,0,0,0,0\right],\left[-3,1,0,0,0,0\right],\left[0,0,0,0,0,0\right],\left[0,-3,0,1,0,0\right],\left[0,0,-2\cdot 3,0,2,0\right]\right],\left[\left[0,0,0,0,0,0\right],\left[0,0,0,0,0,0\right],\left[-1,-3,0,0,0,0\right],\left[0,0,0,0,0,0\right],\left[0,-1,0,-3,0,0\right],\left[0,0,-2,0,-2\cdot 3,0\right]\right],\left[\left[2\cdot 3,-2,0,0,0,0\right],\left[1,2\cdot 3,0,-1,0,0\right],\left[0,0,3,0,-1,0\right],\left[0,2,0,2\cdot 3,0,0\right],\left[0,0,1,0,3,0\right],\left[0,0,0,0,0,0\right]\right]\right]\right):$
 V2 > $\mathrm{ρ2}≔\mathrm{Representation}\left(\mathrm{Alg2},\mathrm{V2},M\right)$

In this example some of the eigenvectors found by the RepresentationEigenvector program are complex and it is not possible to find a real basis in which the representation is upper triangular.

 Alg2 > $\mathrm{Query}\left(\mathrm{ρ2},"Representation"\right)$
 ${\mathrm{true}}$ (2.3)
 Alg2 > $\mathrm{B2},\mathrm{P2},\mathrm{newrho},\mathrm{Part2}≔\mathrm{SolvableRepresentation}\left(\mathrm{ρ2}\right)$
 V2 > $\mathrm{ChangeRepresentationBasis}\left(\mathrm{ρ2},\mathrm{B2},\mathrm{V2}\right)$

To obtain an upper triangular representation with respect to a complex basis, use the optional argument fieldextension = I.

 Alg2 > $\mathrm{B3}≔\mathrm{SolvableRepresentation}\left(\mathrm{ρ2},\mathrm{fieldextension}=I,\mathrm{output}=\left["NewBasis"\right]\right)$
 ${\mathrm{B3}}{:=}\left[{\mathrm{D_x6}}{,}{\mathrm{D_x3}}{-}{I}{}{\mathrm{D_x5}}{,}{\mathrm{D_x3}}{+}{I}{}{\mathrm{D_x5}}{,}{\mathrm{D_x1}}{-}{I}{}{\mathrm{D_x2}}{-}{\mathrm{D_x4}}{,}{\mathrm{D_x1}}{+}{\mathrm{D_x4}}{,}{\mathrm{D_x1}}{+}{I}{}{\mathrm{D_x2}}{-}{\mathrm{D_x4}}\right]$ (2.4)
 V2 > $\mathrm{ChangeRepresentationBasis}\left(\mathrm{ρ2},\mathrm{B3},\mathrm{V2}\right)$

Example 3.

If the name of an algebra is passed to the program SolvableRepresentation, then the assumed representation is the adjoint representation of the algebra (or current frame).

 Alg2 > $\mathrm{L3}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg3},\left[5\right]\right],\left[\left[\left[1,2,1\right],-1\right],\left[\left[1,2,5\right],1\right],\left[\left[1,3,1\right],1\right],\left[\left[1,3,5\right],-1\right],\left[\left[1,4,1\right],2\right],\left[\left[1,4,2\right],1\right],\left[\left[1,4,3\right],1\right],\left[\left[2,3,1\right],-1\right],\left[\left[2,3,5\right],1\right],\left[\left[2,4,3\right],-1\right],\left[\left[2,5,1\right],1\right],\left[\left[2,5,5\right],-1\right],\left[\left[3,4,3\right],1\right],\left[\left[3,4,5\right],-1\right],\left[\left[3,5,1\right],-1\right],\left[\left[3,5,5\right],1\right],\left[\left[4,5,2\right],-1\right],\left[\left[4,5,3\right],-1\right],\left[\left[4,5,5\right],-2\right]\right]\right]\right)$
 ${\mathrm{L3}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e1}}{+}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{-}{\mathrm{e3}}{-}{2}{}{\mathrm{e5}}\right]$ (2.5)
 V2 > $\mathrm{DGsetup}\left(\mathrm{L3}\right):$

The adjoint representation of this algebra is not upper triangular.

 Alg3 > $\mathrm{Adjoint}\left(\right)$
 Alg3 > $B≔\mathrm{SolvableRepresentation}\left(\mathrm{Alg3},\mathrm{output}=\left["NewBasis"\right]\right)$
 ${B}{:=}\left[{\mathrm{e2}}{+}{\mathrm{e3}}{+}{\mathrm{e5}}{,}{\mathrm{e1}}{-}{\mathrm{e5}}{,}{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e4}}\right]$ (2.6)
 Alg3 > $\mathrm{L4}≔\mathrm{LieAlgebraData}\left(B,\mathrm{Alg4}\right)$
 ${\mathrm{L4}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e2}}{+}{\mathrm{e3}}{+}{\mathrm{e4}}\right]$ (2.7)
 Alg3 > $\mathrm{DGsetup}\left(\mathrm{L4}\right):$

Now in this new basis the adjoint representation is upper triangular.

 Alg4 > $\mathrm{Adjoint}\left(\right)$

Example 4.

An example with complex eigenvalues.

 Alg4 > $L≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg5},\left[5\right]\right],\left[\left[\left[1,2,1\right],5\right],\left[\left[1,2,2\right],-5\right],\left[\left[1,2,3\right],-3\right],\left[\left[1,2,5\right],-2\right],\left[\left[1,3,1\right],-1\right],\left[\left[1,3,2\right],1\right],\left[\left[1,3,3\right],-1\right],\left[\left[1,3,5\right],2\right],\left[\left[1,4,1\right],4\right],\left[\left[1,4,2\right],-3\right],\left[\left[1,4,3\right],-3\right],\left[\left[1,4,4\right],-1\right],\left[\left[1,4,5\right],-1\right],\left[\left[1,5,1\right],-4\right],\left[\left[1,5,2\right],5\right],\left[\left[1,5,3\right],3\right],\left[\left[1,5,4\right],-1\right],\left[\left[1,5,5\right],1\right],\left[\left[2,3,1\right],-2\right],\left[\left[2,3,2\right],2\right],\left[\left[2,3,5\right],2\right],\left[\left[2,4,2\right],1\right],\left[\left[2,4,4\right],-1\right],\left[\left[2,5,1\right],-4\right],\left[\left[2,5,2\right],4\right],\left[\left[2,5,3\right],3\right],\left[\left[2,5,5\right],1\right],\left[\left[3,4,1\right],1\right],\left[\left[3,4,2\right],-1\right],\left[\left[3,4,5\right],-1\right],\left[\left[3,5,1\right],-1\right],\left[\left[3,5,2\right],2\right],\left[\left[3,5,3\right],1\right],\left[\left[3,5,4\right],-1\right],\left[\left[4,5,1\right],-3\right],\left[\left[4,5,2\right],3\right],\left[\left[4,5,3\right],3\right]\right]\right]\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{5}{}{\mathrm{e1}}{-}{5}{}{\mathrm{e2}}{-}{3}{}{\mathrm{e3}}{-}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e2}}{-}{\mathrm{e3}}{+}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{4}{}{\mathrm{e1}}{-}{3}{}{\mathrm{e2}}{-}{3}{}{\mathrm{e3}}{-}{\mathrm{e4}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{4}{}{\mathrm{e1}}{+}{5}{}{\mathrm{e2}}{+}{3}{}{\mathrm{e3}}{-}{\mathrm{e4}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e1}}{+}{2}{}{\mathrm{e2}}{+}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{4}{}{\mathrm{e1}}{+}{4}{}{\mathrm{e2}}{+}{3}{}{\mathrm{e3}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e2}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{+}{2}{}{\mathrm{e2}}{+}{\mathrm{e3}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{3}{}{\mathrm{e1}}{+}{3}{}{\mathrm{e2}}{+}{3}{}{\mathrm{e3}}\right]$ (2.8)
 Alg4 > $\mathrm{DGsetup}\left(L\right):$
 Alg5 > $\mathrm{B1},\mathrm{C1}≔\mathrm{SolvableRepresentation}\left(\mathrm{Alg5},\mathrm{output}=\left["NewBasis","Partition"\right]\right)$
 ${\mathrm{B1}}{,}{\mathrm{C1}}{:=}\left[{\mathrm{e1}}{-}{\mathrm{e2}}{-}{\mathrm{e3}}{,}{\mathrm{e1}}{-}{\mathrm{e4}}{-}{\mathrm{e5}}{,}{\mathrm{e2}}{-}{\mathrm{e4}}{,}{\mathrm{e1}}{,}{\mathrm{e2}}\right]{,}\left[{1}{..}{1}{,}{2}{..}{3}{,}{4}{..}{5}\right]$ (2.9)
 Alg5 > $\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{B1},\mathrm{Alg6}\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{4}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{3}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{+}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{3}{}{\mathrm{e1}}{+}{2}{}{\mathrm{e2}}{-}{2}{}{\mathrm{e3}}\right]$ (2.10)

In this new basis the adjoint representation is upper triangular except for a 2x2 "complex" block on the diagonal for ad(e4).

 Alg5 > $\mathrm{Adjoint}\left(\mathrm{L2}\right)$

We rerun this example with the option fieldextension = I

 Alg5 > $\mathrm{B3}≔\mathrm{SolvableRepresentation}\left(\mathrm{Alg5},\mathrm{fieldextension}=I,\mathrm{output}=\left["NewBasis"\right]\right)$
 ${\mathrm{B3}}{:=}\left[{\mathrm{e1}}{-}{\mathrm{e2}}{-}{\mathrm{e3}}{,}{\mathrm{e1}}{-}\left({1}{+}{I}\right){}{\mathrm{e2}}{+}{I}{}{\mathrm{e4}}{-}{\mathrm{e5}}{,}{\mathrm{e1}}{-}\left({1}{-}{I}\right){}{\mathrm{e2}}{-}{I}{}{\mathrm{e4}}{-}{\mathrm{e5}}{,}{\mathrm{e1}}{,}{\mathrm{e2}}\right]$ (2.11)
 Alg5 > $\mathrm{L3}≔\mathrm{LieAlgebraData}\left(\mathrm{B3},\mathrm{Alg7}\right)$
 ${\mathrm{L3}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{4}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{3}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}\left({1}{+}{I}\right){}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}\left({1}{-}{I}\right){}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{3}{}{\mathrm{e1}}{+}{\mathrm{e2}}{+}{\mathrm{e3}}\right]$ (2.12)
 Alg5 > $\mathrm{Adjoint}\left(\mathrm{L3}\right)$

Example 5.

Let be a representation of a nilpotent Lie algebra $\mathrm{𝔤}$ on a vector space $V$. The representation is called a nilrepresentation if each matrix is nilpotent, that is  ${A}^{k}=0$ for some $k.$  Engel's theorem (see, for example, Fulton and Harris, page 125 or Varadarajan, page 189) asserts that if rho is a nilrepresentation, then there is a basis for V for which all the representation matrices are strictly upper triangular.

 Alg5 > $\mathrm{L5}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg5},\left[6\right]\right],\left[\left[\left[1,2,2\right],1\right],\left[\left[1,2,3\right],1\right],\left[\left[1,2,4\right],-1\right],\left[\left[1,2,5\right],1\right],\left[\left[1,3,3\right],-\frac{1}{2}\right],\left[\left[1,3,5\right],\frac{1}{2}\right],\left[\left[1,3,6\right],-\frac{1}{2}\right],\left[\left[1,4,2\right],1\right],\left[\left[1,4,3\right],1\right],\left[\left[1,4,4\right],-1\right],\left[\left[1,4,5\right],1\right],\left[\left[1,5,3\right],\frac{1}{2}\right],\left[\left[1,5,5\right],-\frac{1}{2}\right],\left[\left[1,5,6\right],\frac{1}{2}\right],\left[\left[1,6,3\right],1\right],\left[\left[1,6,5\right],-1\right],\left[\left[1,6,6\right],1\right],\left[\left[2,3,3\right],-\frac{1}{2}\right],\left[\left[2,3,5\right],-\frac{1}{2}\right],\left[\left[2,3,6\right],-\frac{1}{2}\right],\left[\left[2,4,5\right],-1\right],\left[\left[2,6,3\right],\frac{1}{2}\right],\left[\left[2,6,5\right],\frac{1}{2}\right],\left[\left[2,6,6\right],\frac{1}{2}\right],\left[\left[3,4,5\right],1\right],\left[\left[4,6,5\right],1\right]\right]\right]\right)$
 ${\mathrm{L5}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}{-}{\mathrm{e4}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}\frac{{1}}{{2}}{}{\mathrm{e3}}{+}\frac{{1}}{{2}}{}{\mathrm{e5}}{-}\frac{{1}}{{2}}{}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}{-}{\mathrm{e4}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}\frac{{1}}{{2}}{}{\mathrm{e3}}{-}\frac{{1}}{{2}}{}{\mathrm{e5}}{+}\frac{{1}}{{2}}{}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{-}{\mathrm{e5}}{+}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}\frac{{1}}{{2}}{}{\mathrm{e3}}{-}\frac{{1}}{{2}}{}{\mathrm{e5}}{-}\frac{{1}}{{2}}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}\frac{{1}}{{2}}{}{\mathrm{e3}}{+}\frac{{1}}{{2}}{}{\mathrm{e5}}{+}\frac{{1}}{{2}}{}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}\right]$ (2.13)
 Alg5 > $\mathrm{DGsetup}\left(\mathrm{L5}\right):$
 Alg5 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],\mathrm{V5}\right):$
 V5 > $\mathrm{M5}≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[-5,-9,10,-4\right],\left[-4,-7,8,-3\right],\left[-5,-9,10,-4\right],\left[3,5,-6,2\right]\right],\left[\left[-8,-12,14,-6\right],\left[-5,-8,9,-4\right],\left[-9,-14,16,-7\right],\left[0,0,0,0\right]\right],\left[\left[-1,-2,2,-1\right],\left[0,0,0,0\right],\left[0,0,0,0\right],\left[1,2,-2,1\right]\right],\left[\left[-5,-8,9,-4\right],\left[0,0,0,0\right],\left[-5,-8,9,-4\right],\left[-5,-8,9,-4\right]\right],\left[\left[-1,-2,2,-1\right],\left[0,0,0,0\right],\left[-1,-2,2,-1\right],\left[-1,-2,2,-1\right]\right],\left[\left[-2,-4,4,-2\right],\left[-2,-4,4,-2\right],\left[-3,-6,6,-3\right],\left[0,0,0,0\right]\right]\right]\right):$
 V5 > $\mathrm{ρ}≔\mathrm{Representation}\left(\mathrm{Alg5},\mathrm{V5},\mathrm{M5}\right)$

Check that Alg5 is a nilpotent algebra, that rho is a representation, and that rho is a nilrepresentation.

 Alg5 > $\mathrm{Query}\left(\mathrm{Alg5},"Nilpotent"\right)$
 ${\mathrm{true}}$ (2.14)
 Alg5 > $\mathrm{Query}\left(\mathrm{ρ},"Representation"\right)$
 ${\mathrm{true}}$ (2.15)
 Alg5 > $\mathrm{Query}\left(\mathrm{ρ},"NilRepresentation"\right)$
 ${\mathrm{true}}$ (2.16)
 Alg5 > $B≔\mathrm{SolvableRepresentation}\left(\mathrm{Alg5},\mathrm{output}=\left["NewBasis"\right]\right)$
 ${B}{:=}\left[{\mathrm{e2}}{-}{\mathrm{e6}}{,}{\mathrm{e3}}{+}{\mathrm{e6}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e1}}{,}{\mathrm{e2}}\right]$ (2.17)
 Alg5 > $\mathrm{L5a}≔\mathrm{LieAlgebraData}\left(B,\mathrm{Alg5a}\right)$
 ${\mathrm{L5a}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e3}}{-}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}\frac{{1}}{{2}}{}{\mathrm{e2}}{+}\frac{{1}}{{2}}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}\frac{{1}}{{2}}{}{\mathrm{e2}}{+}\frac{{1}}{{2}}{}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e2}}{+}{\mathrm{e3}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}\frac{{1}}{{2}}{}{\mathrm{e2}}{+}\frac{{1}}{{2}}{}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{-}{\mathrm{e3}}{+}{\mathrm{e4}}\right]$ (2.18)

In this new basis the ad matrices are all nilpotent.

 Alg5 > $\mathrm{Adjoint}\left(\mathrm{L5a}\right)$