 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 > $\mathrm{ρ1}≔\mathrm{Representation}\left(\mathrm{alg1},\mathrm{V1},M\right)$
 ${\mathrm{ρ1}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrrr}{8}& {8}& {0}& {0}& {0}\\ {-}{1}& {5}& {6}& {0}& {0}\\ {0}& {-}{2}& {2}& {4}& {0}\\ {0}& {0}& {-}{3}& {-}{1}& {2}\\ {0}& {0}& {0}& {-}{4}& {-}{4}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrrr}{8}& {16}& {0}& {0}& {0}\\ {-}{1}& {4}& {12}& {0}& {0}\\ {0}& {-}{2}& {0}& {8}& {0}\\ {0}& {0}& {-}{3}& {-}{4}& {4}\\ {0}& {0}& {0}& {-}{4}& {-}{8}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrrr}{-}{4}& {-}{8}& {0}& {0}& {0}\\ {1}& {-}{1}& {-}{6}& {0}& {0}\\ {0}& {2}& {2}& {-}{4}& {0}\\ {0}& {0}& {3}& {5}& {-}{2}\\ {0}& {0}& {0}& {4}& {8}\end{array}\right]\right]\right]$ (2.2)

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

 alg1 > $\mathrm{B1},\mathrm{P1},\mathrm{newrho},\mathrm{Part1}≔\mathrm{SolvableRepresentation}\left(\mathrm{ρ1}\right)$
 ${\mathrm{B1}}{,}{\mathrm{P1}}{,}{\mathrm{newrho}}{,}{\mathrm{Part1}}{:=}\left[{\mathrm{D_x1}}{-}\frac{{1}}{{2}}{}{\mathrm{D_x2}}{+}\frac{{1}}{{4}}{}{\mathrm{D_x3}}{-}\frac{{1}}{{8}}{}{\mathrm{D_x4}}{+}\frac{{1}}{{16}}{}{\mathrm{D_x5}}{,}{\mathrm{D_x1}}{-}\frac{{1}}{{4}}{}{\mathrm{D_x3}}{+}\frac{{1}}{{4}}{}{\mathrm{D_x4}}{-}\frac{{3}}{{16}}{}{\mathrm{D_x5}}{,}{\mathrm{D_x1}}{-}\frac{{1}}{{6}}{}{\mathrm{D_x2}}{+}\frac{{1}}{{48}}{}{\mathrm{D_x5}}{,}{\mathrm{D_x1}}{-}\frac{{1}}{{16}}{}{\mathrm{D_x5}}{,}{\mathrm{D_x1}}\right]{,}\left[\begin{array}{ccccc}{1}& {1}& {1}& {1}& {1}\\ {-}\frac{{1}}{{2}}& {0}& {-}\frac{{1}}{{6}}& {0}& {0}\\ \frac{{1}}{{4}}& {-}\frac{{1}}{{4}}& {0}& {0}& {0}\\ {-}\frac{{1}}{{8}}& \frac{{1}}{{4}}& {0}& {0}& {0}\\ \frac{{1}}{{16}}& {-}\frac{{3}}{{16}}& \frac{{1}}{{48}}& {-}\frac{{1}}{{16}}& {0}\end{array}\right]{,}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{ccccc}{4}& {5}& {3}& {-}{1}& {0}\\ {0}& {3}& \frac{{5}}{{3}}& {-}{1}& {0}\\ {0}& {0}& {2}& {9}& {6}\\ {0}& {0}& {0}& {1}& {2}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{ccccc}{0}& {8}& \frac{{10}}{{3}}& {-}{2}& {0}\\ {0}& {0}& {2}& {-}{2}& {0}\\ {0}& {0}& {0}& {12}& {6}\\ {0}& {0}& {0}& {0}& {2}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{ccccc}{0}& {-}{5}& {-}{3}& {1}& {0}\\ {0}& {1}& {-}\frac{{5}}{{3}}& {1}& {0}\\ {0}& {0}& {2}& {-}{9}& {-}{6}\\ {0}& {0}& {0}& {3}& {-}{2}\\ {0}& {0}& {0}& {0}& {4}\end{array}\right]\right]\right]{,}\left[{1}{..}{1}{,}{2}{..}{2}{,}{3}{..}{3}{,}{4}{..}{4}{,}{5}{..}{5}\right]$ (2.3)

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

 V1 > $\mathrm{ChangeRepresentationBasis}\left(\mathrm{ρ1},\mathrm{B1},\mathrm{V1}\right)$
 $\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{ccccc}{4}& {5}& {3}& {-}{1}& {0}\\ {0}& {3}& \frac{{5}}{{3}}& {-}{1}& {0}\\ {0}& {0}& {2}& {9}& {6}\\ {0}& {0}& {0}& {1}& {2}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{ccccc}{0}& {8}& \frac{{10}}{{3}}& {-}{2}& {0}\\ {0}& {0}& {2}& {-}{2}& {0}\\ {0}& {0}& {0}& {12}& {6}\\ {0}& {0}& {0}& {0}& {2}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{ccccc}{0}& {-}{5}& {-}{3}& {1}& {0}\\ {0}& {1}& {-}\frac{{5}}{{3}}& {1}& {0}\\ {0}& {0}& {2}& {-}{9}& {-}{6}\\ {0}& {0}& {0}& {3}& {-}{2}\\ {0}& {0}& {0}& {0}& {4}\end{array}\right]\right]\right]$ (2.4)

Example 2.

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

 alg1 > $\mathrm{L2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg2},\left[3\right]\right],\left[\left[\left[1,3,2\right],-1\right],\left[\left[1,3,1\right],3\right],\left[\left[2,3,1\right],1\right],\left[\left[2,3,2\right],3\right]\right]\right]\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{+}{3}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{3}{}{\mathrm{e2}}\right]$ (2.5)
 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 > $M≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[0,0,0,0,0,0\right],\left[0,0,0,0,0,0\right],\left[-3,1,0\right]\right]\right]\right)$