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( ρ, options)
SolvableRepresentation(Alg, options)
Parameters
ρ - a representation of a solvable Lie algebra 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
Examples
Let rho: 𝔤 → glVbe a representation of a solvable Lie algebra 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 V such that the matrix representation of ρxis upper triangular for all x ∈𝔤.
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 a + bI), the matrix representation will not be upper triangular but will contain the matrix ab−ba 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 ℬ forV 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 P = 1.. n1, n1+1 .. n2, n2+1 .. n3, ... , then the subspaces ℬ1, ..., n1, ℬ1, ..., n2, ℬ1, ..., n3 are ρ−invariant subspaces. If, for example, P = 1.. 1, 2.. 2 , 3.. 3, then all the eigenvectors calculated by RepresentationEigenvector are real. If C = 1..1, 2..3 then the vectors ℬ2 and ℬ3 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.
withDifferentialGeometry:withLieAlgebras:withLibrary:
Example 1.
We define a 5-dimensional representation of a 3-dimensional solvable Lie algebra.
L≔_DGLieAlgebra,alg1,3,1,2,2,1,2,3,2,1
L:=e1,e2=e2,e2,e3=e2
DGsetupL:
DGsetupx1,x2,x3,x4,x5,V1:
M≔mapMatrix,8,8,0,0,0,−1,5,6,0,0,0,−2,2,4,0,0,0,−3,−1,2,0,0,0,−4,−4,8,16,0,0,0,−1,4,12,0,0,0,−2,0,8,0,0,0,−3,−4,4,0,0,0,−4,−8,−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:
ρ1≔Representationalg1,V1,M
ρ1:=e1,88000−156000−224000−3−12000−4−4,e2,816000−1412000−208000−3−44000−4−8,e3,−4−80001−1−600022−400035−200048
We find a new basis for the representation space in which the matrices are all upper triangular.
B1,P1,newrho,Part1≔SolvableRepresentationρ1
B1,P1,newrho,Part1:=D_x1−12D_x2+14D_x3−18D_x4+116D_x5,D_x1−14D_x3+14D_x4−316D_x5,D_x1−16D_x2+148D_x5,D_x1−116D_x5,D_x1,11111−120−160014−14000−1814000116−316148−1160,e1,453−100353−10002960001200000,e2,08103−20002−200001260000200000,e3,0−5−31001−5310002−9−60003−200004,1..1,2..2,3..3,4..4,5..5
To verify this result we use the ChangeRepresentationBasis command to change basis in the representation space.
ChangeRepresentationBasisρ1,B1,V1
e1,453−100353−10002960001200000,e2,08103−20002−200001260000200000,e3,0−5−31001−5310002−9−60003−200004
Example 2.
We define a 6-dimensional representation of a 3-dimensional solvable Lie algebra.
L2≔_DGLieAlgebra,Alg2,3,1,3,2,−1,1,3,1,3,2,3,1,1,2,3,2,3
L2:=e1,e3=−e2+3e1,e2,e3=e1+3e2
DGsetupL2:
DGsetupx1,x2,x3,x4,x5,x6,V2:
M≔mapMatrix,0,0,0,0,0,0,0,0,0,0,0,0,&m
