SolvableRepresentation - Maple Help

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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

Description

• 

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 abba 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 ρinvariant 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 3 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

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

(2.1)

DGsetupL:

alg1 > 

DGsetupx1,x2,x3,x4,x5,V1:

V1 > 

MmapMatrix,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:

V1 > 

ρ1Representationalg1,V1,M

ρ1:=e1,8800015600022400031200044,e2,816000141200020800034400048,e3,4800011600022400035200048

(2.2)

 

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

alg1 > 

B1,P1,newrho,Part1SolvableRepresentationρ1

B1,P1,newrho,Part1:=D_x112D_x2+14D_x318D_x4+116D_x5,D_x114D_x3+14D_x4316D_x5,D_x116D_x2+148D_x5,D_x1116D_x5,D_x1,111111201600141400018140001163161481160,e1,45310035310002960001200000,e2,0810320002200001260000200000,e3,05310015310002960003200004,1..1,2..2,3..3,4..4,5..5

(2.3)

 

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

V1 > 

ChangeRepresentationBasisρ1,B1,V1

e1,45310035310002960001200000,e2,0810320002200001260000200000,e3,05310015310002960003200004

(2.4)

 

Example 2.

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

alg1 > 

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

(2.5)
alg1 > 

DGsetupL2:

Alg2 > 

DGsetup