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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

 

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

alg1 > 

B1,P1,newrho,Part1SolvableRepresentationρ1

 

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

V1 > 

ChangeRepresentationBasisρ1,B1,V1

 

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

(2.2)
alg1 > 

DGsetupL2:

Alg2 > 

DGsetupx1,x2,x3,x4,x5,x6,V2:

V2 > 

MmapMatrix,0,0,0,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,0,3,0,1,0,0,0,0,23,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,2,0,23,0,23,2,0,0,0,0,1,23,0,1,0,0,0,0,3,0,1,0,0,2,0,23,0,0,0,0,1,0,3,0,0,0,0,0,0,0:

V2 > 

ρ2RepresentationAlg2,V2,M

 

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 > 

Queryρ2,Representation

true

(2.3)
Alg2 > 

B2,P2,newrho,Part2SolvableRepresentationρ2

V2 > 

ChangeRepresentationBasisρ2,B2,V2

 

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

Alg2 > 

B3SolvableRepresentationρ2,fieldextension=I,output=NewBasis

B3:=D_x6,D_x3ID_x5,D_x3+ID_x5,D_x1ID_x2D_x4,D_x1+D_x4,D_x1+ID_x2D_x4

(2.4)
V2 > 

ChangeRepresentationBasisρ2,B3,V2

 

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 > 

L3_DGLieAlgebra,Alg3,5,1,2,1,1,1,2,5,1,1,3,1,1,1,3,5,1,1,4,1,2,1,4,2,1,1,4,3,1,2,3,1,1,2,3,5,1,2,4,3,1,2,5,1,1,2,5,5,1,3,4,3,1,3,4,5,1,3,5,1,1,3,5,5,1,4,5,2,1,4,5,3,1,4,5,5,2

L3:=e1,e2=e1+e5,e1,e3=e1e5,e1,e4=2e1+e2+e3,e2,e3=e1+e5,e2,e4=e3,e2,e5=e1e5,e3,e4=e3e5,e3,e5=e1+e5,e4,e5=e2e32e5

(2.5)
V2 > 

DGsetupL3:

 

The adjoint representation of this algebra is not upper triangular.

Alg3 > 

Adjoint

Alg3 > 

BSolvableRepresentationAlg3,output=NewBasis

B:=e2+e3+e5,e1e5,e1,e2,e4

(2.6)
Alg3 > 

L4LieAlgebraDataB,Alg4

L4:=e1,e5=e1,e2,e5=2e2,e3,e4=e2,e3,e5=e1+e2+e3,e4,e5=e1e2+e3+e4

(2.7)
Alg3 > 

DGsetupL4:

 

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

Alg4 > 

Adjoint

 

Example 4.

An example with complex eigenvalues.

Alg4 > 

L_DGLieAlgebra,Alg5,5,1,2,1,5,1,2,2,5,1,2,3,3,1,2,5,2,1,3,1,1,1,3,2,1,1,3,3,1,1,3,5,2,1,4,1,4,1,4,2,3,1,4,3,3,1,4,4,1,1,4,5,1,1,5,1,4,1,5,2,5,1,5,3,3,1,5,4,1,1,5,5,1,2,3,1,2,2,3,2,2,2,3,5,2,2,4,2,1,2,4,4,1,2,5,1,4,2,5,2,4,2,5,3,3,2,5,5,1,3,4,1,1,3,4,2,1,3,4,5,1,3,5,1,1,3,5,2,2,3,5,3,1,3,5,4,1,4,5,1,3,4,5,2,3,4,5,3,3

L:=e1,e2=5e15e23e32e5,e1,e3=e1+e2e3+2e5,e1,e4=4e13e23e3e4e5,e1,e5=4e1+5e2+3e3e4+e5,e2,e3=2e1+2e2+2e5,e2,e4=e2e4,e2,e5=4e1+4e2+3e3+e5,e3,e4=e1e2e5,e3,e5=e1+2e2+e3e4,e4,e5=3e1+3e2+3e3

(2.8)
Alg4 > 

DGsetupL:

Alg5 > 

B1,C1SolvableRepresentationAlg5,output=NewBasis,Partition

B1,C1:=e1e2e3,e1e4e5,e2e4,e1,e2,1..1,2..3,4..5

(2.9)
Alg5 > 

L2LieAlgebraDataB1,Alg6

L2:=e1,e4=4e1,e1,e5=3e1,e2,e4=2e3,e2,e5=e2,e3,e4=e2+2e3,e3,e5=e3,e4,e5=3e1+2e22e3

(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 > 

AdjointL2

 

We rerun this example with the option fieldextension = I

Alg5 > 

B3SolvableRepresentationAlg5,fieldextension=I,output=NewBasis

B3:=e1e2e3,e11+Ie2+Ie4e5,e11Ie2Ie4e5,e1,e2

(2.11)
Alg5 > 

L3LieAlgebraDataB3,Alg7

L3:=e1,e4=4e1,e1,e5=3e1,e2,e4=1+Ie2,e2,e5=e2,e3,e4=1Ie3,e3,e5=e3,e4,e5=3e1+e2+e3

(2.12)
Alg5 > 

AdjointL3

 

Example 5.

Let rho:𝔤  V be a representation of a nilpotent Lie algebra 𝔤 on a vector space V. The representation is called a nilrepresentation if each matrix A=ρx is nilpotent, that is  Ak=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 > 

L5_DGLieAlgebra,Alg5,6,1,2,2,1,1,2,3,1,1,2,4,1,1,2,5,1,1,3,3,12,1,3,5,12,1,3,6,12,1,4,2,1,1,4,3,1,1,4,4,1,1,4,5,1,1,5,3,12,1,5,5,12,1,5,6,12,1,6,3,1,1,6,5,1,1,6,6,1,2,3,3,12,2,3,5,12,2,3,6,12,2,4,5,1,2,6,3,12,2,6,5,12,2,6,6,12,3,4,5,1,4,6,5,1

L5:=e1,e2=e2+e3e4+e5,e1,e3=12e3+12e512e6,e1,e4=e2+e3e4+e5,e1,e5=12e312e5+12e6,e1,e6=e3e5+e6,e2,e3=12e312e512e6,e2,e4=e5,e2,e6=12e3+12e5+12e6,e3,e4=e5,e4,e6=e5

(2.13)
Alg5 > 

DGsetupL5:

Alg5 > 

DGsetupx1,x2,x3,x4,V5:

V5 > 

M5mapMatrix,5,9,10,4,4,7,8,3,5,9,10,4,3,5,6,2,8,12,14,6,5,8,9,4,9,14,16,7,0,0,0,0,1,2,2,1,0,0,0,0,0,0,0,0,1,2,2,1,5,8,9,4,0,0,0,0,5,8,9,4,5,8,9,4,1,2,2,1,0,0,0,0,1,2,2,1,1,2,2,1,2,4,4,2,2,4,4,2,3,6,6,3,0,0,0,0:

V5 > 

ρRepresentationAlg5,V5,M5

 

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

Alg5 > 

QueryAlg5,Nilpotent

true

(2.14)
Alg5 > 

Queryρ,Representation

true

(2.15)
Alg5 > 

Queryρ,NilRepresentation

true

(2.16)
Alg5 > 

BSolvableRepresentationAlg5,output=NewBasis

B:=e2e6,e3+e6,e4,e5,e1,e2

(2.17)
Alg5 > 

L5aLieAlgebraDataB,Alg5a

L5a:=e1,e5=e1+e32e4,e1,e6=12e2+12e4,e2,e5=12e2+12e4,e3,e5=e1e2+e3e4,e3,e6=e4,e4,e5=12e2+12e4,e5,e6=e1+e2e3+e4

(2.18)

 

In this new basis the ad matrices are all nilpotent.

Alg5 > 

AdjointL5a

See Also

DifferentialGeometry

Library

LieAlgebras

Adjoint

ChangeRepresentationBasis

Query