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LieAlgebras[RepresentationEigenvector] - find a simultaneous eigenvector for the representation of a solvable Lie algebra

Calling Sequences

     RepresentationEigenvector(ρ)

Parameters

     ρ       - a representation of a solvable Lie algebra 𝔤 on a vector space V

 

Description

Description

• 

If  ρ: 𝔤  glV is a representation of a solvable Lie algebra 𝔤, then a fundamental theorem due to Lie (see, for example, Fulton and Harris, page 125 or Varadarajan, page 200) asserts that there is a vector Y V  such that  ρxY= λxY for all x 𝔤.  The eigenvector Y  is called a simultaneous eigenvector for the representation .  The vector Y may be complex and will, in general, not be unique.  

• 

 Let e1, e2, ... , en  be the given basis for 𝔤.  The program RepresentationEigenvector(ρ) returns  the list of eigenvalues λe1, λe2, ... , λen and the simultaneous eigenvector Y.

• 

The program RepresentationEigenvector(ρ) works as follows. First a change of basis is made using the program AscendingIdealsBasis so that in the new basis f1 , f2, ... , fn the subalgebra spanned by f1 , f2, ... , fk is an ideal in the subalgebra f1 , f2, ... , fk+1. Then an eigenspace E1V  for the the linear transformation ρf1 is found. This space is invariant under ρf2 so the program next finds a subspace E2  E1  which is a eigenspace for  ρf1 and so on.

 

V > 

with(DifferentialGeometry): with(LieAlgebras): with(Library):

 

Example 1.

First we define a representation of a 4 dimensional algebra.

L1 := LieAlgebraData([[x2, x4] = x1, [x3 ,x4] = x3], [x1, x2, x3, x4], Alg1);

L1:=e2,e4=e1,e3,e4=e3

(2.1)

DGsetup(L1);

Lie algebra: Alg1

(2.2)
Alg1 > 

DGsetup([x1, x2, x3, x4], V);

frame name: V

(2.3)
V > 

M1 := [Matrix([[0, 0, 0, 0], [0, 0, 0, - 1], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, - 1], [0, 0, 0, 0]]), Matrix([[0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix([[- 1, 0, 0, 0], [0, 0, - 1, 0], [0, 0, 0, 0], [0, 0, 0, 0]])]:

V > 

rho1 := Representation(Alg1, V, M1);

ρ1:=e1,0000000100000000,e2,0000000000010000,e3,0001000000000000,e4,1000001000000000

(2.4)

 

Calculate a common eigenvector. The program returns the eigenvalues and the eigenvectors.

Alg1 > 

RepresentationEigenvector(rho1);

0,0,0,1,D_x1

(2.5)
V > 

seq(ApplyRepresentation(rho1, v, D_x1), v = [e1, e2, e3, e4]);

0D_x1,0D_x1,0D_x1,D_x1

(2.6)

 

We remark that this representation admits an invariant vector Dx2 which is also a simultaneous eigenvector with eigenvalues 0, 0,0,0.

V > 

Invariants(rho1);

D_x2

(2.7)

 

Example 2.

When there is more than one simultaneous eigenvector, the one found by the procedure RepresentationEigenvector may depend upon the basis chosen for the representation space.  For example, if we change basis for the representation space in Example 1, the program returns a vector with eigenvalues 0,0,0,0 - that is, an invariant vector.

V > 

P := Matrix([[1, 0, 1, 0], [0, 1, - 1, 0], [2, 0, 1, - 2], [1, 2, 0, 1]]);

P:=1010011020121201

(2.8)
V > 

rho2 := ChangeRepresentationBasis(rho1, P, "Range", V);

ρ2:=e1,4804360348042402,e2,1201120112011201,e3,51005480448043603,e4,1309810076120887054

(2.9)
Alg1 > 

y := RepresentationEigenvector(rho2);

y:=0,0,0,0,D_x134D_x2D_x3+12D_x4

(2.10)
V > 

seq(ApplyRepresentation(rho2, v, y[2][1]), v = [e1, e2, e3, e4]);

0D_x1,0D_x1,0D_x1,0D_x1

(2.11)

 

Example 3.

Here we give a simple example where one of the eigenvalues is complex.

V > 

L := Retrieve("Winternitz", 1, [3, 7], Alg2);

L:=e1,e3=_ae1e2,e2,e3=_ae2+e1

(2.12)
V > 

DGsetup(L):

Alg2 > 

DGsetup([x1, x2, x3], V2):

V2 > 

rho3 := Representation(Alg2, V2, Adjoint(Alg2));

ρ3:=e1,00_a001000,e2,00100_a000,e3,_a101_a0000

(2.13)
Alg2 > 

y := RepresentationEigenvector(rho3);

y:=0,0,_a+I,D_x1ID_x2

(2.14)
V2 > 

seq(ApplyRepresentation(rho3, v, y[2][1]), v = [e1, e2, e3]);

0D_x1,0D_x1,_aID_x1+1+I_aD_x2

(2.15)

 

Example 4.

We redo Example 1 with a change of basis for the algebra.  We also set infolevel[RepresentationEigenvector] = 2 so that the sequence of common eigenvectors is displayed.

V2 > 

ChangeFrame(Alg1):

Alg1 > 

B := evalDG([e1 - e3 + 2*e4, e3, e2 + 2*e4, e1 - e3 + e4]);

B:=e1e3+2e4,e3,e2+2e4,e1e3+e4

(2.16)
Alg1 > 

L4 := LieAlgebraData(B, Alg4);

L4:=e1,e2=2e2,e1,e3=2e14e24e4,e1,e4=e2,e2,e3=2e2,e2,e4=e2,e3,e4=e1+3e2+2e4

(2.17)
Alg1 > 

DGsetup(L4, [f], [alpha]);

Lie algebra: Alg4

(2.18)
Alg4 > 

rho4 := ChangeRepresentationBasis(rho2, B, Alg4);

ρ4:=f1,3518187271414532161681910103,f2,51005480448043603,f3,252181719214132321617132109,f4,221891171471201680121051

(2.19)
Alg4 > 

infolevel[RepresentationEigenvector] := 3:

Alg4 > 

y := RepresentationEigenvector(rho4);

The common eigenspace for the first 1 vectors is [-D_x1+D_x4, D_x3, -2*D_x1+D_x2]

The common eigenspace for the first 2 vectors is [-4/3*D_x1+D_x2+4/3*D_x3-2/3*D_x4]
The common eigenspace for the first 3 vectors is [-4/3*D_x1+D_x2+4/3*D_x3-2/3*D_x4]

The common eigenspace for the first 4 vectors is [-4/3*D_x1+D_x2+4/3*D_x3-2/3*D_x4]

y:=0,0,0,0,D_x134D_x2D_x3+12D_x4

(2.20)
V > 

seq(ApplyRepresentation(rho4, f||i, y[2][1]) &minus (y[1][i] &mult y[2][1]), i = 1..4);

0D_x1,0D_x1,0D_x1,0D_x1

(2.21)

See Also

DifferentialGeometry

Library

LieAlgebras

ApplyRepresentation

ChangeBasis

infolevel

Invariants

Representation

Retrieve