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LieAlgebras[RepresentationEigenvector] - find a simultaneous eigenvector for the representation of a solvable Lie algebra
Calling Sequences
RepresentationEigenvector(rho)
Parameters
rho - a representation of a solvable Lie algebra g on a vector space V
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Description
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If rho: g -> gl(V) 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 in V such that rho(x)(y) = lamba_x y for all x in g. The eigenvector y may be complex and will, in general, not be unique. The program RepresentationEigenvector(rho) returns one such vector.
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The program RepresentationEigenvector(rho) works as follows: First a change of basis is made using the program AscendingIdealsBasis so that in the new basis [f_1, f_2, ..., f_n] the vectors [f_1, f_2, ..., f_k] form an ideal in [f_1, f_2, ..., f_(k + 1)]. Then an eigenspace E1 for rho(f1) is found. This space is invariant under rho(f2) so the program next finds a subspace E2 in E1 which is an eigenspace for rho(f2) and so on.
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Examples
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Example 1.
First we define a representation of a 4 dimensional algebra.
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![L1 := LieAlgebraData([[x2, x4] = x1, [x3, x4] = x3], [x1, x2, x3, x4], Alg1)](/support/helpjp/helpview.aspx?si=6630/file05830/math90.png)
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![_DG([["LieAlgebra", Alg1, [4, table( [ ] )]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]])](/support/helpjp/helpview.aspx?si=6630/file05830/math93.png)
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| (2.2) |
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Alg1 >
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![DGsetup([x1, x2, x3, x4], V)](/support/helpjp/helpview.aspx?si=6630/file05830/math104.png)
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| (2.3) |
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![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]])]](/support/helpjp/helpview.aspx?si=6630/file05830/math111.png)
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Calculate a common eigenvector. The program returns the eigenvalues and the eigenvectors.
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Alg1 >
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![[0, 0, 0, -1], [_DG([["vector", V, []], [[[1], 1]]])]](/support/helpjp/helpview.aspx?si=6630/file05830/math129.png)
| (2.5) |
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![seq(ApplyRepresentation(rho1, v, D_x1), v = [e1, e2, e3, e4])](/support/helpjp/helpview.aspx?si=6630/file05830/math133.png)
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![_DG([["vector", V, []], [[[1], 0]]]), _DG([["vector", V, []], [[[1], 0]]]), _DG([["vector", V, []], [[[1], 0]]]), _DG([["vector", V, []], [[[1], -1]]])](/support/helpjp/helpview.aspx?si=6630/file05830/math136.png)
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We remark that this representation admits an invariant vector D_x2, which is also a simultaneous eigenvector with eigenvalues [0, 0, 0, 0].
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![[_DG([["vector", V, []], [[[2], 1]]])]](/support/helpjp/helpview.aspx?si=6630/file05830/math151.png)
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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.
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![P := Matrix([[1, 0, 1, 0], [0, 1, -1, 0], [2, 0, 1, -2], [1, 2, 0, 1]])](/support/helpjp/helpview.aspx?si=6630/file05830/math165.png)
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| (2.8) |
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| (2.9) |
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Alg1 >
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![[0, 0, 0, 0], [_DG([["vector", V, []], [[[1], 1], [[2], -3/4], [[3], -1], [[4], 1/2]]])]](/support/helpjp/helpview.aspx?si=6630/file05830/math182.png)
| (2.10) |
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![seq(ApplyRepresentation(rho2, v, y[2][1]), v = [e1, e2, e3, e4])](/support/helpjp/helpview.aspx?si=6630/file05830/math186.png)
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![_DG([["vector", V, []], [[[1], 0]]]), _DG([["vector", V, []], [[[1], 0]]]), _DG([["vector", V, []], [[[1], 0]]]), _DG([["vector", V, []], [[[1], 0]]])](/support/helpjp/helpview.aspx?si=6630/file05830/math189.png)
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Example 3.
Here we give a simple example where one of the eigenvalues is complex.
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![L := Retrieve("Winternitz", 1, [3, 7], Alg2)](/support/helpjp/helpview.aspx?si=6630/file05830/math199.png)
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![_DG([["LieAlgebra", Alg2, [3]], [[[1, 3, 2], -1], [[1, 3, 1], a], [[2, 3, 1], 1], [[2, 3, 2], a]]])](/support/helpjp/helpview.aspx?si=6630/file05830/math202.png)
| (2.12) |
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Alg2 >
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![DGsetup([x1, x2, x3], V2)](/support/helpjp/helpview.aspx?si=6630/file05830/math210.png)
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V2 >
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![rho3 := [[e1, Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = a, (2, 1) = 0, (2, 2) = 0, (2, 3) = -1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0})], [e2, Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = a, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0})], [e3, Matrix(3, 3, {(1, 1) = -a, (1, 2) = -1, (1, 3) = 0, (2, 1) = 1, (2, 2) = -a, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0})]]](/support/helpjp/helpview.aspx?si=6630/file05830/math217.png)
| (2.13) |
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Alg2 >
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![[0, 0, -a+I], [_DG([["vector", V2, []], [[[1], 1], [[2], -I]]])]](/support/helpjp/helpview.aspx?si=6630/file05830/math224.png)
| (2.14) |
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![seq(ApplyRepresentation(rho3, v, y[2][1]), v = [e1, e2, e3])](/support/helpjp/helpview.aspx?si=6630/file05830/math228.png)
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![_DG([["vector", V2, []], [[[1], 0]]]), _DG([["vector", V2, []], [[[1], 0]]]), _DG([["vector", V2, []], [[[1], -a+I], [[2], 1+I*a]]])](/support/helpjp/helpview.aspx?si=6630/file05830/math231.png)
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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.
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V2 >
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Alg1 >
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![B := evalDG([e1-e3+2*e4, e3, e2+2*e4, e1-e3+e4])](/support/helpjp/helpview.aspx?si=6630/file05830/math247.png)
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![[_DG([["vector", Alg1, []], [[[1], 1], [[3], -1], [[4], 2]]]), _DG([["vector", Alg1, []], [[[3], 1]]]), _DG([["vector", Alg1, []], [[[2], 1], [[4], 2]]]), _DG([["vector", Alg1, []], [[[1], 1], [[3], -1], [[4], 1]]])]](/support/helpjp/helpview.aspx?si=6630/file05830/math250.png)
| (2.16) |
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Alg1 >
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![_DG([["LieAlgebra", Alg4, [4, table( [ ] )]], [[[1, 2, 2], -2], [[1, 3, 1], 2], [[1, 3, 2], -4], [[1, 3, 4], -4], [[1, 4, 2], 1], [[2, 3, 2], 2], [[2, 4, 2], 1], [[3, 4, 1], -1], [[3, 4, 2], 3], [[3, 4, 4], 2]]])](/support/helpjp/helpview.aspx?si=6630/file05830/math257.png)
| (2.17) |
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Alg1 >
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![DGsetup(L4, [f], [alpha])](/support/helpjp/helpview.aspx?si=6630/file05830/math261.png)
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| (2.18) |
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Alg4 >
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| (2.19) |
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Alg4 >
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![infolevel[RepresentationEigenvector] := 3](/support/helpjp/helpview.aspx?si=6630/file05830/math275.png)
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Alg4 >
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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]
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![[0, 0, 0, 0], [_DG([["vector", V, []], [[[1], 1], [[2], -3/4], [[3], -1], [[4], 1/2]]])]](/support/helpjp/helpview.aspx?si=6630/file05830/math289.png)
| (2.20) |
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![seq(`&minus`(ApplyRepresentation(rho4, f || i, y[2][1]), `&mult`(y[1][i], y[2][1])), i = 1 .. 4)](/support/helpjp/helpview.aspx?si=6630/file05830/math293.png)
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![_DG([["vector", V, []], [[[1], 0]]]), _DG([["vector", V, []], [[[1], 0]]]), _DG([["vector", V, []], [[[1], 0]]]), _DG([["vector", V, []], [[[1], 0]]])](/support/helpjp/helpview.aspx?si=6630/file05830/math296.png)
| (2.21) |
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