RepresentationEigenvector - Maple Help

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

Calling Sequences

RepresentationEigenvector(${\mathbf{ρ}}$)

Parameters

$\mathrm{ρ}$       - a representation of a solvable Lie algebra $\mathrm{𝔤}$ on a vector space $V$

Description

 • If   is a representation of a solvable Lie algebra $\mathrm{𝔤}$, 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  such that  $\mathrm{ρ}$for all .  The eigenvector  is called a simultaneous eigenvector for the representation .  The vector may be complex and will, in general, not be unique.
 • Let  be the given basis for $𝔤$.  The program RepresentationEigenvector(${\mathbf{ρ}}$) returns  the list of eigenvalues  and the simultaneous eigenvector $Y$.
 • The program RepresentationEigenvector(${\mathbf{ρ}}$) works as follows. First a change of basis is made using the program AscendingIdealsBasis so that in the new basis the subalgebra spanned by  is an ideal in the subalgebra . Then an eigenspace for the linear transformation is found. This space is invariant under $\mathrm{\rho }\left({f}_{2}\right)$ so the program next finds a subspace  which is a eigenspace for  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);
 ${\mathrm{L1}}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (2.1)
 > DGsetup(L1);
 ${\mathrm{Lie algebra: Alg1}}$ (2.2)
 Alg1 > DGsetup([x1, x2, x3, x4], V);
 ${\mathrm{frame name: V}}$ (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);
 ${\mathrm{ρ1}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrrr}{-}{1}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.4)

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

 Alg1 > RepresentationEigenvector(rho1);
 $\left[{0}{,}{0}{,}{0}{,}{-}{1}\right]{,}\left[{\mathrm{D_x1}}\right]$ (2.5)
 V > seq(ApplyRepresentation(rho1, v, D_x1), v = [e1, e2, e3, e4]);
 ${0}{}{\mathrm{D_x1}}{,}{0}{}{\mathrm{D_x1}}{,}{0}{}{\mathrm{D_x1}}{,}{-}{\mathrm{D_x1}}$ (2.6)

We remark that this representation admits an invariant vector ${\mathrm{D}}_{{x}^{2}}$ which is also a simultaneous eigenvector with eigenvalues .

 V > Invariants(rho1);
 $\left[{\mathrm{D_x2}}\right]$ (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 $\left[0,0,0,0\right]$ - that is, an invariant vector.

 V > P := Matrix([[1, 0, 1, 0], [0, 1, - 1, 0], [2, 0, 1, - 2], [1, 2, 0, 1]]);
 ${P}{:=}\left[\begin{array}{rrrr}{1}& {0}& {1}& {0}\\ {0}& {1}& {-}{1}& {0}\\ {2}& {0}& {1}& {-}{2}\\ {1}& {2}& {0}& {1}\end{array}\right]$ (2.8)
 V > rho2 := ChangeRepresentationBasis(rho1, P, "Range", V);
 ${\mathrm{ρ2}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrr}{-}{4}& {-}{8}& {0}& {-}{4}\\ {3}& {6}& {0}& {3}\\ {4}& {8}& {0}& {4}\\ {-}{2}& {-}{4}& {0}& {-}{2}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrr}{1}& {2}& {0}& {1}\\ {-}{1}& {-}{2}& {0}& {-}{1}\\ {-}{1}& {-}{2}& {0}& {-}{1}\\ {1}& {2}& {0}& {1}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrr}{5}& {10}& {0}& {5}\\ {-}{4}& {-}{8}& {0}& {-}{4}\\ {-}{4}& {-}{8}& {0}& {-}{4}\\ {3}& {6}& {0}& {3}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrrr}{-}{13}& {0}& {-}{9}& {8}\\ {10}& {0}& {7}& {-}{6}\\ {12}& {0}& {8}& {-}{8}\\ {-}{7}& {0}& {-}{5}& {4}\end{array}\right]\right]\right]$ (2.9)
 Alg1 > y := RepresentationEigenvector(rho2);
 ${y}{:=}\left[{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{\mathrm{D_x1}}{-}\frac{{3}}{{4}}{}{\mathrm{D_x2}}{-}{\mathrm{D_x3}}{+}\frac{{1}}{{2}}{}{\mathrm{D_x4}}\right]$ (2.10)
 V > seq(ApplyRepresentation(rho2, v, y[2][1]), v = [e1, e2, e3, e4]);
 ${0}{}{\mathrm{D_x1}}{,}{0}{}{\mathrm{D_x1}}{,}{0}{}{\mathrm{D_x1}}{,}{0}{}{\mathrm{D_x1}}$ (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);
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{_a}}{}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{_a}}{}{\mathrm{e2}}{+}{\mathrm{e1}}\right]$ (2.12)
 V > DGsetup(L):
 Alg2 > DGsetup([x1, x2, x3], V2):
 V2 > rho3 := Representation(Alg2, V2, Adjoint(Alg2));
 ${\mathrm{ρ3}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{ccc}{0}& {0}& {\mathrm{_a}}\\ {0}& {0}& {-}{1}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{ccc}{0}& {0}& {1}\\ {0}& {0}& {\mathrm{_a}}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{ccc}{-}{\mathrm{_a}}& {-}{1}& {0}\\ {1}& {-}{\mathrm{_a}}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.13)
 Alg2 > y := RepresentationEigenvector(rho3);
 ${y}{:=}\left[{0}{,}{0}{,}{-}{\mathrm{_a}}{+}{I}\right]{,}\left[{\mathrm{D_x1}}{-}{I}{}{\mathrm{D_x2}}\right]$ (2.14)
 V2 > seq(ApplyRepresentation(rho3, v, y[2][1]), v = [e1, e2, e3]);
 ${0}{}{\mathrm{D_x1}}{,}{0}{}{\mathrm{D_x1}}{,}{-}\left({\mathrm{_a}}{-}{I}\right){}{\mathrm{D_x1}}{+}\left({1}{+}{I}{}{\mathrm{_a}}\right){}{\mathrm{D_x2}}$ (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]);
 ${B}{:=}\left[{\mathrm{e1}}{-}{\mathrm{e3}}{+}{2}{}{\mathrm{e4}}{,}{\mathrm{e3}}{,}{\mathrm{e2}}{+}{2}{}{\mathrm{e4}}{,}{\mathrm{e1}}{-}{\mathrm{e3}}{+}{\mathrm{e4}}\right]$ (2.16)
 Alg1 > L4 := LieAlgebraData(B, Alg4);
 ${\mathrm{L4}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e1}}{-}{4}{}{\mathrm{e2}}{-}{4}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{+}{3}{}{\mathrm{e2}}{+}{2}{}{\mathrm{e4}}\right]$ (2.17)
 Alg1 > DGsetup(L4, [f], [alpha]);
 ${\mathrm{Lie algebra: Alg4}}$ (2.18)
 Alg4 > rho4 := ChangeRepresentationBasis(rho2, B, Alg4);
 ${\mathrm{ρ4}}{:=}\left[\left[{\mathrm{f1}}{,}\left[\begin{array}{rrrr}{-}{35}& {-}{18}& {-}{18}& {7}\\ {27}& {14}& {14}& {-}{5}\\ {32}& {16}& {16}& {-}{8}\\ {-}{19}& {-}{10}& {-}{10}& {3}\end{array}\right]\right]{,}\left[{\mathrm{f2}}{,}\left[\begin{array}{rrrr}{5}& {10}& {0}& {5}\\ {-}{4}& {-}{8}& {0}& {-}{4}\\ {-}{4}& {-}{8}& {0}& {-}{4}\\ {3}& {6}& {0}& {3}\end{array}\right]\right]{,}\left[{\mathrm{f3}}{,}\left[\begin{array}{rrrr}{-}{25}& {2}& {-}{18}& {17}\\ {19}& {-}{2}& {14}& {-}{13}\\ {23}& {-}{2}& {16}& {-}{17}\\ {-}{13}& {2}& {-}{10}& {9}\end{array}\right]\right]{,}\left[{\mathrm{f4}}{,}\left[\begin{array}{rrrr}{-}{22}& {-}{18}& {-}{9}& {-}{1}\\ {17}& {14}& {7}& {1}\\ {20}& {16}& {8}& {0}\\ {-}{12}& {-}{10}& {-}{5}& {-}{1}\end{array}\right]\right]\right]$ (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]
 ${y}{:=}\left[{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{\mathrm{D_x1}}{-}\frac{{3}}{{4}}{}{\mathrm{D_x2}}{-}{\mathrm{D_x3}}{+}\frac{{1}}{{2}}{}{\mathrm{D_x4}}\right]$ (2.20)
 V > seq(ApplyRepresentation(rho4, f||i, y[2][1]) &minus (y[1][i] &mult y[2][1]), i = 1..4);
 ${0}{}{\mathrm{D_x1}}{,}{0}{}{\mathrm{D_x1}}{,}{0}{}{\mathrm{D_x1}}{,}{0}{}{\mathrm{D_x1}}$ (2.21)