find the infinitesimal isotropy filtration for a Lie algebra of vector fields - Maple Programming Help

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GroupActions[IsotropyFiltration] - find the infinitesimal isotropy filtration for a Lie algebra of vector fields

Calling Sequences

IsotropyFiltration(Gamma, pt, option)

Parameters

Gamma     - a list of vector fields on a manifold $M$

pt        - a list of equations  specifying the coordinates of point

option    - the optional argument output = O, where O is a list containing the keywords "Vector" and/or the name of an initialized abstract algebra for the Lie algebra of vector fields Gamma.

Description

 • Let be a Lie algebra of vector fields on a manifold  and letThe isotropy filtration of  the  Lie algebra of vector fields is the decreasing nested sequence of subalgebras  defined by

{  the coefficients of and their derivatives all vanish to order at $p$ }.

Note that if and , then .  The subalgebra is called the isotropy subalgebra of at $p$.

 • The command IsotropyFiltration(Gamma, pt) returns a list of list of vector fields, the first list gives a basis for the second list gives a basis for ${\mathrm{\Gamma }}_{p}^{1}$ and so on.
 • The command IsotropyFiltration is part of the DifferentialGeometry:-GroupActions package.  It can be used in the form IsotropyFiltration(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-IsotropyFiltration(...).

Examples

 > with(DifferentialGeometry): with(GroupActions): with(Library): with(LieAlgebras):

Example 1.

First we obtain a Lie algebra of vector fields from the paper by Gonzalez-Lopez, Kamran, Olver in the DifferentialGeometry Library using the Retrieve command and then we compute the isotropy filtration.

 > DGsetup([x, y], M);
 ${\mathrm{frame name: M}}$ (2.1)
 M > Gamma := Retrieve("Gonzalez-Lopez", 1, [27,4], manifold = M);
 ${\mathrm{Γ}}{:=}\left[{\mathrm{D_x}}{,}{2}{}{x}{}{\mathrm{D_x}}{+}{4}{}{y}{}{\mathrm{D_y}}{,}{{x}}^{{2}}{}{\mathrm{D_x}}{+}{4}{}{x}{}{y}{}{\mathrm{D_y}}{,}{\mathrm{D_y}}{,}{x}{}{\mathrm{D_y}}{,}{{x}}^{{2}}{}{\mathrm{D_y}}{,}{{x}}^{{3}}{}{\mathrm{D_y}}{,}{{x}}^{{4}}{}{\mathrm{D_y}}\right]$ (2.2)

We calculate the isotropy filtration as a subalgebra of $\mathrm{Γ}$.

 M > F1 := IsotropyFiltration(Gamma, [x = 0, y = 0]);
 ${\mathrm{F1}}{:=}\left[\left[{2}{}{x}{}{\mathrm{D_x}}{+}{4}{}{y}{}{\mathrm{D_y}}{,}{{x}}^{{2}}{}{\mathrm{D_x}}{+}{4}{}{x}{}{y}{}{\mathrm{D_y}}{,}{x}{}{\mathrm{D_y}}{,}{{x}}^{{2}}{}{\mathrm{D_y}}{,}{{x}}^{{3}}{}{\mathrm{D_y}}{,}{{x}}^{{4}}{}{\mathrm{D_y}}\right]{,}\left[{{x}}^{{2}}{}{\mathrm{D_x}}{+}{4}{}{x}{}{y}{}{\mathrm{D_y}}{,}{{x}}^{{2}}{}{\mathrm{D_y}}{,}{{x}}^{{3}}{}{\mathrm{D_y}}{,}{{x}}^{{4}}{}{\mathrm{D_y}}\right]{,}\left[{{x}}^{{3}}{}{\mathrm{D_y}}{,}{{x}}^{{4}}{}{\mathrm{D_y}}\right]{,}\left[{{x}}^{{4}}{}{\mathrm{D_y}}\right]{,}\left[{}\right]\right]$ (2.3)

Example 2.

We continue with Example 1. Here we calculate the isotropy filtration as a subalgebra of the abstract Lie algebra defined by $\mathrm{Γ}$. To this end, we first calculate the structure constants for $\mathrm{Γ}$and initialize the result as Alg1.

 M > L := LieAlgebraData(Gamma, Alg1);
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{3}{}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{4}{}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{4}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{4}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{4}{}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{3}{}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}\right]$ (2.4)
 M > DGsetup(L):

Now re-run the IsotropyFiltration command with the third argument output = [Alg1].

 Alg1 > F := IsotropyFiltration(Gamma, [x = 0, y = 0], output = [Alg1]);
 ${F}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}{,}{\mathrm{e8}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}{,}{\mathrm{e8}}\right]{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{,}\left[{\mathrm{e8}}\right]{,}\left[{}\right]\right]$ (2.5)

We check that F does indeed define a filtration of the Lie algebra (note that there is an index shift  ${\mathrm{Γ}}_{p}^{k}$ = F[k+1]).

 Alg1 > BracketOfSubspaces(F[1], F[1]);
 $\left[{2}{}{\mathrm{e3}}{,}{-}{2}{}{\mathrm{e5}}{,}{2}{}{\mathrm{e7}}{,}{4}{}{\mathrm{e8}}{,}{-}{3}{}{\mathrm{e6}}\right]$ (2.6)
 Alg1 > BracketOfSubspaces(F[1], F[2]);
 $\left[{2}{}{\mathrm{e3}}{,}{2}{}{\mathrm{e7}}{,}{4}{}{\mathrm{e8}}{,}{3}{}{\mathrm{e6}}\right]$ (2.7)
 Alg1 > BracketOfSubspaces(F[1], F[3]);
 $\left[{2}{}{\mathrm{e7}}{,}{4}{}{\mathrm{e8}}\right]$ (2.8)
 Alg1 > BracketOfSubspaces(F[1], F[4]);
 $\left[{4}{}{\mathrm{e8}}\right]$ (2.9)
 Alg1 > BracketOfSubspaces(F[2], F[3]);
 $\left[{-}{\mathrm{e8}}\right]$ (2.10)
 Alg1 > BracketOfSubspaces(F[2], F[4]);
 $\left[{}\right]$ (2.11)

All these brackets can be checked at once with Query/"filtration".

 Alg1 > Query(F, "Filtration");
 ${\mathrm{true}}$ (2.12)