find the infinitesimal isotropy subalgebra of a Lie algebra of vector fields and the representation of the isotropy subalgebra on the tangent space - Maple Programming Help

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GroupActions[IsotropySubalgebra] - find the infinitesimal isotropy subalgebra of a Lie algebra of vector fields and the representation of the isotropy subalgebra on the tangent space

Calling Sequences

IsotropySubalgebra(Gamma, p, option)

Parameters

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

p         - a list of equations  specifying the coordinates of point

option    - the optional argument output = O, where O is a list containing the keywords "Vector", "Representation", 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 subalgebra  of the Lie algebra of vector fields at the point is defined by . The Lie bracket of vector fields defines a natural representation of ${\mathrm{Γ}}_{p}$ on the tangent space  by  for  , and $\stackrel{‾}{Y}$ any vector field on $M$ such that . The representation $\mathrm{ρ}$ is called the linear isotropy representation.
 • IsotropySubalgebra(Gamma, p) returns a list of vectors whose span defines the isotropy subalgebra ${\mathrm{\Gamma }}_{p}$as a subalgebra of  $\mathrm{Γ}$.
 • With output = ["Vector", "Representation"], two lists are returned. The first is a list of vectors giving the isotropy subalgebra ${\mathrm{Γ}}_{p}$as a subalgebra of  and the second is the list of matrices defining the linear isotropy representation with respect to the standard basis for ${T}_{p}M$.
 • Let algname be the name of the abstract Lie algebra created from $\mathrm{Γ}$. With output = ["Vector", algname], the second list returned gives the isotropy subalgebra as a subalgebra of the abstract Lie algebra $\mathrm{𝔤}$.
 • The command IsotropySubalgebra is part of the DifferentialGeometry:-GroupActions package.  It can be used in the form IsotropySubalgebra(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-IsotropySubalgebra(...).

Examples

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

Example 1.

We use the Retrieve command to obtain a Lie algebra of vector fields in the paper by Gonzalez-Lopez, Kamran, and Olver from the DifferentialGeometry Library. We compute the isotropy subalgebra and isotropy representation at the points  and

 > DGsetup([x, y], M):
 M > G := Retrieve("Gonzalez-Lopez", 1, [5], manifold = M);
 ${G}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_x}}{}{x}{-}{\mathrm{D_y}}{}{y}{,}{y}{}{\mathrm{D_x}}{,}{x}{}{\mathrm{D_y}}\right]$ (2.1)
 M > L := LieAlgebraData(G, Alg1);
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}\right]$ (2.2)
 M > DGsetup(L);
 ${\mathrm{Lie algebra: Alg1}}$ (2.3)
 Alg1 > MultiplicationTable("LieTable");
 $\left[\begin{array}{ccccccc}{}& {|}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e4}}& {\mathrm{e5}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e1}}& {|}& {0}& {0}& {\mathrm{e1}}& {0}& {\mathrm{e2}}\\ {\mathrm{e2}}& {|}& {0}& {0}& {-}{\mathrm{e2}}& {\mathrm{e1}}& {0}\\ {\mathrm{e3}}& {|}& {-}{\mathrm{e1}}& {\mathrm{e2}}& {0}& {-}{2}{}{\mathrm{e4}}& {2}{}{\mathrm{e5}}\\ {\mathrm{e4}}& {|}& {0}& {-}{\mathrm{e1}}& {2}{}{\mathrm{e4}}& {0}& {-}{\mathrm{e3}}\\ {\mathrm{e5}}& {|}& {-}{\mathrm{e2}}& {0}& {-}{2}{}{\mathrm{e5}}& {\mathrm{e3}}& {0}\end{array}\right]$ (2.4)

We illustrate some different possible outputs from the IsotropySubalgebra program.

 Alg1 > Iso1 := IsotropySubalgebra(G, [x = 0, y = 0]);
 ${\mathrm{Iso1}}{:=}\left[{\mathrm{D_x}}{}{x}{-}{\mathrm{D_y}}{}{y}{,}{y}{}{\mathrm{D_x}}{,}{x}{}{\mathrm{D_y}}\right]$ (2.5)
 M > Iso1, A1 := IsotropySubalgebra(G, [x = 0, y = 0], output = ["Vector", Alg1]);
 ${\mathrm{Iso1}}{,}{\mathrm{A1}}{:=}\left[{\mathrm{D_x}}{}{x}{-}{\mathrm{D_y}}{}{y}{,}{y}{}{\mathrm{D_x}}{,}{x}{}{\mathrm{D_y}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]$ (2.6)
 Alg1 > Iso1, A1, S1 := IsotropySubalgebra(G, [x = 0, y = 0], output = ["Vector", Alg1,"Representation"]);
 ${\mathrm{Iso1}}{,}{\mathrm{A1}}{,}{\mathrm{S1}}{:=}\left[{\mathrm{D_x}}{}{x}{-}{\mathrm{D_y}}{}{y}{,}{y}{}{\mathrm{D_x}}{,}{x}{}{\mathrm{D_y}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]{,}\left[\left[\begin{array}{rr}{-}{1}& {0}\\ {0}& {1}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {-}{1}\\ {0}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {0}\\ {-}{1}& {0}\end{array}\right]\right]$ (2.7)
 Alg1 > A1 := IsotropySubalgebra(G, [x = 0, y = 0], output = [Alg1]);
 ${\mathrm{A1}}{:=}\left[{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]$ (2.8)
 Alg1 > Iso2, A2, S2 := IsotropySubalgebra(G, [x = 1, y = 1], output = ["Vector", Alg1,"Representation"]);
 ${\mathrm{Iso2}}{,}{\mathrm{A2}}{,}{\mathrm{S2}}{:=}\left[{-}\left({y}{-}{1}\right){}{\mathrm{D_x}}{,}{-}\left({x}{-}{1}\right){}{\mathrm{D_y}}{,}\left({-}{y}{+}{x}\right){}{\mathrm{D_x}}{+}\left({-}{y}{+}{x}\right){}{\mathrm{D_y}}\right]{,}\left[{\mathrm{e1}}{-}{\mathrm{e4}}{,}{\mathrm{e2}}{-}{\mathrm{e5}}{,}{\mathrm{e3}}{-}{\mathrm{e4}}{+}{\mathrm{e5}}\right]{,}\left[\left[\begin{array}{rr}{0}& {1}\\ {0}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {0}\\ {1}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{-}{1}& {1}\\ {-}{1}& {1}\end{array}\right]\right]$ (2.9)

Note that the vectors in Iso2 all vanish at

It is apparent from the multiplication table that the pair Alg1, S1 is a symmetric pair with respect to the complementary subspace. We can check this with the command Query/"SymmetricPair".

 Alg1 > Query(A1, [e1, e2], "SymmetricPair");
 ${\mathrm{true}}$ (2.10)

The linear isotropy representation can be converted to a representation.

 Alg1 > L2 := LieAlgebraData(A1, iso1);
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e1}}\right]$ (2.11)
 Alg1 > DGsetup(L2);
 ${\mathrm{Lie algebra: iso1}}$ (2.12)
 iso1 > rho := Representation(iso1, M, S1);
 ${\mathrm{ρ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rr}{-}{1}& {0}\\ {0}& {1}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rr}{0}& {-}{1}\\ {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rr}{0}& {0}\\ {-}{1}& {0}\end{array}\right]\right]\right]$ (2.13)
 iso1 > Query(rho, "Representation");
 ${\mathrm{true}}$ (2.14)