find the action of a solvable Lie group on a manifold from its infinitesimal generators - Maple Help

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GroupActions[Action] - find the action of a solvable Lie group on a manifold from its infinitesimal generators

Calling Sequences

Action(Gamma, G, options)

Parameters

Gamma     - a list, a basis for a Lie algebras of vector fields on a manifold M

G         - a Maple name or string, the name of a coordinate system for the abstract Lie group defined by Gamma

options   - output = O, where O is a list of keywords "ManifoldToManifold", "GroupToManifold", "LieGroup", "Basis"

Description

 • Let $G$ be a Lie group with multiplication * and identity $e$. An action of $G$ on a manifold is a smooth map such that and for all  and .  For a given action $\mathrm{μ},$define

 by and  by .

The infinitesimal generators for the action is the Lie algebra of vector fields ${\mathrm{Γ}}_{\mathrm{μ}}$ on $M$ defined by the pushforward by ${\mathrm{\mu }}_{2,x}$ of the right invariant vector fields on $G$. The infinitesimal generators can also be computed by differentiating the components of the map ${\mathrm{\mu }}_{2,x}$with respect to the group parameters $a$ and evaluating the results at the identity.

 • The command Action(Gamma, G) calculates the group action $\mathrm{μ}$ such that . The program returns the transformation ${\mathrm{\mu }}_{1,a}$. With the keyword list O = ["GroupToManifold"], the transformation ${\mathrm{\mu }}_{2,x}$ is returned.
 • In the course of finding the action $\mathrm{μ}$, the Action procedure calculates the abstract Lie algebra $\mathrm{𝔤}$ defined by the vector fields $\mathrm{Γ}$. If the adjoint representation for is not upper triangular, then a call to the program SolvableRepresentation is made to find a new basis for $\mathrm{𝔤}$ (and hence $\mathrm{Γ}$) in which the adjoint representation is upper triangular. This new basis can be retrieved by adding the keyword "Basis" to the keyword list O. The action procedure also uses the LieGroup command to find the Lie group module for the Lie algebra $\mathrm{𝔤}$. This module can be returned by adding the keyword "LieGroup" to the keyword list O.
 • The default value for O is O = ["ManifoldToManifold"]. The option O = ["all"] is equivalent to O = ["ManifoldToManifold", "GroupToManifold", "LieGroup", "Basis"].
 • The command Action is part of the DifferentialGeometry:-GroupActions package. It can be used in the form Action(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-Action(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{GroupActions}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$$\mathrm{with}\left(\mathrm{Library}\right):$

Example 1.

First define a 2-dimensional manifold M with coordinates $\left[x,y\right].$

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right):$

On M, define a 3-dimensional Lie algebra of vector fields $\mathrm{Γ}$.

 M > $\mathrm{Γ}≔\mathrm{evalDG}\left(\left[\mathrm{D_x},\mathrm{D_y},y\mathrm{D_x}\right]\right)$
 ${\mathrm{Γ}}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{y}{}{\mathrm{D_x}}\right]$ (2.1)

We need a 3-dimensional manifold to represent the abstract Lie group defined by $\mathrm{Γ}$.

 M > $\mathrm{LieAlgebraData}\left(\mathrm{Γ}\right)$
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (2.2)
 M > $\mathrm{DGsetup}\left(\left[\mathrm{z1},\mathrm{z2},\mathrm{z3}\right],G\right):$
 G > $\mathrm{μ1}≔\mathrm{Action}\left(\mathrm{Γ},G\right)$
 ${\mathrm{μ1}}{:=}\left[{x}{=}{y}{}{\mathrm{z3}}{+}{\mathrm{z2}}{}{\mathrm{z3}}{+}{x}{+}{\mathrm{z1}}{,}{y}{=}{\mathrm{z2}}{+}{y}\right]$ (2.3)

Use the InfinitesimalTransformation command to find the infinitesimal generators for this action. Note that they are precisely the vectors we began with.

 M > $\mathrm{newGamma}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{μ1},\left[\mathrm{z1},\mathrm{z2},\mathrm{z3}\right]\right)$
 ${\mathrm{newGamma}}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{y}{}{\mathrm{D_x}}\right]$ (2.4)

Example 2.

We continue with Example 1 but this time present the vector fields in a different order.

 M > $\mathrm{Γ2}≔\mathrm{evalDG}\left(\left[y\mathrm{D_x},\mathrm{D_x},\mathrm{D_y}\right]\right)$
 ${\mathrm{Γ2}}{:=}\left[{y}{}{\mathrm{D_x}}{,}{\mathrm{D_x}}{,}{\mathrm{D_y}}\right]$ (2.5)
 M > $\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{Γ2},\mathrm{Alg2}\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}\right]$ (2.6)
 M > $\mathrm{DGsetup}\left(\mathrm{L2}\right)$
 ${\mathrm{Lie algebra: Alg2}}$ (2.7)

In this case the adjoint representation is not upper triangular. The Action program will force us back to the basis of Example 1.  This change of basis can be obtained using the output option.

 Alg2 > $\mathrm{Adjoint}\left(\right)$
 $\left[\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {-}{1}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {1}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]$ (2.8)
 Alg2 > $\mathrm{μ1},B≔\mathrm{Action}\left(\mathrm{Γ2},G,\mathrm{output}=\left["ManifoldToManifold","Basis"\right]\right)$
 ${\mathrm{μ1}}{,}{B}{:=}\left[{x}{=}{y}{}{\mathrm{z2}}{+}{x}{+}{\mathrm{z1}}{,}{y}{=}{\mathrm{z3}}{+}{y}\right]{,}\left[\left[{0}{,}{1}{,}{0}\right]{,}\left[{1}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{1}\right]\right]$ (2.9)
 M > $\mathrm{newGamma}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{μ1},\left[\mathrm{z1},\mathrm{z2},\mathrm{z3}\right]\right)$
 ${\mathrm{newGamma}}{:=}\left[{\mathrm{D_x}}{,}{y}{}{\mathrm{D_x}}{,}{\mathrm{D_y}}\right]$ (2.10)

This basis for the infinitesimal generators agrees with the basis whose components are given by the list B.

 M > $\mathrm{map}\left(\mathrm{DGzip},B,\mathrm{Γ2},"plus"\right)$
 $\left[{\mathrm{D_x}}{,}{y}{}{\mathrm{D_x}}{,}{\mathrm{D_y}}\right]$ (2.11)

Example 3.

We take an example from the Lie algebras of vector fields in the paper by Gonzalez-Lopez, Kamran, Olver. The Lie algebra of vector fields in this paper are part of the DifferentialGeometry Library.

 M > $\mathrm{DGsetup}\left(\left[x,y\right],M\right):$
 M > $\mathrm{Γ3}≔\mathrm{Retrieve}\left("Gonzalez-Lopez",1,\left[22,17\right],\mathrm{manifold}=M\right)$
 ${\mathrm{Γ3}}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{x}{}{\mathrm{D_y}}{,}\frac{{1}}{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y}}{,}{{ⅇ}}^{{x}}{}{\mathrm{D_y}}\right]$ (2.12)
 M > $\mathrm{DGsetup}\left(\left[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5}\right],\mathrm{G3}\right)$
 ${\mathrm{frame name: G3}}$ (2.13)
 G3 > $\mathrm{μ}≔\mathrm{Action}\left(\mathrm{Γ3},\mathrm{G3}\right)$
 ${\mathrm{μ}}{:=}\left[{x}{=}{\mathrm{z5}}{+}{x}{,}{y}{=}{{ⅇ}}^{{\mathrm{z5}}{+}{x}}{}{\mathrm{z1}}{+}{\mathrm{z2}}{+}{\mathrm{z3}}{}{x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{}{\mathrm{z4}}{+}{y}\right]$ (2.14)
 M > $\mathrm{InfinitesimalTransformation}\left(\mathrm{μ},\left[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5}\right]\right)$
 $\left[{{ⅇ}}^{{x}}{}{\mathrm{D_y}}{,}{\mathrm{D_y}}{,}{x}{}{\mathrm{D_y}}{,}\frac{{1}}{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y}}{,}{\mathrm{D_x}}\right]$ (2.15)

Example 4.

We take an example from the Lie algebras of vector fields in the book by Petrov.

 M > $\mathrm{DGsetup}\left(\left[x,y,u,v\right],\mathrm{M4}\right):$
 M4 > $\mathrm{Γ4}≔\mathrm{Retrieve}\left("Petrov",1,\left[32,6\right],\mathrm{manifold}=\mathrm{M4}\right)$
 ${\mathrm{Γ4}}{:=}\left[{\mathrm{D_y}}{,}{\mathrm{D_u}}{,}{\mathrm{D_u}}{}{u}{+}{\mathrm{D_y}}{}{y}{-}{\mathrm{D_x}}{,}{\mathrm{D_u}}{}{y}{-}{\mathrm{D_y}}{}{u}\right]$ (2.16)
 M4 > $\mathrm{DGsetup}\left(\left[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4}\right],\mathrm{G4}\right)$
 ${\mathrm{frame name: G4}}$ (2.17)
 G4 > $\mathrm{μ}≔\mathrm{Action}\left(\mathrm{Γ4},\mathrm{G4}\right)$
 ${\mathrm{μ}}{:=}\left[{x}{=}{-}{\mathrm{z3}}{+}{x}{,}{y}{=}{-}{\mathrm{sin}}{}\left({\mathrm{z4}}\right){}{{ⅇ}}^{{\mathrm{z3}}}{}{u}{+}{\mathrm{cos}}{}\left({\mathrm{z4}}\right){}{{ⅇ}}^{{\mathrm{z3}}}{}{y}{+}{\mathrm{z1}}{,}{u}{=}{\mathrm{sin}}{}\left({\mathrm{z4}}\right){}{{ⅇ}}^{{\mathrm{z3}}}{}{y}{+}{\mathrm{cos}}{}\left({\mathrm{z4}}\right){}{{ⅇ}}^{{\mathrm{z3}}}{}{u}{+}{\mathrm{z2}}{,}{v}{=}{v}\right]$ (2.18)
 M4 > $\mathrm{InfinitesimalTransformation}\left(\mathrm{μ},\left[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4}\right]\right)$
 $\left[{\mathrm{D_y}}{,}{\mathrm{D_u}}{,}{\mathrm{D_u}}{}{u}{+}{\mathrm{D_y}}{}{y}{-}{\mathrm{D_x}}{,}{\mathrm{D_u}}{}{y}{-}{\mathrm{D_y}}{}{u}\right]$ (2.19)