Let $G$ be a Lie group with multiplication * and identity $e$. An action of $G$ on a manifold $M$is a smooth map $\mathrm{\μ} \:G\to \.M$such that $\mathrm{\μ}\left(e\, x\right) \= x$and $\mathrm{\μ}\left(a\*b\,x\right) \= \mathrm{\μ}\left(a\, \mathrm{\μ}\left(b\, x\right)\right)$for all $a\, b \in G$ and $x \in M$.  For a given action $\mathrm{\μ}\,$define

The command Action(Gamma, G) calculates the group action $\mathrm{\μ}$ such that ${\mathrm{\Γ}}_{\mathrm{\μ}} \= \mathrm{\Γ}$. 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 $\mathrm{\𝔤}\mathit{ }$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(...).

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

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

$\mathrm{Gamma}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\,\mathrm{D\_y}\,y\mathrm{D\_x}\right]\right)$

$\mathrm{\Γ}:=\left[\mathrm{D\_x}\,\mathrm{D\_y}\,y\mathrm{D\_x}\right]$

$\mathrm{LieAlgebraData}\left(\mathrm{Gamma}\right)$

$\left[\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\right]$

$\mathrm{DGsetup}\left(\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{z3}\right]\,G\right)\:$

$\mathrm{\μ1}≔\mathrm{Action}\left(\mathrm{Gamma}\,G\right)$

$\mathrm{\μ1}:=\left[x\=y\mathrm{z3}\+\mathrm{z2}\mathrm{z3}\+x\+\mathrm{z1}\,y\=\mathrm{z2}\+y\right]$

$\mathrm{newGamma}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{\μ1}\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{z3}\right]\right)$

$\mathrm{newGamma}:=\left[\mathrm{D\_x}\,\mathrm{D\_y}\,y\mathrm{D\_x}\right]$

$\mathrm{\Γ2}≔\mathrm{evalDG}\left(\left[y\mathrm{D\_x}\,\mathrm{D\_x}\,\mathrm{D\_y}\right]\right)$

$\mathrm{\Γ2}:=\left[y\mathrm{D\_x}\,\mathrm{D\_x}\,\mathrm{D\_y}\right]$

$\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{\Γ2}\,\mathrm{Alg2}\right)$

$\mathrm{L2}:=\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=-\mathrm{e2}\right]$

$\mathrm{DGsetup}\left(\mathrm{L2}\right)$

$\mathrm{Lie\; algebra:\; Alg2}$

$\mathrm{Adjoint}\left(\right)$

$\mathrm{\μ1},B≔\mathrm{Action}\left(\mathrm{\Γ2}\,G\,\mathrm{output}=\left[''ManifoldToManifold''\,''Basis''\right]\right)$

$\mathrm{\μ1}\,B:=\left[x\=y\mathrm{z2}\+x\+\mathrm{z1}\,y\=\mathrm{z3}\+y\right]\,\left[\left[0\,1\,0\right]\,\left[1\,0\,0\right]\,\left[0\,0\,1\right]\right]$

$\mathrm{newGamma}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{\μ1}\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{z3}\right]\right)$

$\mathrm{newGamma}:=\left[\mathrm{D\_x}\,y\mathrm{D\_x}\,\mathrm{D\_y}\right]$

$\mathrm{map}\left(\mathrm{DGzip}\,B\,\mathrm{\Γ2}\,''plus''\right)$

$\left[\mathrm{D\_x}\,y\mathrm{D\_x}\,\mathrm{D\_y}\right]$

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

$\mathrm{\Γ3}≔\mathrm{Retrieve}\left(''Gonzalez-Lopez''\,1\,\left[22\,17\right]\,\mathrm{manifold}=M\right)$

$\mathrm{\Γ3}:=\left[\mathrm{D\_x}\,\mathrm{D\_y}\,x\mathrm{D\_y}\,\frac{1}{2}{x}^2\mathrm{D\_y}\,{\ⅇ}^x\mathrm{D\_y}\right]$

$\mathrm{DGsetup}\left(\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{z3}\,\mathrm{z4}\,\mathrm{z5}\right]\,\mathrm{G3}\right)$

$\mathrm{frame\; name:\; G3}$

$\mathrm{\mu }≔\mathrm{Action}\left(\mathrm{\Γ3}\,\mathrm{G3}\right)$

$\mathrm{\μ}:=\left[x\=\mathrm{z5}\+x\,y\={\ⅇ}^{\mathrm{z5}\+x}\mathrm{z1}\+\mathrm{z2}\+\mathrm{z3}x\+\frac{1}{2}{x}^2\mathrm{z4}\+y\right]$

$\mathrm{InfinitesimalTransformation}\left(\mathrm{\mu }\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{z3}\,\mathrm{z4}\,\mathrm{z5}\right]\right)$

$\left[{\ⅇ}^x\mathrm{D\_y}\,\mathrm{D\_y}\,x\mathrm{D\_y}\,\frac{1}{2}{x}^2\mathrm{D\_y}\,\mathrm{D\_x}\right]$

$\mathrm{DGsetup}\left(\left[x\,y\,u\,v\right]\,\mathrm{M4}\right)\:$

$\mathrm{\Γ4}≔\mathrm{Retrieve}\left(''Petrov''\,1\,\left[32\,6\right]\,\mathrm{manifold}=\mathrm{M4}\right)$

$\mathrm{\Γ4}:=\left[\mathrm{D\_y}\,\mathrm{D\_u}\,\mathrm{D\_u}u\+\mathrm{D\_y}y-\mathrm{D\_x}\,\mathrm{D\_u}y-\mathrm{D\_y}u\right]$

$\mathrm{DGsetup}\left(\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{z3}\,\mathrm{z4}\right]\,\mathrm{G4}\right)$

$\mathrm{frame\; name:\; G4}$

$\mathrm{\mu }≔\mathrm{Action}\left(\mathrm{\Γ4}\,\mathrm{G4}\right)$

$\mathrm{\μ}:=\left[x\=-\mathrm{z3}\+x\,y\=-\sin \left(\mathrm{z4}\right){\ⅇ}^{\mathrm{z3}}u\+\cos \left(\mathrm{z4}\right){\ⅇ}^{\mathrm{z3}}y\+\mathrm{z1}\,u\=\sin \left(\mathrm{z4}\right){\ⅇ}^{\mathrm{z3}}y\+\cos \left(\mathrm{z4}\right){\ⅇ}^{\mathrm{z3}}u\+\mathrm{z2}\,v\=v\right]$

$\mathrm{InfinitesimalTransformation}\left(\mathrm{\mu }\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{z3}\,\mathrm{z4}\right]\right)$

$\left[\mathrm{D\_y}\,\mathrm{D\_u}\,\mathrm{D\_u}u\+\mathrm{D\_y}y-\mathrm{D\_x}\,\mathrm{D\_u}y-\mathrm{D\_y}u\right]$