Let G be a Lie group with multiplication * and identity e.  An action of G on a manifold M is a smooth map mu: G x M -> M such that mu(e, x) = x and mu(a*b, x) = mu(a, mu(b, x)) for all a, b in G and x in M.  Define mu1_a: M -> M and mu2_x: G -> M by mu1_a(x) = mu(a, x) and mu2_x(a) = mu(a, x).  The infinitesimal generators for the action mu is the list of vector fields Gamma_mu defined by the pushforward by mu2_x of the right invariant vector fields on G.  Gamma_mu can also be computed by differentiating the components of the map mu1_a with respect to the group parameters a and evaluating the results at the identity.

The command Action(Gamma, G) calculates the group action mu such that Gamma_mu = Gamma.  The program returns the transformations mu1_a. With the keyword list O = ["GroupToManifold"], the transformation mu2_x is returned. In the course of finding the action mu, the Action procedure calculates the abstract Lie algebra g defined by the vector fields Gamma.  If the adjoint representation is not upper triangular, then a call to the program SolvableRepresentation is made to find a new basis for g (and hence Gamma) 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 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(...).