The DifferentialGeometry:-GroupActions package provides basic symbolic capabilities for working with Lie groups, transformation groups and infinitesimal transformation groups.

For a given Lie algebra of vector fields (that is, an infinitesimal transformation group) on a manifold, important geometric information is provided by the (infinitesimal) isotropy subalgebras, their representations on the tangent space and the isotropy filtration. This information can be calculated with the GroupActions package.

The infinitesimal isometries of any metric tensor can be calculated and conversely, all metric tensors invariant with respect to a given infinitesimal transformation group can be found.  More generally, the infinitesimal symmetries of any collection of vector fields, differential forms, tensor fields or connections can be found.

A global Lie group can be calculated for any solvable Lie algebra.  The r-parameter transformation group for a given r-dimensional solvable infinitesimal transformation group can be determined.

Invariants and differential invariants for group actions can also be determined by the method of moving frames.

The GroupActions package is a subpackage of the DifferentialGeometry package. Each command in the GroupActions package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

Action: find the action of a solvable Lie group on a manifold from its infinitesimal generators.

InfinitesimalPseudoGroupNormalizer: find the normalizer of a finite dimensional Lie algebra of vector fields in an (infinite-dimensional) pseudo-Lie algebra of vector fields

InfinitesimalSymmetriesOfGeometricObjectFields: find the infinitesimal symmetries (vector fields) for a collection of vector fields, differential forms or tensors.

InvariantGeometricObjectFields: find the vector fields, differential forms, or tensors which are invariant with respect to a Lie algebra of vector fields.

InvariantVectorsAndForms: calculate a basis of left and right invariant vector fields and differential 1-forms on a Lie group.

IsotropyFiltration: find the infinitesimal isotropy filtration for a Lie algebra of vector fields.

IsotropySubalgebra: find the infinitesimal isotropy subalgebra of a Lie algebra of vector fields and infinitesimal isotropy representation.

LieGroup: create a module defining a Lie group.

LiesThirdTheorem: find a Lie algebra of pointwise independent vector fields with prescribed structure equations (solvable algebras only).

MatrixGroup: find the matrix group defined by a matrix algebra or by a matrix of 1-forms

MovingFrames: a small package for the method of moving frames.