Example 1.

Example 2.

We retrieve a 5-dimensional Lie algebra appearing in the paper by \011J. Patera, R. T. Sharp,  and P. Winternitz, Invariants of real low dimensional Lie algebras, Journal of Mathematical Physics, Vol 17, No 6 (1976), 966--994.

At this point we can immediately initialize this Lie algebra and perform computations.  For example, using the LeviDecomposition command we find that the algebra [5, 40] in ["Winternitz", 1] admits a non-trivial Levi decomposition with a 2 dimensional radical [e4, e5] and a semi-simple subalgebra [e1, e2, e3].

Example 3.

We retrieve a 5 dimensional Lie algebra of vector fields in the plane appearing in the paper by A. Gonzalez-Lopex, N. Kamran and P. J. Olver, Lie algebras of vector fields in the real plane , Proc. London Math Soc. Vol 64 (1992), 339--368

We can immediately do computations with this 5-dimensional Lie algebra of vector fields.  For example, let us prolong these vector fields to the 1st order jet spaces J^1(R, R) with the Prolong command and compute the isotropy subalgebra (with the IsotropySubalgebra command) for this prolonged infinitesimal action at a generic point [ x = a, y[ ] = b, y[1] = c].

Since the infinitesimal isotropy of this 5 dimension Lie algebra of vector fields is 2 dimensional, the generic orbits for the corresponding group action are 3 dimensional and therefore, since J^1(R, R) has dimension 3, there are no differential invariants for this action on the 1-jets.

Example 4.

The DifferentialGeometry Library contains the lists of ordinary differential equations from the book by Kamke. Let us retrieve one such equation, use the procedure PDEtools:-Infinitesimals to find the infinitesimal symmetries of this equation, and then use the LieAlgebras command LieAlgebraData to calculate the structure equations for this Lie algebra.

We use the convert command with the keyword DGvector to convert the output of the PDEtools:-Infinitesimals programs to the Differential Geometry vector field format.

Let us find the Levi decomposition for this 6 dimensional symmetry algebra.

Example 5.

The DifferentialGeometry Library contains detailed information on some of the space-time metrics found in the books by Hawking and Ellis and Stephani, Kramer et. al.  

First define a spacetime manifold.

Retrieve the metric and other fields (if present):

Retrieve a null tetrad for the space-time.

Retrieve the Petrov type.

Retrieve  the Killing vectors

The keyword argument output accepts a list of table indices chosen from ["Authors", "BasePoints",  "Comments", "Coordinates", "CosmologicalConstant", "Domains", "Fields", "IsometryDimension , "KillingEquations", "KillingVectors", "NewtonConstant", "NullTetrad", "OrthonormalTetrad", "Parameters", "PetrovType", "PlebanskiPetrovType", "PrimaryDescription", "OrbitDimension", "OrbitType", "Reference", "SegreType", "SideConditionsAssuming",  "SideConditionsSimplify", "SecondaryDescription", "TertiaryDescription" ].

Example 6.

Many table entries in the DifferentialGeometry library contain arbitrary parameters.  These parameters can be assigned specific values by first adding the optional argument parameters = "yes" to the calling sequence for the Retrieve command and using the elements of the returned  parameter list in conjunction with the eval command.

This Lie algebra depends upon parameters [p, q, s].  Use the following command to assign the values p =3, q = 7, s= K in the structure equations L.

DifferentialGeometry, GroupActions, JetCalculus, Library, LieAlgebras,  Tools, DGinfo, Infinitesimals, IsotropySubalgebra, LeviDecomposition, LieAlgebraData, Prolong