Example 1.

First we initialize a Lie algebra.

Now define coordinates for the dual of the Lie algebra.

Calculate the infinitesimal generators for the co-adjoint action.

The center of the Lie algebra  is trivial and therefore the structure equations for the Lie algebra  are the same as those for .

The vector fields  may be calculated directly using the Adjoint and convert/DGvector commands. For example, we obtain the last vector in as follows.

Example 2.

First we initialize a 4-dimensional Lie algebra.

Now define coordinates for the dual of the Lie algebra.

Calculate the infinitesimal generators for the co-adjoint action.

In this example, the Lie algebra has a non-trivial center  and now the structure equations for  are those for the quotient of by its center.

Example 3.

The invariants for the co-adjoint action are called generalized Casimir operators (See  J. Patera,  R. T. Sharp , P. Winternitz  and H. Zassenhaus,  Invariants of real low dimensional Lie algebras,  J. Math. Phys. vol 17, No 6, June 1976, 966--994).

We calculate the generalized Casimir operators for the Lie algebra [5,12] from this article. First use the Retrieve command to obtain the structure equations for this algebra and initialize the Lie algebra.

Calculate the infinitesimal generators for the co-adjoint action.

We use the  InvariantGeometricObjectFields command to calculate the functions which invariant under the group generated by .

Functional combinations of these invariants give the formulas for the generalized Casimir operators in the Patera, Sharp, et al. paper.

DifferentialGeometry, GroupActions, Library,  LieAlgebras, convert/DGvector, LieAlgebraData, Adjoint, Retrieve, InvariantGeometricObjectFields