Example 1.

We obtain a Lie algebra from the DifferentialGeometry library using the Retrieve command and initialize it.

We define a manifold M of dimension 4 (the same dimension as the Lie algebra).

We calculate the structure equations for the Lie algebra of vector fields Gamma1 and check that these structure equations coincide with those for Alg1.

Example 2.

We re-work the previous example in a more complicated basis.  In this basis the adjoint representation is not upper triangular, in which case LiesThirdTheorem first calls the program SolvableRepresentation to find a basis for the algebra in which the adjoint representation is upper triangular. (Remark:  It is almost always useful, when working with solvable algebras, to transform to a basis where the adjoint representation is upper triangular.)

Example 3.

Here is an example where one of the adjoint matrices has complex eigenvalues. The Lie algebra contains parameters p and b.

Example 4.

We calculate the Maurer-Cartan matrix  of 1-forms for a solvable matrix algebra, namely the matrices defining the adjoint representation for Alg1 from Example 1.

Note that the elements of  this matrix   coincide with the appropriate linear combinations of the forms in the list  from Example 1.

DifferentialGeometry, GroupActions, Library, LieAlgebras, Representation, Adjoint, LieAlgebraData, Representation, Retrieve, SolvableRepresentation