Example 1.

First we retrieve a Lie algebra from the DifferentialGeometry Library and initialize it.

The center of Alg1 is trivial and therefore the adjoint representation is faithful.  The command Adjoint will return the list of adjoint matrices for the Lie algebra Alg1.

To define a representation of Alg1 using the matrices M, we shall need to define a representation space V.

We can verify that this is really a representation of Alg1 with the Query command.

Apply the representation to some vectors in Alg1:

Apply the linear transformations to some vectors in V:

Example 2.

The infinitesimal automorphisms or derivations of a Lie algebra g define a matrix Lie algebra which is automatically a representation.  In this case we can take g to be the representation space.

The Derivations command calculates the derivations on the Lie algebra Alg2.

We calculate the structure equations for this matrix algebra and initialize the result.

Example 3.

If Gamma is a Lie algebra of vector fields on a manifold M, then the isotropy subalgebra at a given point admits a natural representation, defined by the Lie bracket, on the tangent space of M. To illustrate the definition of this representation, we first obtain a Lie algebra of vector fields from the DifferentialGeometry Library with the Retrieve command. Then we use the IsotropySubalgebra from the GroupActions package to calculate the isotropy subalgebra and its representation.

Example 4.

Here is 4 dimensional faithful representation of an indecomposable 4-dimensional Lie algebra with a center.

DifferentialGeometry, GroupActions, Library, Adjoint, Center, Derivations, IsotropySubalgebra, Query, Retrieve