For convenience the following matrix algebras can also be constructed.

For the definitions and examples of all these algebras, see Details for SimpleLieAlgebraData.

Example 1.

We obtain the explicit matrix representation for , the 8 dimensional Lie algebra of trace-free matrices.

First create the data for using the SimpleLieAlgebraData command.

Initialize this Lie algebra.

Now get the explicit matrices defining .

The notation for the basis for the abstract Lie algebra was constructed to match this list of matrices:

The structure equations for the Lie algebra defined by the matrices  coincides exactly with the Lie algebra structure equations generated by the SimpleLieAlgebraData command in equation (2.1).

One can see by inspection that all the matrices in (2.3) are trace-free. This can also be verified using the Query command.

To obtain the standard vector field representation for first define a manifold with coordinates

We get the desired vector fields with the second calling sequence.

Again the stucture equations for are identical to those in equations (2.1)or (2.5).

Example 2.

In this example we construct 2 different matrix representations for the Lorentz Lie algebra  .  This is the 6-dimensional Lie algebra of 4×4 matrices  which are skew-symmetric with respect to a signature  quadratic form , that is,  There are two standard forms foreither

     or  

which give rise to two forms for the matrices Either form can be generated.  The default is since this form is better for calculating the Cartan subalgebra, the root space decomposition, the Cartan decomposition and so on.

Here are the defining matrices for  with respect to

To get the alternative form for using add the keyword argument to the arguments for SimpleLieAlgebraData.

Here are the defining matrices for with respect to

We check that these matrices satisfy the defining equations  and  respectively.

Example 3.

We give the standard representation for the Lie algebra of  skew-Hermitian matrices.

To calculate the structure equations for this list of matrices, as a real Lie algebra, include the keyword argument " in the calling sequence for the LieAlgebraData command.

We obtain the same structure equations as in (2.18).

We check that these matrices are skew-Hermitian but they are not all trace-free.

DifferentialGeometry,  LieAlgebras, LieAlgebraData, Query, Representation,  SimpleLieAlgebraData