Example 1.

We define a representation and make a change of basis for the representation space.

Define the new basis for the representation space.

Compute the representation rho in the basis Fr.

We can use the Query command to check that  phi1 is a representation of Alg1.

Check, by example, that the matrices for phi1 are correct.  We apply rho(e1) to Fr[1] and express the result as a linear combination of the vectors Fr.  This should give the first column of the matrix for e1 in phi1.

Example 2.

We obtain the same change of basis as in Example 1 using the other calling sequence for the procedure ChangeBasis.  We take the matrix A to be the matrix whose columns are the coefficients of the new basis in terms of the old.

Example 3.

Now we make a change of basis in the Lie algebra.  First we use the LieAlgebraData command to create the Lie algebra in the new basis.

Example 4.

We obtain the same change of basis as in Example 3 using the other calling sequence for the procedure ChangeBasis.  We take the matrix A to be the matrix whose columns are the coefficients of the new basis in terms of the old.

DifferentialGeometry, LieAlgebras, ChangeFrame, GetComponents, LieAlgebraData, Query