Example 1.
We define a 5-dimensional representation of a 3-dimensional solvable Lie algebra.
We find a new basis for the representation space in which the matrices are all upper triangular.
To verify this result we use the ChangeRepresentationBasis command to change basis in the representation space.
Example 2.
We define a 6-dimensional representation of a 3-dimensional solvable Lie algebra.
In this example some of the eigenvectors found by the RepresentationEigenvector program are complex and it is not possible to find a real basis in which the representation is upper triangular.
To obtain an upper triangular representation with respect to a complex basis, use the optional argument fieldextension = I.
Example 3.
If the name of an algebra is passed to the program SolvableRepresentation, then the assumed representation is the adjoint representation of the algebra (or current frame).
The adjoint representation of this algebra is not upper triangular.
Now in this new basis the adjoint representation is upper triangular.
Example 4.
An example with complex eigenvalues.
In this new basis the adjoint representation is upper triangular except for a 2x2 "complex" block on the diagonal for ad(e4).
We rerun this example with the option fieldextension = I
Example 5.
Let be a representation of a nilpotent Lie algebra on a vector space . The representation is called a nilrepresentation if each matrix is nilpotent, that is for some Engel's theorem (see, for example, Fulton and Harris, page 125 or Varadarajan, page 189) asserts that if rho is a nilrepresentation, then there is a basis for V for which all the representation matrices are strictly upper triangular.
Check that Alg5 is a nilpotent algebra, that rho is a representation, and that rho is a nilrepresentation.
In this new basis the ad matrices are all nilpotent.
