LieAlgebras[SolvableRepresentation] - given a representation of a solvable algebra, find a basis for the representation space in which the representation matrices are upper triangular matrices
SolvableRepresentation( ρ, options)
ρ - a representation of a solvable Lie algebra 𝔤 on a vector space V
alg - a string or name, the name of a initialized solvable Lie algebra
options - the keyword argument output = O, where O is a list with members "NewBasis", ChangeOfBasisMatrix", "TransformedMatrices", "Partition"; the keyword argument fieldextension = I
Let rho: 𝔤 → glVbe a representation of a solvable Lie algebra 𝔤 on a vector space V. A corollary of Lie's fundamental theorem for solvable Lie algebras (see RepresentationEigenvector) implies that there always exists a basis (possibly complex) for V such that the matrix representation of ρxis upper triangular for all x ∈𝔤.
The program SolvableRepresentation(rho) uses the program RepresentationEigenvector to construction such a basis. In the case when the RepresentationEigenvector program returns a complex eigenvector (with associated complex eigenvalue a + bI), the matrix representation will not be upper triangular but will contain the matrix ab−ba on the diagonal (similar to the real Jordan form of a matrix).
For the second calling sequence, the program SolvableRepresentation is applied to the adjoint representation of the algebra Alg.
The output is a 4-element sequence. The 1st element is a new basis ℬ forV in which the representation is upper triangular, the 2nd element is the change of basis matrix, the 3rd element is the representation in the new basis. The 4th element P gives the partition defining the size of the diagonal block matrices. If P = 1.. n1, n1+1 .. n2, n2+1 .. n3, ... , then the subspaces ℬ1, ..., n1, ℬ1, ..., n2, ℬ1, ..., n3 are ρ−invariant subspaces. If, for example, P = 1.. 1, 2.. 2 , 3.. 3, then all the eigenvectors calculated by RepresentationEigenvector are real. If C = 1..1, 2..3 then the vectors ℬ2 and ℬ3 are the real and imaginary parts of a complex eigenvector. The precise form of the output can be specified by the user with the keyword argument output = O, where O is a list with members "NewBasis", ChangeOfBasisMatrix", "TransformedMatrices", "Partition".
With the option fieldextension = I, a complex basis will be returned (if needed) which puts the representation into upper triangular form.
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.
We define a 6-dimensional representation of a 3-dimensional solvable Lie algebra.