SolvableRepresentation - Maple Help

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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

Calling Sequences

     SolvableRepresentation( ρ, options)

     SolvableRepresentation(Alg, 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 abba 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 ρinvariant 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 3 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.




Example 1.

We define a 5-dimensional representation of a 3-dimensional solvable Lie algebra.





alg1 > 


V1 > 


V1 > 





We find a new basis for the representation space in which the matrices are all upper triangular.

alg1 > 





To verify this result we use the ChangeRepresentationBasis command to change basis in the representation space.

V1 > 





Example 2.

We define a 6-dimensional representation of a 3-dimensional solvable Lie algebra.

alg1 >