LieAlgebrasRepresentationEigenvector - find a simultaneous eigenvector for the representation of a solvable Lie algebra Calling Sequences RepresentationEigenvector( ρ ) Parameters ρ - a representation of a solvable Lie algebra 𝔤 on a vector space V Description. If ρ : 𝔤 → gl V is a representation of a solvable Lie algebra 𝔤, then a fundamental theorem due to Lie (see, for example, page 125 or, page 200) asserts that there is a vector Y ∈ V such that ρ x Y = λ x Y for all x ∈ 𝔤. The eigenvector Y is called a simultaneous eigenvector for the representation. The vector Y may be complex and will, in general, not be unique.
Let e 1 , e 2 ,. . , e n be the given basis for 𝔤. The program RepresentationEigenvector( ρ ) returns the list of eigenvalues λ e 1 , λ e 2 ,. . , λ e n and the simultaneous eigenvector Y. The program RepresentationEigenvector( ρ ) works as follows. First a change of basis is made using the program so that in the new basis f 1 , f 2 ,.
. , f n the subalgebra spanned by f 1 , f 2 ,. . , f k is an ideal in the subalgebra f 1 , f 2 ,.
The View From Eigenvector Support For Mac Download
. , f k + 1. Then an eigenspace E 1 ⊂ V for the the linear transformation ρ f 1 is found. This space is invariant under ρ f 2 so the program next finds a subspace E 2 ⊂ E 1 which is a eigenspace for ρ f 1 and so on. V with(DifferentialGeometry): with(LieAlgebras): with(Library): Example 1. First we define a representation of a 4 dimensional algebra.