The Geometry of the Full Kostant-Toda Lattice

New method for constructing semi-invariants and integrals of the full symmetric $\mathfrak{s}\mathfrak{l}_n $ Toda lattice

“…We used semi-invariants (30), which are Plücker coordinates (71) in the corresponding projective spaces, to construct the full noncommutative family of integrals expressed exactly in terms of Lax operator matrix (12) and in terms of eigenvalue and eigenvector matrices (75) of arbitrary rank.…”

“…It was discovered in [12], [18], [19] that in the case n = 4, there are two involutive families of integrals of motion in the full Kostant-Toda lattice. We show that the same also holds for the full symmetric Toda lattice.…”

“…The existence of different families of involutive integrals of the full Kostant-Toda lattice was first noted for the case n = 4 in [12]. It was shown in [14]- [16] that this system is integrable in the noncommutative sense (see [17]).…”

“…It was shown in [12] that the integrals of motion of the full Kostant-Toda lattice can be expressed via the homogeneous coordinates of some projective spaces. For this, an embedding in the flag space and the dynamics with respect to one-parameter subgroups induced by the isospectral integrals of motion were considered (see Appendix B).…”

“…They were first found for the full symmetric Toda lattice in [10], where the chopping procedure was used. Constructing the additional integrals and semi-invariants was studied in [6], [10]- [12]. We note that a general formula for calculating the integrals of motion without using the chopping procedure was recently found in [13] and was further used to generalize the Toda lattice to the quantum case.…”