Semi-invariants and Integrals of the Full Symmetric sl(n) Toda Lattice

Abstract

We consider the full symmetric version of the Lax operator of the Toda lattice which is known as the full symmetric Toda lattice. The phase space of this system is the generic orbit of the coadjoint action of the Borel subgroup B^+(n) of SL(n,R). This system is integrable. We propose a new method of constructing semi-invariants and integrals of the full symmetric Toda lattice. Using only the Toda equations for the Lax eigenvector matrix we prove the existence of the semi-invariants which are Plucker coordinates in the corresponding projective spaces. Then we use these semi-invariants to construct the integrals. It is known that the full symmetric sl(n) Toda lattice has additional integrals which can be produced by Kostant procedure except for the integrals which can be derived by the chopping procedure. Altogether these integrals constitute a full set of the independent non-involutive integrals. Yet the unsolved complicated technical problem is their explicit derivation since Kostant procedure has crucial computational complexities even for low-rank Lax matrices and is practically unapplicable for higher ranks. Our new approach provides a resolution of this problem and results in simple explicit formulae for the full set of independent semi-invariants and integrals expressed both in terms of the Lax matrix and its eigenvector and eigenvalue matrices of the full symmetric sl(n) Toda lattice without using the chopping and Kostant procedures. We also describe the structure of the additional integrals of motion as functions on the flag space modulo the Toda flows and show how Plucker coordinates of different projective spaces define different families of the additional integrals. In this paper we present detailed proofs of the propositions of [24].