The integrability of the periodic Full Kostant-Toda on a simple Lie algebra

TL;DR

This paper establishes the Liouville integrability of the periodic Full Kostant-Toda lattice on simple Lie algebras by constructing conserved quantities using R-matrix Poisson brackets.

Contribution

It introduces a new integrable system on simple Lie algebras and proves its integrability using invariant functions and R-matrix techniques.

Findings

The system is Hamiltonian with a Poisson structure from an R-matrix.

A large family of conserved quantities is constructed.

The system's integrability is proven for all simple Lie algebras.

Abstract

We define the periodic Full Kostant-Toda on every simple Lie algebra, and show its Liouville integrability. More precisely we show that this lattice is given by a Hamiltonian vector field, associated to a Poisson bracket which results from an R-matrix. We construct a large family of constant of motion which we use to prove the Liouville integrability of the system with the help of several results on simple Lie algebras, R-matrix, invariant functions and root systems.