The integrability of the 2-Toda lattice on a simple Lie algebra

TL;DR

This paper establishes the Liouville integrability of the 2-Toda lattice defined on any simple Lie algebra by constructing a large family of conserved quantities using R-matrix structures.

Contribution

It generalizes the 2-Toda lattice to all simple Lie algebras and proves its integrability through a novel construction of constants of motion.

Findings

The 2-Toda lattice is integrable on all simple Lie algebras.

A large family of conserved quantities is constructed.

Liouville integrability is proven using Lie algebraic structures.

Abstract

We define the 2-Toda lattice on every simple Lie algebra g, and we show its Liouville integrability. We show that this lattice is given by a pair of Hamiltonian vector fields, associated with a Poisson bracket which results from an R-matrix of the underlying Lie algebra. We construct a big family of constants of motion which we use to prove the Liouville integrability of the system. We achieve the proof of their integrability by using several results on simple Lie algebras, R-matrices, invariant functions and root systems.