so(p,q) Toda Systems

TL;DR

This paper introduces a new integrable Toda-type Hamiltonian system linked to the real Lie algebra so(p,q), providing Lax pairs, Poisson structures, and demonstrating Liouville integrability, extending classical Toda lattice models.

Contribution

It defines a novel Toda system associated with so(p,q), constructs Lax pairs, and proves its integrability, connecting it to classical Toda lattices through transformations.

Findings

Constructed Lax pair representations for the system

Proved Liouville integrability of the new system

Established connections to classical Toda lattices

Abstract

We define an integrable hamiltonian system of Toda type associated with the real Lie algebra $\so{p}{q}$. As usual there exists a periodic and a non-periodic version. We construct, using the root space, two Lax pair representations and the associated Poisson tensors. We prove Liouville integrability and examine the multi-hamiltonian structure. The system is a projection of a canonical $A_n$ type Toda lattice via a Flaschka type transformation. It is also obtained via a complex change of variables from the classical Toda lattice.