Poisson quasi-Nijenhuis deformations of the canonical PN structure

TL;DR

This paper introduces a method to deform Poisson-Nijenhuis structures into Poisson quasi-Nijenhuis structures using closed 2-forms, with applications to Toda lattice models.

Contribution

It provides a new deformation technique for Poisson-Nijenhuis manifolds and applies it to Toda lattice structures, linking different integrable systems.

Findings

Deformation from Poisson-Nijenhuis to Poisson quasi-Nijenhuis structures achieved.

Application to Toda lattice models demonstrates the method's utility.

Establishes connections between open and closed Toda lattice structures.

Abstract

We present a result which allows us to deform a Poisson-Nijenhuis manifold into a Poisson quasi-Nijenhuis manifold by means of a closed 2-form. Under an additional assumption, the deformed structure is also Poisson-Nijenhuis. We apply this result to show that the canonical Poisson-Nijenhuis structure on R^2n gives rise to both the Poisson-Nijenhuis structure of the open (or non periodic) n-particle Toda lattice, introduced by Das and Okubo [6], and the Poisson quasi-Nijenhuis structure of the closed (or periodic) n-particle Toda lattice, described in our recent work [7].