Poisson quasi-Nijenhuis manifolds, closed Toda lattices, and generalized recursion relations

TL;DR

This paper develops involutivity theorems for Poisson quasi-Nijenhuis manifolds, applying them to closed Toda lattices and extending recursion relations, thereby linking geometric structures with integrable systems.

Contribution

It introduces new involutivity theorems for PqN manifolds and connects these to Toda lattices and generalized recursion relations, expanding the geometric understanding of integrable systems.

Findings

Involutivity theorems for PqN manifolds established.

Application to closed Toda lattices of various types demonstrated.

Relation between geometric structures and recursion relations clarified.

Abstract

We present two involutivity theorems in the context of Poisson quasi-Nijenhuis %(PqN) manifolds. The second one stems from recursion relations that generalize the so called Lenard-Magri relations on a bi-Hamiltonian manifold. We apply these results to the closed (or periodic) Toda lattices of type $A_n^{(1)}$, $C_n^{(1)}$, $A_{2n}^{(2)}$ and, for the ones of type $A^{(1)}_n$, we show how this geometrical setting relates to their bi-Hamiltonian representation and to their recursion relations.