# Integrable systems around the world, September 14-16, 2020

Sep 14 – 16, 2020
SISSA
Europe/Rome timezone

## Poisson quasi-Nijenhuis manifolds and the Toda system

Sep 15, 2020, 10:00 AM
50m
Chairman: Gregorio Falqui

#### Chairman: Gregorio Falqui

Scientific Programme

### Speaker

Marco Pedroni (Università di Bergamo)

### Description

The notion of Poisson quasi-Nijenhuis manifold, introduced by Stiénon and Xu, generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this talk, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically, the Poisson quasi-Nijenhuis structure is largely too general to ensure Liouville integrability of a system.
However, we present a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds, and we give sufficient conditions such that the spectral invariants of the "quasi-Nijenhuis recursion operator'' of a Poisson quasi-Nijenhuis manifold (obtained by deforming a Poisson-Nijenhuis structure) are in involution. Then we prove that the closed (or periodic) n-particle Toda lattice, along with its relation with the open (or non periodic) Toda system, can be framed in such a geometrical structure. These results have been obtained in collaboration with Gregorio Falqui, Igor Mencattini, and Giovanni Ortenzi.

### Primary author

Marco Pedroni (Università di Bergamo)