Beyond recursion operators

TL;DR

This paper explores the generalization of classical integrable systems using Haantjes structures, extending the bi-hamiltonian formalism by replacing recursion operators with families of commuting Haantjes operators.

Contribution

It introduces the use of Haantjes manifolds and symplectic-Haantjes structures as a novel framework for integrable systems, broadening the classical approach.

Findings

Haantjes torsion generalizes Nijenhuis torsion.

Haantjes manifolds extend bi-hamiltonian structures.

Families of commuting Haantjes operators replace recursion operators.

Abstract

We briefly recall the history of the Nijenhuis torsion of (1,1)-tensors on manifolds and of the lesser-known Haantjes torsion. We then show how the Haantjes manifolds of Magri and the symplectic-Haantjes structures of Tempesta and Tondo generalize the classical approach to integrable systems in the bi-hamiltonian and symplectic-Nijenhuis formalisms, the sequence of powers of the recursion operator being replaced by a family of commuting Haantjes operators.