# Beyond recursion operators

@article{KosmannSchwarzbach2019BeyondRO, title={Beyond recursion operators}, author={Y. Kosmann-Schwarzbach}, journal={arXiv: History and Overview}, year={2019}, pages={167-180} }

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.

#### 6 Citations

Higher Haantjes Brackets and Integrability

- Mathematics
- 2018

We propose a new, infinite class of brackets generalizing the Frölicher– Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular,… Expand

Haantjes Algebras of the Lagrange Top

- Mathematics, Physics
- 2018

We study a symplectic-Haantjes manifold and a Poisson–Haantjes manifold for the Lagrange top and compute a set of Darboux–Haantjes coordinates. Such coordinates are separation variables for the… Expand

A New Class of Generalized Haantjes Tensors and Nilpotency

- Mathematics
- 2018

We propose a new infinite class of generalized binary tensor fields. The first representative of this class is the known Fr\"olicher--Nijenhuis bracket. Also, this new family of tensors reduces to… Expand

Haantjes algebras and diagonalization

- Mathematics, Physics
- 2021

Abstract We introduce the notion of Haantjes algebra: It consists of an assignment of a family of operator fields on a differentiable manifold, each of them with vanishing Haantjes torsion. They are… Expand

A New family of higher-order Generalized Haantjes Tensors, Nilpotency and Integrability

- Mathematics
- 2018

We propose a new infinite class of generalized binary tensor fields, whose first representative of is the known Frolicher--Nijenhuis bracket. This new family of tensors reduces to the generalized… Expand

#### References

SHOWING 1-10 OF 30 REFERENCES

Haantjes Structures for the Jacobi-Calogero Model and the Benenti Systems

- Mathematics, Physics
- 2016

In the context of the theory of symplectic-Haantjes manifolds, we construct the Haantjes structures of generalized Stackel systems and, as a particular case, of the quasi-bi-Hamiltonian systems. As… Expand

Algebraic identity for the Schouten tensor and bi-Hamiltonian systems

- Mathematics
- 2007

Abstract A ( 0 , 3 )-tensor T i j k is introduced in an invariant form. Algebraic identities are derived that connect the Schouten ( 2 , 1 )-tensor S i j k and tensor T i j k with the Nijenhuis… Expand

NijenhuisG-manifolds and Lenard bicomplexes: A new approach to KP systems

- Mathematics
- 1988

We suggest a method to extend the theory of recursion operators to integrable Hamiltonian systems in two-space dimensions, like KP systems. The approach aims to stress the conceptual unity of the… Expand

General algebraic identities for the Nijenhuis and Haantjes tensors

- Mathematics
- 2004

We obtain general algebraic identities for the Nijenhuis and Haantjes tensors on an arbitrary manifold?Mn. For n=3 we derive special algebraic identities connected with the Cartan-Killing form?(u,v)H.

Poisson-Nijenhuis structures

- Mathematics
- 1990

We study the deformation, defined by a Nijenhuis operator, and the dualization, defined by a Poisson bivector, of the Lie bracket of vector fields on a manifold and, more generally, of the Lie… Expand

Recursion Operators and Frobenius Manifolds

- Mathematics, Physics
- 2012

In this note I exhibit a "discrete homotopy" which joins the category of F-manifolds to the category of Poisson-Nijenhuis manifolds, passing through the category of Frobenius manifolds.

Compatible Structures on Lie Algebroids and Monge-Ampère Operators

- Mathematics
- 2008

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures… Expand

Haantjes Manifolds of Classical Integrable Systems

- Mathematics
- 2014

A general theory of classical integrable systems is proposed, based on the geometry of the Haantjes tensor. We introduce the class of symplectic-Haantjes manifolds (or $\omega \mathcal{H}$ manifold),… Expand

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

- Physics, Mathematics
- 2005

The integrability of an m-component system of hydrodynamic type, ut = V(u)ux, by the generalized hodograph method requires the diagonalizability of the m × m matrix V(u). This condition is known to… Expand

A Simple model of the integrable Hamiltonian equation

- Mathematics
- 1978

A method of analysis of the infinite‐dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested. This analysis is based… Expand