Higher Haantjes Brackets and Integrability

Tempesta, Piergiulio; Tondo, Giorgio

doi:10.1007/s00220-021-04233-5

Cited by 7 publications

(5 citation statements)

References 26 publications

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“…Consequently, this distribution, being the intersection of distributions which are involutive due to Theorem 1 [22], is also involutive.…”

Section: Generalized Haantjes Algebrasmentioning

confidence: 94%

“…In this section, for the sake of clarity, we shall briefly review some of the main algebro-geometric properties of the new class of generalized Nijenhuis tensors introduced in [20] and further studied in [22].…”

Section: The Generalized Nijenhuis Tensors and Block-diagonalizationmentioning

confidence: 99%

“…It is also useful, from an applicative point of view, to consider the explicit action of a generalized Nijenhuis torsion on a pair of generalized eigenvectors of A. In [22] the following formula has been proved by induction over the integers m ≥ 2:…”

Section: 1mentioning

confidence: 99%

“…In this article, we address this problem with a tensorial approach, based on the notion of Nijenhuis and Haantjes tensors and their generalizations, recently introduced in [22]. The interest in the geometry of Nijenhuis and Haantjes tensors recently has considerably increased.…”

Section: Introductionmentioning

confidence: 99%

“…The class of P H manifolds has been introduced in [24]. A new, infinite family of higher-order torsions of level m, generalizing both the Nijenhuis and the Haantjes torsions, have been defined and discussed in [22]. These torsions can also be derived from a family of higher-order Haantjes brackets, which generalize the Frölicher-Nijenhuis (FN) one.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Nijenhuis Torsions and block-diagonalization of operator fields

Nozaleda¹,

Tempesta²,

Tondo³

2022

Preprint

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The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhuis and Haantjes, offers new tools for the study of normal forms of operator fields. We propose a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level l vanishes, there exists a local chart where all operators can be simultaneously block-diagonalized. We also introduce the notion of generalized Haantjes algebra, consisting of operators with a vanishing higher-level torsion, as a new algebraic structure naturally generalizing standard Haantjes algebras. Contents 1. Introduction 2. Preliminaries on the Nijenhuis and Haantjes geometry 3. The generalized Nijenhuis tensors and block-diagonalization 3.1. Eigen-distributions and spectral properties of generalized Nijenhuis operators 3.2. Block-diagonalization 4. Generalized Haantjes algebras 4.1. Definitions 4.2. Cyclic generalized Haantjes algebras 5. Generalized Nijenhuis torsions and simultaneous block-diagonalization 6. Block-diagonalization of generalized Haantjes algebras 6.1. A level-three generalized Haantjes algebra 6.2. Block-diagonalization 7. A level-four generalized Haantjes algebra 7.1. Spectral analysis 7.2. Block-diagonalization Acknowledgement References

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“…Consequently, this distribution, being the intersection of distributions which are involutive due to Theorem 1 [22], is also involutive.…”

Section: Generalized Haantjes Algebrasmentioning

confidence: 94%

Section: The Generalized Nijenhuis Tensors and Block-diagonalizationmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Generalized Nijenhuis Torsions and block-diagonalization of operator fields

Nozaleda¹,

Tempesta²,

Tondo³

2022

Preprint

Self Cite

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Generalized Nijenhuis Torsions and Block-Diagonalization of Operator Fields

2023

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Partial separability and symplectic-Haantjes manifolds

Reyes,

Tempesta,

Tondo

2024

Annali di Matematica

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A theory of partial separability for classical Hamiltonian systems is proposed in the context of Haantjes geometry. As a general result, we show that the knowledge of a non-semisimple symplectic-Haantjes manifold for a given Hamiltonian system is sufficient to construct sets of coordinates (called Darboux-Haantjes coordinates) that allow both the partial separability of the associated Hamilton-Jacobi equations and the block-diagonalization of the operators of the corresponding Haantjes algebra. We also introduce a novel class of Hamiltonian systems, characterized by the existence of a generalized Stäckel matrix, which by construction are partially separable. They widely generalize the known families of partially separable Hamiltonian systems. The new systems can be described in terms of semisimple but non-maximal-rank symplectic-Haantjes manifolds.

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Higher Haantjes Brackets and Integrability

Cited by 7 publications

References 26 publications

Generalized Nijenhuis Torsions and block-diagonalization of operator fields

Generalized Nijenhuis Torsions and block-diagonalization of operator fields

Generalized Nijenhuis Torsions and Block-Diagonalization of Operator Fields

Partial separability and symplectic-Haantjes manifolds

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