Haantjes algebras and diagonalization

Tempesta, Piergiulio; Tondo, Giorgio

doi:10.1016/j.geomphys.2020.103968

Cited by 8 publications

(14 citation statements)

References 25 publications

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“…A simple reduction of this family, obtained when each representative depends on a pair ( A, A) of copies of the same operator field, coincides (up to a constant) with the family of generalized torsions defined independently (and from a different perspective) by Kosmann-Schwarzbach in [13] and by ourselves in an early, preprint 2017 version of [23].…”

Section: Introductionmentioning

confidence: 89%

“…If we take A = B, then the previous family of brackets reduces to the generalized torsions proposed independently in [13] and in [23]. Here we remind the main definition of that construction, according to the formulation proposed in [23], since it will be crucial in the subsequent discussion.…”

Section: Lemma 10 the Expression In Local Coordinates Of The Haantjes Brackets Of Level M For M ≥ 2 Reads (H (M)mentioning

confidence: 99%

“…Many more examples of Haantjes operators, relevant in classical mechanics and in Riemannian geometry, can be found for instance in [21][22][23][24][25].…”

Section: Preliminaries On the Nijenhuis And Haantjes Geometriesmentioning

confidence: 99%

“…[4,9]) and in the study of infinite-dimensional integrable systems, in connection with the celebrated WDVV equations of associativity and the theory of Dubrovin-Frobenius manifolds [15][16][17]. In [22][23][24][25], the notion of Haantjes algebras and the related ones of ωH and PH manifolds have been proposed as a natural setting for the formulation of the theory of finite-dimensional integrable Hamiltonian systems.…”

Section: Introductionmentioning

confidence: 99%

“…Our second goal is to clarify the geometric meaning of the "generalized Nijenhuis torsions" of higher level introduced, with different formulations, in [13,23]. In Sect.…”

Section: Introductionmentioning

confidence: 99%

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Higher Haantjes Brackets and Integrability

Tempesta

Tondo

2021

Commun. Math. Phys.

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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, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We also prove that the vanishing of a higher-level Nijenhuis torsion of an operator field is a sufficient condition for the integrability of its eigen-distributions. This result (which does not require any knowledge of the spectral properties of the operator) generalizes the celebrated Haantjes theorem. The same vanishing condition also guarantees that the operator can be written, in a local chart, in a block-diagonal form.

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Section: Introductionmentioning

confidence: 89%

Section: Lemma 10 the Expression In Local Coordinates Of The Haantjes Brackets Of Level M For M ≥ 2 Reads (H (M)mentioning

confidence: 99%

“…Many more examples of Haantjes operators, relevant in classical mechanics and in Riemannian geometry, can be found for instance in [21][22][23][24][25].…”

Section: Preliminaries On the Nijenhuis And Haantjes Geometriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Our second goal is to clarify the geometric meaning of the "generalized Nijenhuis torsions" of higher level introduced, with different formulations, in [13,23]. In Sect.…”

Section: Introductionmentioning

confidence: 99%

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Higher Haantjes Brackets and Integrability

Tempesta

Tondo

2021

Commun. Math. Phys.

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Generalized Nijenhuis Torsions and Block-Diagonalization of Operator Fields

2023

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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 prove a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level m 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.

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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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Haantjes algebras and diagonalization

Cited by 8 publications

References 25 publications

Higher Haantjes Brackets and Integrability

Higher Haantjes Brackets and Integrability

Generalized Nijenhuis Torsions and Block-Diagonalization of Operator Fields

Partial separability and symplectic-Haantjes manifolds

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