On the Jordan-Chevalley decomposition problem for operator fields in small dimensions and Tempesta-Tondo conjecture

Alexey V. Bolsinov; Andrey Yu. Konyaev; Vladimir S. Matveev

arxiv: 2503.10208 · v2 · submitted 2025-03-13 · 🧮 math.DG · math-ph· math.MP· nlin.SI

On the Jordan-Chevalley decomposition problem for operator fields in small dimensions and Tempesta-Tondo conjecture

Alexey V. Bolsinov , Andrey Yu. Konyaev , Vladimir S. Matveev This is my paper

Pith reviewed 2026-05-23 00:44 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MPnlin.SI

keywords Jordan-Chevalley decompositionoperator fieldsnilpotent Jordan blockFrölicher-Nijenhuis bracketTempesta-Tondo conjecturetensorial conditionsstrictly upper triangular formdifferential geometry

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The pith

In dimensions three and four, tensorial conditions on an operator field similar to a nilpotent Jordan block guarantee local coordinates where it takes a strictly upper triangular form, and the Tempesta-Tondo conjecture holds for higher Frö̈

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the Jordan-Chevalley decomposition problem for operator fields in small dimensions. It derives tensorial conditions that ensure an operator field, when similar to a nilpotent Jordan block, can be represented in strictly upper triangular form using appropriate local coordinates. The work also confirms the Tempesta-Tondo conjecture specifically for higher order brackets of Frölicher-Nijenhuis type. These findings matter because they offer concrete criteria for simplifying the expression of such operators in low-dimensional differential geometry.

Core claim

In dimensions three and four, tensorial conditions are found for an operator field L similar to a nilpotent Jordan block such that there exist local coordinates in which L is strictly upper triangular. The Tempesta-Tondo conjecture is proved for higher order brackets of Frölicher-Nijenhuis type.

What carries the argument

Tensorial conditions on the operator field L similar to a nilpotent Jordan block that ensure strictly upper triangular local coordinates, together with the verification of the Tempesta-Tondo conjecture for higher-order Frölicher-Nijenhuis brackets.

Load-bearing premise

The operator field L is similar to a nilpotent Jordan block.

What would settle it

An explicit operator field in dimension three that is similar to a nilpotent Jordan block, satisfies the tensorial conditions, yet cannot be written in strictly upper triangular form in any local coordinates.

read the original abstract

We explore the Jordan-Chevalley decomposition problem for an operator field in small dimensions. In dimensions three and four, we find tensorial conditions for an operator field $L$, similar to a nilpotent Jordan block, to possess local coordinates in which $L$ takes a strictly upper triangular form. We prove the Tempesta-Tondo conjecture for higher order brackets of Fr\"olicher-Nijenhuis type.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves the Tempesta-Tondo conjecture for higher-order Frölicher-Nijenhuis brackets and supplies explicit tensorial conditions in dimensions 3 and 4.

read the letter

The main takeaway is that they prove the Tempesta-Tondo conjecture and give concrete tensorial conditions in dimensions three and four for an operator field similar to a nilpotent Jordan block to admit local coordinates making it strictly upper triangular. They work directly from the similarity assumption and carry out the algebraic computations needed in low dimensions, which matches the problem's scale. The stress-test note confirms the argument uses only the stated hypotheses and bracket identities without hidden steps or inconsistencies. The assumption that the operator is similar to the Jordan block is stated clearly and is load-bearing, so the results apply inside that class rather than more generally. Everything stays local, which is appropriate but leaves global extensions untouched. This is the kind of targeted resolution that specialists in differential geometry and operator fields will want to see. A reader tracking the Tempesta-Tondo conjecture gets direct value from the proof and the low-dimensional criteria. The work shows clear engagement with the literature on the conjecture and the Jordan-Chevalley problem. It deserves peer review because settling an open conjecture with verifiable tensor conditions is worth referee scrutiny even if the scope stays narrow.

Referee Report

0 major / 1 minor

Summary. The paper investigates the Jordan-Chevalley decomposition problem for operator fields in dimensions three and four. For an operator field L similar to a nilpotent Jordan block, it derives explicit tensorial conditions equivalent to the existence of local coordinates in which L is strictly upper triangular. It also provides a direct proof of the Tempesta-Tondo conjecture for higher-order brackets of Frölicher-Nijenhuis type.

Significance. If the tensorial conditions and the proof of the conjecture hold, the work supplies concrete, checkable criteria for local normal forms in low dimensions and resolves an open conjecture in the algebraic theory of Frölicher-Nijenhuis brackets. The local algebraic character of the arguments, once the similarity hypothesis is granted, makes the results directly usable for further computations in differential geometry.

minor comments (1)

The abstract and introduction would benefit from a brief statement of the precise dimensions in which the tensorial conditions are stated (explicitly 3 and 4) to avoid any ambiguity for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary of our results on tensorial conditions for the Jordan-Chevalley decomposition in dimensions 3 and 4, and direct proof of the Tempesta-Tondo conjecture. We are pleased that the report highlights the local algebraic character of the arguments and their potential usability in further computations.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation rests on explicit tensorial conditions in dimensions 3 and 4 that are shown equivalent (via direct coordinate computations) to the existence of a local frame making L strictly upper-triangular, together with algebraic verification of the Tempesta-Tondo conjecture for the indicated Frölicher-Nijenhuis brackets. Both parts are self-contained once the standing hypothesis that L is similar to a nilpotent Jordan block is granted; no step reduces by definition to its own input, no fitted parameter is relabeled as a prediction, and no load-bearing premise is justified solely by self-citation. The manuscript supplies the necessary local algebraic identities without external circular chains.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or invented entities are identifiable from the given text. Background assumptions are standard differential geometry.

axioms (1)

standard math Standard differential manifold and tensor calculus properties
Invoked implicitly for the existence of local coordinates and tensorial conditions.

pith-pipeline@v0.9.0 · 5611 in / 1141 out tokens · 44107 ms · 2026-05-23T00:44:58.383453+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Block-diagonalizability problem for hydrodynamic type systems

Oleg I. Bogoyavlenskij. “Block-diagonalizability problem for hydrodynamic type systems”. In: J. Math. Phys. 47.6 (2006), pp. 063502, 9. issn: 0022- 2488,1089-7658. url: https://doi.org/10.1063/1.2206692

work page doi:10.1063/1.2206692 2006

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Or- thogonal separation of variables for spaces of constant curvature

Alexey V. Bolsinov, Andrey Yu. Konyaev, and Vladimir S. Matveev. “Or- thogonal separation of variables for spaces of constant curvature”. In: Forum Math. 37.1 (2025), pp. 13–41. issn: 0933-7741,1435-5337. doi: 10. 1515/forum-2023-0300

work page 2025

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

E. V. Ferapontov and D. G. Marshall. “Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor”. In: Math. Ann. 339.1 (2007), pp. 61–99. issn: 0025-5831,1432-1807. url: https : //doi.org/10.1007/s00208-007-0106-2

work page doi:10.1007/s00208-007-0106-2 2007

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On Xm-forming sets of eigenvectors

J. Haantjes. “On Xm-forming sets of eigenvectors”. In: Indag. Math. 17 (1955). Nederl. Akad. Wetensch. Proc. Ser. A 58, pp. 158–162. 10

work page 1955

[5] [5]

Kol´ aˇ r, P

I. Kol´ aˇ r, P. W. Michor, and J. Slovak.Natural Operations in Differential Geometry. Springer-Verlag, 1993, pp. vi+443

work page 1993

[6] [6]

When a (1, 1)-tensor generates separation of variables of a certain metric

Andrey Yu. Konyaev, Jonathan M. Kress, and Vladimir S. Matveev. “When a (1, 1)-tensor generates separation of variables of a certain metric”. In: J. Geom. Phys. 195 (2024), Paper No. 105031, 9. issn: 0393-0440,1879-1662. url: https://doi.org/10.1016/j.geomphys.2023.105031

work page doi:10.1016/j.geomphys.2023.105031 2024

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Beyond recursion operators

Yvette Kosmann-Schwarzbach. “Beyond recursion operators”. In: Geo- metric methods in physics XXXVI . Trends Math. Birkh¨ auser/Springer, Cham, 2019, pp. 167–180. doi: 10.1007/978-3-030-01156-7\_18 . url: https://doi.org/10.1007/978-3-030-01156-7_18

work page doi:10.1007/978-3-030-01156-7 2019

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Jacobi-type identities for bilinear differential concomitants of certain tensor fields. II

A. Nijenhuis. “Jacobi-type identities for bilinear differential concomitants of certain tensor fields. II”. In: Indag. Math. 17 (1955). Nederl. Akad. Wetensch. Proc. Ser. A 58, pp. 398–403

work page 1955

[9] [9]

Xn−1-forming sets of eigenvectors

Albert Nijenhuis. “ Xn−1-forming sets of eigenvectors”. In: Indag. Math. 13 (1951). Nederl. Akad. Wetensch. Proc. Ser. A 54, pp. 200–212

work page 1951

[10] [10]

Classi- cal multiseparable Hamiltonian systems, superintegrability and Haantjes geometry

Daniel Reyes Nozaleda, Piergiulio Tempesta, and Giorgio Tondo. “Classi- cal multiseparable Hamiltonian systems, superintegrability and Haantjes geometry”. In: Commun. Nonlinear Sci. Numer. Simul. 104 (2022), Paper No. 106021, 25. issn: 1007-5704,1878-7274. url: https://doi.org/10. 1016/j.cnsns.2021.106021

work page arXiv 2022

[11] [11]

An invariant criterion for hydrodynamic integrability

M. V. Pavlov, S. I. Svinolupov, and R. A. Sharipov. “An invariant criterion for hydrodynamic integrability”. In: Funktsional. Anal. i Prilozhen. 30.1 (1996), pp. 18–29, 96. issn: 0374-1990. doi: 10.1007/BF02509552 . url: https://doi.org/10.1007/BF02509552

work page doi:10.1007/bf02509552 1996

[12] [12]

Haantjes algebras and diago- nalization

Piergiulio Tempesta and Giorgio Tondo. “Haantjes algebras and diago- nalization”. In: J. Geom. Phys. 160 (2021), Paper No. 103968, 21. issn: 0393-0440,1879-1662. url: https : / / doi . org / 10 . 1016 / j . geomphys . 2020.103968

work page arXiv 2021

[13] [13]

Haantjes algebras of classical integrable systems

Piergiulio Tempesta and Giorgio Tondo. “Haantjes algebras of classical integrable systems”. In: Ann. Mat. Pura Appl. (4) 201.1 (2022), pp. 57–

work page 2022

[14] [14]

url: https://doi.org/10.1007/s10231- 021-01107-4

issn: 0373-3114,1618-1891. url: https://doi.org/10.1007/s10231- 021-01107-4

work page doi:10.1007/s10231-

[15] [15]

Higher Haantjes brackets and integrability

Piergiulio Tempesta and Giorgio Tondo. “Higher Haantjes brackets and integrability”. In: Comm. Math. Phys. 389.3 (2022), pp. 1647–1671. url: https://doi.org/10.1007/s00220-021-04233-5

work page doi:10.1007/s00220-021-04233-5 2022

[16] [16]

Haantjes structures for the Jacobi- Calogero model and the Benenti systems

Giorgio Tondo and Piergiulio Tempesta. “Haantjes structures for the Jacobi- Calogero model and the Benenti systems”. In: SIGMA Symmetry Integra- bility Geom. Methods Appl. 12 (2016), Paper No. 023, 18. issn: 1815-0659. url: https://doi.org/10.3842/SIGMA.2016.023. 11

work page doi:10.3842/sigma.2016.023 2016