Pith. sign in

REVIEW 19 references

On the Moser trick for Lie subalgebras and foliations

T0 review · reviewed 2026-06-25 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Necessary and sufficient condition established for smooth triviality of Lie subalgebra deformations

desk verdict The paper claims a necessary and sufficient condition for smooth triviality of Lie subalgebra deformations plus a direct Moser trick proof for foliations, but the abstract alone leaves the actual statements and proofs uncheckable. read the letter →

arxiv 2606.25848 v1 pith:R3B2MWPC submitted 2026-06-24 math.DG math.RAmath.RT

Ilias Ermeidis This is my paper · ORCID
classification math.DGmath.RAmath.RT
keywords LiesubalgebrasMosertrickfoliationsidealssmoothdeformationstrivialitysubalgebroids
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to determine the precise circumstances under which a smooth one-parameter family of Lie subalgebras admits a smooth trivialization by automorphisms. It derives a necessary and sufficient condition for this to occur and obtains an analogous criterion for Lie ideals. The argument rests on a direct proof of the Moser trick in the setting of foliations, which then serves as the foundation for generalizing the triviality results to Lie subalgebroids.

What carries the argument

Direct adaptation of the Moser trick to foliations induced by Lie subalgebras, used to construct isotopies that trivialize the deformation

What would settle it

A concrete smooth deformation of a Lie subalgebra where the necessary and sufficient condition is satisfied yet no smooth family of automorphisms trivializes it, or where the condition fails yet a trivialization exists.

Watch

Extended reading notes

Core claim

Given a smooth deformation of a Lie subalgebra, a necessary and sufficient condition is established for its smooth triviality. An analogous criterion holds for Lie ideals. A direct proof of the Moser trick for foliations is presented, forming the basis for extending these results to general Lie subalgebroids.

Load-bearing premise

The deformation varies smoothly in the natural topology on the space of Lie subalgebras, and the associated foliation permits the direct Moser isotopy construction.

Editorial extensions

If this is right

  • Lie ideals satisfy an analogous necessary and sufficient condition for smooth triviality under deformation.
  • The direct Moser construction for foliations extends the triviality criterion to general Lie subalgebroids.
  • These conditions characterize when a deformation in the space of Lie subalgebras can be undone by a smooth family of automorphisms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The direct Moser approach may simplify explicit calculations in deformation problems for concrete foliations.
  • Similar triviality criteria could be sought for related structures such as Lie algebroids or Poisson manifolds.
  • The method might connect to questions of stability in geometric structures where Lie bracket preservation is required.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 0 minor

Summary. The paper establishes a necessary and sufficient condition for the smooth triviality of a smooth deformation of a Lie subalgebra, derives an analogous criterion for Lie ideals, and gives a direct proof of the Moser trick for foliations as the basis for extension to general Lie subalgebroids.

Significance. If the necessary-and-sufficient condition is independent of the deformation data and the direct Moser construction is valid without hidden hypotheses, the result would supply a concrete tool for analyzing triviality in deformations of Lie structures and foliations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report and for accurately summarizing the main results of the paper. No specific major comments were provided in the report, so there are no individual points requiring point-by-point responses at this stage. We remain available to address any questions or clarifications the referee may have.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified from available text

full rationale

The abstract states that a necessary and sufficient condition for smooth triviality is established and a direct proof of the Moser trick is given, but supplies no equations, parameter fits, self-citations, or derivation steps. Without visible self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations, no circular step can be exhibited by quoting the paper. The result is therefore scored as self-contained on the supplied material.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are stated or derivable from the given text.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the Moser trick for Lie subalgebras and foliations." pith.science (2026). https://pith.science/paper/R3B2MWPC

@misc{pith2026260625848,
  author       = {Pith},
  title        = {Pith review of: On the Moser trick for Lie subalgebras and foliations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R3B2MWPC}},
  note         = {Machine review of arXiv:2606.25848}
}
read the original abstract

Given a smooth deformation of a Lie subalgebra, we establish a necessary and sufficient condition for its smooth triviality and derive an analogous criterion for Lie ideals. We then give a direct proof of the Moser trick for foliations, which forms the basis for extending this result to general Lie subalgebroids.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 1 canonical work pages

  1. [1]

    Rigidity and gauge-equivalences of Lie subalgebroids

    Panagiotis Batakidis and Ilias Ermeidis. Rigidity and gauge-equivalences of Lie subalgebroids. Work in progress

  2. [2]

    Lectures on characteristic classes and foliations , pages 1–94

    Raoul Bott. Lectures on characteristic classes and foliations , pages 1–94. Springer Berlin Heidelberg, Berlin, Hei- delberg, 1972

    1972

  3. [3]

    Stability of Lie group homomorphisms and Lie subgroups

    Cristian Camilo Cárdenas and Ivan Struchiner. Stability of Lie group homomorphisms and Lie subgroups. J. Pure Appl. Algebra, 224(3):1280–1296, 2020

    2020

  4. [4]

    Deformations of Lie brackets: cohomological aspects

    Marius Crainic and Ieke Moerdijk. Deformations of Lie brackets: cohomological aspects. J. Eur. Math. Soc. (JEMS) , 10(4):1037–1059, 2008. 16 ILIAS ERMEIDIS

    2008

  5. [5]

    A survey on stability and rigidity results for Lie algebras

    Marius Crainic, Florian Schätz, and Ivan Struchiner. A survey on stability and rigidity results for Lie algebras. Indag. Math. (N.S.) , 25(5):957–976, 2014

    2014

  6. [6]

    On deformations of compact foliations

    Matias del Hoyo and Rui Loja Fernandes. On deformations of compact foliations. Proc. Amer. Math. Soc. , 147(10):4555–4561, 2019

    2019

  7. [7]

    Deformations of Ideals in Lie Algebroids

    Ilias Ermeidis. Deformations of Ideals in Lie Algebroids . PhD thesis, Georg-August Universität Göttingen, 2025

    2025

  8. [8]

    Deformations of ideals in Lie algebras

    Ilias Ermeidis and Madeleine Jotz. Deformations of ideals in Lie algebras. Preprint, arXiv:2412.20600 [math.DG] (2024), 2024

    work page arXiv 2024

Show all 19 references
  1. [9]

    Simultaneous deformations of algebras and morphisms via derived brackets

    Yaël Frégier and Marco Zambon. Simultaneous deformations of algebras and morphisms via derived brackets. J. Pure Appl. Algebra , 219(12):5344–5362, 2015

    2015

  2. [10]

    James L. Heitsch. A cohomology for foliated manifolds. Comment. Math. Helv. , 50:197–218, 1975

    1975

  3. [11]

    Higher homotopies and Maurer-Cartan algebras: quasi-Lie-Rinehart, Gerstenhaber, and Batalin-Vilkovisky algebras

    Johannes Huebschmann. Higher homotopies and Maurer-Cartan algebras: quasi-Lie-Rinehart, Gerstenhaber, and Batalin-Vilkovisky algebras. In The breadth of symplectic and Poisson geometry , volume 232 of Progr. Math., pages 237–302. Birkhäuser Boston, Boston, MA, 2005

    2005

  4. [12]

    Saber Jafarpour and Andrew D. Lewis. Time-varying vector fields and their flows . SpringerBriefs Math. Cham: Springer, 2014

    2014

  5. [13]

    Simultaneous deformations of a Lie algebroid and its Lie subalgebroid

    Xiang Ji. Simultaneous deformations of a Lie algebroid and its Lie subalgebroid. J. Geom. Phys. , 84:8–29, 2014

    2014

  6. [14]

    Deformations of linear Lie brackets

    Pier Paolo La Pastina and Luca Vitagliano. Deformations of linear Lie brackets. Pacific J. Math. , 303(1):265–298, 2019

    2019

  7. [15]

    John M. Lee. Introduction to smooth manifolds , volume 218 of Graduate Texts in Mathematics . Springer, New York, second edition, 2013

    2013

  8. [16]

    R. W. Richardson, Jr. A rigidity theorem for subalgebras of Lie and associative algebras. Illinois J. Math. , 11:92–110, 1967

    1967

  9. [17]

    R. W. Richardson, Jr. Deformations of subalgebras of Lie algebras. Journal of Differential Geometry , 3(3-4):289 – 308, 1969

    1969

  10. [18]

    Gauge equivalences for foliations and pre-symplectic structures

    Florian Schätz and Marco Zambon. Gauge equivalences for foliations and pre-symplectic structures. Commun. Contemp. Math. , 23(7):Paper No. 2050067, 23, 2021

    2021

  11. [19]

    On the strong homotopy Lie-Rinehart algebra of a foliation

    Luca Vitagliano. On the strong homotopy Lie-Rinehart algebra of a foliation. Commun. Contemp. Math. , 16(6):1450007, 49, 2014. Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece Email address : ermeidis95@gmail.com

    2014

Pith tools

Reviewed June 25, 2026 · model on record in the stance chip above.