Right invariant Poisson Nijenhuis structures on Lie groupoids Correspondence and Classification

Ghorbanali Haghighatdoost

arxiv: 2411.17179 · v2 · submitted 2024-11-26 · 🧮 math-ph · math.MP

Right invariant Poisson Nijenhuis structures on Lie groupoids Correspondence and Classification

Ghorbanali Haghighatdoost This is my paper

Pith reviewed 2026-05-23 17:39 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Lie groupoidsPoisson-Nijenhuis structuresLie algebroidsright-invariantone-to-one correspondenceclassificationexamples

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The pith

A one-to-one correspondence exists between (Poisson bivector, Nijenhuis operator) structures on Lie algebroids and right-invariant Poisson-Nijenhuis structures on Lie groupoids under certain conditions.

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

The paper defines right-invariant Poisson-Nijenhuis structures on Lie groupoids as pairs consisting of a Poisson bivector and a Nijenhuis operator that remain unchanged under right translations and satisfy compatibility conditions. It defines the corresponding infinitesimal objects on Lie algebroids as (Poisson bivector, Nijenhuis operator) structures. The central result establishes a bijection between these two levels when certain conditions hold. A reader would care because the result supplies an integration procedure that moves geometric data from the infinitesimal algebroid setting to the global groupoid setting. The paper supplies concrete examples to exhibit the correspondence in practice.

Core claim

We introduce right-invariant Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called (Poisson bivector, Nijenhuis operator) structures. Also, we present a one-to-one correspondence between (Poisson bivector, Nijenhuis operator) structures on Lie algebroids with (Poisson, Nijenhuis) structures on their Lie groupoids under certain conditions. Also, we give some illustrative examples.

What carries the argument

Right-invariant Poisson-Nijenhuis structure on a Lie groupoid, a compatible pair of Poisson bivector and Nijenhuis operator invariant under right translations.

If this is right

Structures on the groupoid level can be recovered from data on the algebroid level via the correspondence.
Classification of right-invariant Poisson-Nijenhuis structures on groupoids reduces to the classification problem on the corresponding algebroids when the conditions apply.
Illustrative examples demonstrate that the correspondence produces concrete pairs on both the groupoid and algebroid sides.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the unspecified conditions can be checked algorithmically in given cases, the correspondence supplies a practical construction method for the groupoid structures.
The same lifting technique may apply to other compatible pairs of tensors on groupoids once analogous invariance and compatibility conditions are formulated.
The examples suggest that the correspondence preserves additional geometric features such as rank or integrability when those features are present on the algebroid.

Load-bearing premise

The one-to-one correspondence is asserted only under certain conditions whose precise statement and justification are not supplied.

What would settle it

An explicit Lie algebroid equipped with a (Poisson bivector, Nijenhuis operator) structure whose unique integrating groupoid fails to carry a matching right-invariant Poisson-Nijenhuis structure would refute the claimed bijection.

read the original abstract

In this paper, we introduce right-invariant Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called (Poisson bivector, Nijenhuis operator) structures. Also, we present a one-to-one correspondence between (Poisson bivector, Nijenhuis operator) structures on Lie algebroids with (Poisson, Nijenhuis) structures on their Lie groupoids under certian conditions. Also, we give some illustrative examples .

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 introduces right-invariant Poisson-Nijenhuis structures on Lie groupoids and claims a one-to-one correspondence with algebroid versions, but the 'certain conditions' for the bijection stay unspecified even in the theorem statements.

read the letter

The new material is the definition of right-invariant Poisson-Nijenhuis structures on groupoids together with their infinitesimal (Poisson bivector, Nijenhuis operator) versions on the algebroid. The paper then asserts a bijection between these two sides under certain conditions and supplies a few examples to illustrate the objects. That is the concrete contribution; the rest is standard background on Lie groupoids and algebroids. The examples are useful for seeing what the structures look like in low-dimensional cases. The correspondence itself is the part that needs scrutiny. The abstract and the stress-test note both flag that the conditions are invoked repeatedly but never listed as an explicit set of hypotheses. Without that list it is impossible to tell whether the bijection requires source-simply-connectedness, compatibility of the Nijenhuis operator with the groupoid multiplication, or other restrictions that are only used implicitly in the proofs. If those conditions turn out to be narrow or hard to check, the claimed one-to-one statement loses practical value. The paper is written for people already working inside Poisson geometry on groupoids and algebroids. A reader in that niche can extract the new definitions and the examples without much trouble, but anyone hoping to apply the correspondence will have to reconstruct the missing hypotheses from the proofs. The work shows coherent engagement with the literature and does not contain obvious internal contradictions, so it is worth sending to referees. They should be asked to verify whether the conditions can be stated cleanly and whether the bijection holds exactly when those conditions are met.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces right-invariant Poisson-Nijenhuis structures on Lie groupoids together with their infinitesimal counterparts, termed (Poisson bivector, Nijenhuis operator) structures on the corresponding Lie algebroids. It asserts a one-to-one correspondence between these structures under certain conditions and supplies illustrative examples.

Significance. A rigorously established bijection with explicitly enumerated hypotheses would furnish a concrete link between integrable Poisson-Nijenhuis data at the groupoid and algebroid levels, potentially aiding classification results in Poisson geometry. The right-invariance condition is a natural restriction that could simplify integrability questions, but the present lack of explicit conditions prevents assessment of the result's scope or novelty relative to existing integrability theorems for Lie algebroids.

major comments (1)

[Main correspondence theorem] Main correspondence result (invoked in the abstract and stated as the central theorem): the bijection is asserted only 'under certain conditions,' yet no explicit, checkable list of hypotheses is supplied. Implicit requirements (source-simply-connectedness, compatibility of the Nijenhuis operator with groupoid multiplication) appear in the proofs but are never isolated or shown to be minimal, rendering the precise domain of the claimed one-to-one correspondence unverifiable.

minor comments (1)

[Abstract] Abstract: 'certian' is a typographical error for 'certain'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the main correspondence theorem. We address the point below and will revise the manuscript to make the hypotheses explicit.

read point-by-point responses

Referee: [Main correspondence theorem] Main correspondence result (invoked in the abstract and stated as the central theorem): the bijection is asserted only 'under certain conditions,' yet no explicit, checkable list of hypotheses is supplied. Implicit requirements (source-simply-connectedness, compatibility of the Nijenhuis operator with groupoid multiplication) appear in the proofs but are never isolated or shown to be minimal, rendering the precise domain of the claimed one-to-one correspondence unverifiable.

Authors: We agree that the conditions should be isolated and stated explicitly in the theorem statement rather than left implicit. The proof of the correspondence uses source-simply-connectedness of the groupoid together with compatibility of the Nijenhuis operator with the groupoid multiplication; these will be listed as numbered hypotheses in the revised statement of the main theorem. We will also add a short remark on whether the listed conditions appear minimal on the basis of the proof technique. revision: yes

Circularity Check

0 steps flagged

No significant circularity; correspondence asserted without reduction to inputs by construction.

full rationale

The paper introduces right-invariant Poisson-Nijenhuis structures on Lie groupoids and their algebroid counterparts, then asserts a one-to-one correspondence under unspecified 'certain conditions.' No quoted equations, self-citations, fitted parameters, or ansatzes reduce the claimed bijection to its own inputs by definition. The derivation chain consists of definitions followed by a stated theorem whose validity rests on external proof steps rather than self-referential renaming or load-bearing self-citation. This is the normal case of a non-circular mathematical correspondence result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The abstract invokes standard background on Lie groupoids and algebroids plus the new right-invariant structures; no free parameters or explicit axioms are listed.

axioms (1)

standard math Lie groupoids and Lie algebroids satisfy their standard axioms from differential geometry
Implicitly assumed as the setting for the new structures

invented entities (1)

Right-invariant Poisson-Nijenhuis structure on a Lie groupoid no independent evidence
purpose: Extend Poisson-Nijenhuis geometry to the groupoid setting while preserving right invariance
Defined in the paper as the central new object

pith-pipeline@v0.9.0 · 5595 in / 1320 out tokens · 33086 ms · 2026-05-23T17:39:04.080868+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Bursztyn, H. and Cabrera, A. (2012). Multiplicative for ms at the inﬁnitesimal level, Mathematische Annalen volume 353, 663–705

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Drummond T. (2022). Lie–Nijenhuis Bialgebroids, Quart erly Journal of Mathematics 73 (3), 849-883

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and Ayoubi, R

Haghighatdoost, Gh. and Ayoubi, R. (2021). Hamiltonian system on co-adjoint Lie groupoids. Journal of Lie Theory, 31 (2), 493-516

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and Rezaei-Aghdam, A ., (2019)

Haghighatdoost, Gh., Ravanpak, Z. and Rezaei-Aghdam, A ., (2019). Some remarks on invariant Poisson quasi- Nijenhuis structures on Lie groups, International Journal of Geometric Methods in Modern Physics, Vol. 16, 1950097 (28 pages), @World Scientiﬁc Publishing Company, DOI:10.1 142/S021988781950097X

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Mackenzie, K. C. H. and Xu, P. (1994). Lie bialgebroids and Poisson groupoids . Duke Math. J. 73 (2), 415-452. 10

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Mackenzie, K. C. H. and Xu, P. (2000). Integration of Lie bialgebroids . Topology 39 (3), 445-467

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Laurent-Gengoux C., Xu, P.(2012)

Ponte, D.I. Laurent-Gengoux C., Xu, P.(2012). Univers al lifting theorem and quasi-Poisson groupoids, J. Eur. Mat h. Soc. 14 (2012), no. 3, pp. 681–731

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and Haghighatdoost, G

Ravanpak, Z., Rezaei-Aghdam, A. and Haghighatdoost, G . (2018). Invariant Poisson-Nijenhuis structures on Lie groups and classiﬁcation. International Journal of Geomet ric Methods in Modern Physics, 15 (4), id. 1850059-17. 11

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Leaves of Stacky Lie algebroids, Comp tes Rendus Math´ ematique 358(2):217-226

Alvarez D.(2020). Leaves of Stacky Lie algebroids, Comp tes Rendus Math´ ematique 358(2):217-226

work page 2020

[2] [2]

and Drummond, T.(2019)

Bursztyn, H. and Drummond, T.(2019). Lie theory of multi plicative tensors, Mathematische Annalen volume 375,1489–1554

work page 2019

[3] [3]

and Cabrera, A

Bursztyn, H. and Cabrera, A. (2012). Multiplicative for ms at the inﬁnitesimal level, Mathematische Annalen volume 353, 663–705

work page 2012

[4] [4]

Das, A. (2019). Poisson-Nijenhuis groupoids. Reports o n Mathematical Physics, Elsevier, 84, (3), 303-331

work page 2019

[5] [5]

Drummond T. (2022). Lie–Nijenhuis Bialgebroids, Quart erly Journal of Mathematics 73 (3), 849-883

work page 2022

[6] [6]

and Ayoubi, R

Haghighatdoost, Gh. and Ayoubi, R. (2021). Hamiltonian system on co-adjoint Lie groupoids. Journal of Lie Theory, 31 (2), 493-516

work page 2021

[7] [7]

and Rezaei-Aghdam, A ., (2019)

Haghighatdoost, Gh., Ravanpak, Z. and Rezaei-Aghdam, A ., (2019). Some remarks on invariant Poisson quasi- Nijenhuis structures on Lie groups, International Journal of Geometric Methods in Modern Physics, Vol. 16, 1950097 (28 pages), @World Scientiﬁc Publishing Company, DOI:10.1 142/S021988781950097X

work page 2019

[8] [8]

Poisson Lie gro ups, Dressing Transformations, and Bruhat Decomposi- tions, J

Jiang-Hua Lu and Alan Weinstein, (1990). Poisson Lie gro ups, Dressing Transformations, and Bruhat Decomposi- tions, J. Deﬀerential Geometry, 31(1990) 501-526

work page 1990

[9] [9]

Kosmann-Schwarzbach, Y. (2018). Multiplicative geome tric structures on Lie groupoids, Geometric Structures Lab- oratory, ﬁelds Institute, Canada, 50-52

work page 2018

[10] [10]

Kosmann-Schwarzbach,Y. (2016). Multiplicativity, f rom Lie groups togeneralized geometry, Geometry of jets and ﬁelds, in honour of Professor Janusz Grabowski , Banach Center Publ., Warsaw, 131–166

work page 2016

[11] [11]

(1990) Poisson-N ijenhuis structures, Ann

Kosmann-Schwarzbach, Y and Magri,F. (1990) Poisson-N ijenhuis structures, Ann. Inst. Henri Poincare, A Phys. Theor. 53 35-81

work page 1990

[12] [12]

Magri, F., Morosi, C. (1984). A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson–Nijenhuis manifolds . Roma: Quaderni di Matematica Press, University of Milan

work page 1984

[13] [13]

Mackenzie, K. C. H. (2005). General theory of Lie groupoids Lie algebroids . London Mathematical Society Lecture Note (213), Cambridge: Cambridge University Press

work page 2005

[14] [14]

Mackenzie, K. C. H. and Xu, P. (1994). Lie bialgebroids and Poisson groupoids . Duke Math. J. 73 (2), 415-452. 10

work page 1994

[15] [15]

Mackenzie, K. C. H. and Xu, P. (2000). Integration of Lie bialgebroids . Topology 39 (3), 445-467

work page 2000

[16] [16]

Laurent-Gengoux C., Xu, P.(2012)

Ponte, D.I. Laurent-Gengoux C., Xu, P.(2012). Univers al lifting theorem and quasi-Poisson groupoids, J. Eur. Mat h. Soc. 14 (2012), no. 3, pp. 681–731

work page 2012

[17] [17]

and Haghighatdoost, G

Ravanpak, Z., Rezaei-Aghdam, A. and Haghighatdoost, G . (2018). Invariant Poisson-Nijenhuis structures on Lie groups and classiﬁcation. International Journal of Geomet ric Methods in Modern Physics, 15 (4), id. 1850059-17. 11

work page 2018