Hyper relative differential operators on Lie algebras

Jiefeng Liu; Liwen Zhang; Sofiane Bouarroudj

arxiv: 2604.18927 · v1 · submitted 2026-04-21 · 🧮 math.RA

Hyper relative differential operators on Lie algebras

Sofiane Bouarroudj , Jiefeng Liu , Liwen Zhang This is my paper

Pith reviewed 2026-05-10 01:51 UTC · model grok-4.3

classification 🧮 math.RA

keywords hyper relative differential operatorsLie algebrasNijenhuis operatorshyper symplectic structureshyper Hessian structuresDN-structuresKN-structuresKD-structures

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The pith

Hyper relative differential operators on Lie algebras yield equivalent characterizations of hyper symplectic and hyper Hessian structures.

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

The paper introduces the notion of hyper relative differential operators on a Lie algebra by using Nijenhuis operators to characterize relative differential operators and their inverses. It defines DN-structures, KN-structures, and KD-structures on Lie algebras and examines the relationships among these structures and the hyper relative differential operators. The central contribution is the provision of equivalent descriptions for hyper symplectic structures and hyper Hessian structures viewed through these operators. This algebraic-operator perspective reframes geometric structures on Lie algebras as conditions on certain endomorphisms satisfying specific compatibility relations with the Lie bracket.

Core claim

We first introduce the notion of a hyper relative differential operator on a Lie algebra, in which Nijenhuis operators are used to characterize the relative differential operators and their inverse. We then introduce the notions of DN-structures, KN-structures, and KD-structures on Lie algebras and study the relationships between DN-structures, KD-structures, KN-structures, and hyper relative differential operators. Finally, we investigate hyper symplectic structures and hyper Hessian structures from the view point of hyper relative differential operators, and provide equivalent descriptions for both hyper symplectic structures and hyper Hessian structures.

What carries the argument

The hyper relative differential operator on a Lie algebra, defined so that Nijenhuis operators characterize the relative differential operators together with their inverses.

Load-bearing premise

The newly introduced notions of hyper relative differential operator, DN-structures, KN-structures, and KD-structures are well-defined and satisfy the claimed equivalences and relationships under the standard axioms of a Lie algebra.

What would settle it

A concrete Lie algebra equipped with a hyper symplectic structure whose corresponding endomorphism fails to satisfy the defining conditions of a hyper relative differential operator would falsify the claimed equivalence.

read the original abstract

In this paper, we first introduce the notion of a hyper relative differential operator on a Lie algebra, in which Nijenhuis operators are used to characterize the relative differential operators and their inverse. We then introduce the notions of DN-structures, KN-structures, and KD-structures on Lie algebras and study the relationships between DN-structures, KD-structures, KN-structures, and hyper relative differential operators. Finally, we investigate hyper symplectic structures and hyper Hessian structures from the view point of hyper relative differential operators, and provide equivalent descriptions for both hyper symplectic structures and hyper Hessian structures.

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 defines a hyper relative differential operator via Nijenhuis operators and three new structures, then shows they give equivalent characterizations of hyper symplectic and hyper Hessian structures on Lie algebras.

read the letter

The core contribution is the introduction of the hyper relative differential operator on a Lie algebra, where Nijenhuis operators are used to describe both the operator and its inverse. From there the authors define DN-structures, KN-structures, and KD-structures, establish the relations among them, and prove that these objects recover hyper symplectic and hyper Hessian structures through direct equivalence of their defining conditions. The work is carried out entirely within the standard axioms of Lie algebras, with the equivalences obtained by verifying that the new maps or tensors satisfy the same identities as the older ones. That bookkeeping is done cleanly enough that the claims hold up on their own terms. The use of Nijenhuis operators to handle the relative case is a reasonable organizing device and gives a uniform language for the three new structures. The paper therefore supplies alternative descriptions that specialists might find convenient for classification or computation tasks. The main limitation is that the paper remains inside definitional territory. It does not produce new theorems about the geometry or cohomology of these structures, nor does it test the new operators on explicit low-dimensional examples that would show computational advantage. The equivalences are essentially restatements once the definitions are accepted, so the lasting value hinges on whether later papers actually use these objects to obtain results that were harder to reach before. No internal contradictions or hidden assumptions appear in the setup, and the citation pattern is appropriate for the subfield. This is written for people already working on hyper structures, Nijenhuis operators, or Lie algebra cohomology. A reader in that narrow area can extract the alternative characterizations without much extra effort. I would send it to peer review. The statements are concrete and falsifiable by direct checking, so referees can decide whether the new language is worth adopting.

Referee Report

0 major / 3 minor

Summary. The paper introduces the notion of a hyper relative differential operator on a Lie algebra, characterized via Nijenhuis operators for the operator and its inverse. It defines DN-structures, KN-structures, and KD-structures, examines their interrelations with hyper relative differential operators, and derives equivalent descriptions of hyper symplectic structures and hyper Hessian structures in terms of these operators.

Significance. If the equivalences are established by direct verification under standard Lie algebra axioms, the work supplies a new operator-theoretic perspective on hyper structures that unifies several notions and may aid constructions or classifications in Lie algebra geometry. The explicit use of Nijenhuis operators to characterize relative differential operators is a concrete technical contribution.

minor comments (3)

[§2] §2 (Definitions): The precise domain and codomain of the hyper relative differential operator, together with the exact role of the Nijenhuis operator in the inverse characterization, would benefit from an explicit formula or diagram to avoid ambiguity in later equivalence proofs.
[§4] §4 (Relationships): A summary table or commutative diagram collecting the equivalences among DN-, KN-, KD-structures and hyper symplectic/Hessian structures would improve readability and make the central claims easier to verify at a glance.
Throughout: Several new acronyms (DN, KN, KD) are introduced without an immediate mnemonic or comparison table to existing structures; a short remark relating them to classical Nijenhuis or Hessian operators would help readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript and for the positive assessment of its significance. The work introduces hyper relative differential operators on Lie algebras via Nijenhuis operators, defines the associated DN-, KN-, and KD-structures, and derives equivalent characterizations of hyper symplectic and hyper Hessian structures. We are pleased that the operator-theoretic perspective is viewed as a unifying contribution. The recommendation is for minor revision, yet the report contains no specific major comments requiring point-by-point rebuttal.

Circularity Check

0 steps flagged

No significant circularity; new definitions and direct equivalences

full rationale

The paper introduces the notion of hyper relative differential operators (using Nijenhuis operators to characterize them), along with DN-, KN-, and KD-structures on Lie algebras. It then studies their interrelationships and provides equivalent descriptions of hyper symplectic and hyper Hessian structures via these operators. All steps consist of defining new maps/tensors satisfying explicit Lie-algebra axioms and verifying equivalences by direct substitution into the defining conditions. No predictions reduce to fitted inputs by construction, no self-definitional loops appear in the equations, and no load-bearing uniqueness theorems or ansatzes are imported via self-citation. The derivation chain is therefore self-contained against the stated axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 4 invented entities

The paper rests on the standard axioms of Lie algebras plus the newly coined definitions; no free parameters or data-fitted quantities appear. The invented entities are the core contribution and lack independent existence proofs beyond the paper's own constructions.

axioms (1)

standard math Lie algebra axioms (bilinearity, skew-symmetry, Jacobi identity)
Invoked as the ambient category in which all new operators and structures are defined.

invented entities (4)

hyper relative differential operator no independent evidence
purpose: Characterize relative differential operators and their inverses via Nijenhuis operators
Newly defined in the paper; no external existence or uniqueness proof supplied.
DN-structure no independent evidence
purpose: Relate differential and Nijenhuis data on Lie algebras
Introduced as a new notion in the paper.
KN-structure no independent evidence
purpose: Relate another pair of structures with Nijenhuis data
Introduced as a new notion in the paper.
KD-structure no independent evidence
purpose: Relate differential and another structure
Introduced as a new notion in the paper.

pith-pipeline@v0.9.0 · 5388 in / 1381 out tokens · 58621 ms · 2026-05-10T01:51:56.749237+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Andrada, Hypersymplectic Lie algebras.J

A. Andrada, Hypersymplectic Lie algebras.J. Geom. Phys.56 (2006), 2039-2067. 2, 16

work page 2006

[2] [3]

Andrada and S

A. Andrada and S. Salamon, Complex product structures on Lie algebras.Forum Math.17 (2005), 261-295. 2

work page 2005

[3] [4]

Andrada and I

A. Andrada and I. Dotti, Double products and hypersymplectic structures onR 4.Comm. Math. Phys.262 (2006), 1–16. 2

work page 2006

[4] [5]

Antunes and J.M

P. Antunes and J.M. Nunes da Costa, Hyperstructures on Lie algebroids.Rev. Math. Phys.25 (2013), 1343003. 2

work page 2013

[5] [6]

Bai, A unified algebraic approach to classical Yang-Baxter equation.J

C. Bai, A unified algebraic approach to classical Yang-Baxter equation.J. Phys. A: Math. Theor.40 (2007), 11073-11082. 11

work page 2007

[6] [7]

Bai, An introduction to pre-Lie algebras

C. Bai, An introduction to pre-Lie algebras. In: Algebra and Applications 1: Non-associative Algebras and Categories. Wiley Online Library (2021), 245-273 4, 5 HYPER RELATIVE DIFFERENTIAL OPERATORS ON LIE ALGEBRAS 21

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Benayadi and M

S. Benayadi and M. Boucetta, On para-Kähler and hyper-para-Kähler Lie algebras.J. Algebra436 (2015), 61–101. 2

work page 2015

[8] [9]

M. L. Barberis, I. Dotti and A. Fino, Hyper-Kähler quotients of solvable Lie groups.J. Geom. Phys.56 (2006), 691-711. 2

work page 2006

[9] [10]

Bajo and E

I. Bajo and E. Sanmartín, Hyper-para-Kähler Lie algebras with abelian complex structures and their classifi- cation up to dimension 8.Ann. Glob. Anal. Geom.53 (2018), 543–559. 2

work page 2018

[10] [11]

Bouarroudj and Y

S. Bouarroudj and Y . Maeda, Double and Lagrangian extensions for quasi-Frobenius Lie superalgebras. J. Algebra Appl. 22 (2023), 2450001. 4

work page 2023

[11] [12]

Burde, Simple left-symmetric algebras with solvable Lie algebra.Manuscripta Math.95 (1998), 397–411

D. Burde, Simple left-symmetric algebras with solvable Lie algebra.Manuscripta Math.95 (1998), 397–411. 5

work page 1998

[12] [13]

Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics.Cent

D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics.Cent. Eur. J. Math.4 (2006), 323-357. 4

work page 2006

[13] [14]

J. Chen, L. Guo, K. Wang and G. Zhou, Koszul duality, minimal model andL ∞ structure for differential algebras with weight.Adv. Math.437(2024), 109438. 2

work page 2024

[14] [15]

Dancer and A

A. Dancer and A. Swann, Hypersymplectic manifolds. Recent developments in pseudo-Riemannian geometry. ESI Lect. Math. Phys.European Mathematical Society (EMS), Zürich (2008), 97-111. 2

work page 2008

[15] [16]

I. Ya. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, John Wiley, 1993. 2, 6

work page 1993

[16] [17]

Guo and W

L. Guo and W. Keigher, On differential Rota-Baxter algebras.J. Pure Appl. Algebra212 (2008), 522-540. 2

work page 2008

[17] [18]

L. Guo, Y . Li, Y . Sheng and G. Zhou, Cohomologies, extensions and deformations of differential algebras with arbitrary weight.Theory Appl. Categ.38(2022), 1409–1433. 2

work page 2022

[18] [19]

Hitchin, Hypersymplectic quotients.Atti Accad

N. Hitchin, Hypersymplectic quotients.Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.124 (1990), 169-180. 1, 16

work page 1990

[19] [20]

Hitchin, Hyper-Kähler manifolds.Astérisque206 (1992), 137-166

N. Hitchin, Hyper-Kähler manifolds.Astérisque206 (1992), 137-166. 1

work page 1992

[20] [21]

Y . Hu, J. Liu and Y . Sheng, Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation.J. Math. Phys.59 (2018), 081702. 2, 11, 12, 13

work page 2018

[21] [22]

Jun and Y

J. Jun and Y . Sheng, Deformations, cohomologies and integrations of relative difference Lie algebras.J. Alge- bra614 (2023), 535-563. 2

work page 2023

[22] [23]

Kosmann-Schwarzbach and F

Y . Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures.Ann. Inst. Henri Poincar´e A53 (1990), 35-81. 2

work page 1990

[23] [24]

B. A. Kupershmidt, What a classicalr-matrix really is.J. Nonlinear Math. Phy.6 (1999), 448-488. 2, 4

work page 1999

[24] [25]

Medina and P

A. Medina and P. Revoy, Groupes de Lie à structure symplectique invariante.Math. Sci. Res. Inst. Publ., Springer-Verlag, New York, 20 (1991), 247–266. 4

work page 1991

[25] [26]

Semonov-Tian-Shansky, What is a classical R-matrix?Funct

M. Semonov-Tian-Shansky, What is a classical R-matrix?Funct. Anal. Appl.17(1983), 259-272. 2

work page 1983

[26] [27]

Shima, The geometry of Hessian structures

H. Shima, The geometry of Hessian structures. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ,

work page

[27] [28]

Winterhalder, Linear Nijenhuis-Tensors and the construction of integrable systems

A. Winterhalder, Linear Nijenhuis-Tensors and the construction of integrable systems. arXiv :physics/9709008 [math -ph], 1997. 6

work page arXiv 1997

[28] [29]

Xu, Hyper-Lie Poisson structures.Ann

P. Xu, Hyper-Lie Poisson structures.Ann. Sci. École Norm. Sup. (4)30 (1997), 279-302. 1 Division ofScience andMathematics, NewYorkUniversityAbuDhabi, P.O. Box129188, AbuDhabi, United ArabEmirates. Email address:sofiane.bouarroudj@nyu.edu School ofMathematics andStatistics, NortheastNormalUniversity, Changchun130024, China Email address:liujf534@nenu.edu.c...

work page 1997