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arxiv: 1907.05004 · v1 · pith:4R5QDSAQnew · submitted 2019-07-11 · 🧮 math.SG · math.DG

Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids and Hom-Dirac structures on Hom-Courant algebroids

Tomoya Nakamura This is my paper

Pith reviewed 2026-05-24 22:58 UTC · model grok-4.3

classification 🧮 math.SG math.DG
keywords Hom-Lie algebroidHom-Poisson-Nijenhuis structureHom-Dirac structureHom-Courant algebroidPoisson isomorphismMaurer-Cartan equation
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The pith

Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids admit hierarchies and Maurer-Cartan relations parallel to the classical case, together with a correspondence to Poisson isomorphisms.

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

The paper defines Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids and Hom-Dirac structures on Hom-Courant algebroids. It proves that these objects obey the same formal relations and construction rules as the ordinary Poisson-Nijenhuis and Dirac structures. In particular the definitions yield infinite hierarchies of compatible structures and link Hom-Dirac structures to Maurer-Cartan equations. A bijection is established between Poisson structures on the base manifold and Poisson isomorphisms of those structures. This extension keeps the algebraic machinery intact while moving into the Hom setting.

Core claim

We introduce the notions of Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis structures on a Hom-Lie algebroid and the notion of Hom-Dirac structures on a Hom-Courant algebroid. We show that these structures satisfy similar properties to structures non 'Hom-'version. For example, there exists the hierarchy of a Hom-Poisson-Nijenhuis structure and we have a relation between Hom-Dirac structures and Maurer-Cartan type equation. Moreover we show that there exists a one-to-one correspondence between the pairs consisting of a Poisson structure on M and a Poisson isomorphism for it.

What carries the argument

The Hom-Poisson-Nijenhuis structure consisting of a Hom-Poisson bivector and a compatible Hom-Nijenhuis endomorphism that generates further structures in a hierarchy.

Load-bearing premise

Foundational definitions and basic properties of Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids remain valid when Hom-Poisson-Nijenhuis and Hom-Dirac structures are imposed.

What would settle it

A specific Hom-Lie algebroid with a Hom-Poisson-Nijenhuis structure whose generated sequence fails to remain compatible would contradict the hierarchy claim.

read the original abstract

In this paper, we develop the theory of Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids introduced by Cai, Liu and Sheng. Specifically, we introduce the notions of Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis structures on a Hom-Lie algebroid and the notion of Hom-Dirac structures on a Hom-Courant algebroid. We show that these structures satisfy similar properties to structures non "Hom-"version. For example, there exists the hierarchy of a Hom-Poisson-Nijenhuis structure and we have a relation between Hom-Dirac structures and Maurer-Cartan type equation. Moreover we show that there exists a one-to-one correspondence between the pairs consisting of a Poisson structure on $M$ and a Poisson isomorphism for it.

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.

Referee Report

0 major / 1 minor

Summary. The paper develops the theory of Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids by defining Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids together with Hom-Dirac structures on Hom-Courant algebroids. It asserts that the new structures obey the same formal properties as their non-Hom analogues, including the existence of a hierarchy, a relation between Hom-Dirac structures and Maurer-Cartan equations, and a bijection between pairs consisting of a Poisson structure on the base manifold M and a Poisson isomorphism.

Significance. If the claimed analogies are verified by the proofs, the work supplies a systematic extension of the Cai-Liu-Sheng framework that preserves the algebraic and geometric relations central to Poisson and Dirac geometry. The explicit bijection result links the Hom-setting back to classical Poisson geometry in a concrete way and may be useful for deformation questions.

minor comments (1)
  1. [Abstract] Abstract: the phrasing of the final sentence on the one-to-one correspondence is imprecise; a clearer statement of the objects being placed in bijection would help readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; new structures and properties derived from independent prior axioms

full rationale

The paper introduces Hom-Poisson, Hom-Nijenhuis, Hom-Poisson-Nijenhuis and Hom-Dirac structures on top of the Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids previously defined by Cai, Liu and Sheng. It then proves that these new structures satisfy analogous properties (hierarchies, Maurer-Cartan relations, and a bijection between Poisson structures on M and Poisson isomorphisms). All derivations rest on the external axioms of the foundational Hom-structures; no equations reduce the claimed results to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The cited prior work is independent and supplies the non-circular base.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the prior definitions of Hom-Lie algebroids supplied by Cai et al.; no free parameters or new physical entities are introduced.

axioms (1)
  • domain assumption Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids as defined by Cai, Liu and Sheng satisfy the required bracket and anchor axioms.
    The paper states it develops the theory introduced by them and builds all new structures on top of those definitions.

pith-pipeline@v0.9.0 · 5690 in / 1392 out tokens · 46860 ms · 2026-05-24T22:58:25.540811+00:00 · methodology

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Reference graph

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