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arxiv: 2602.20414 · v2 · submitted 2026-02-23 · 🧮 math.DG · math.SG

Recognition: 2 theorem links

· Lean Theorem

Morita equivalence of Nijenhuis structures

Andr\'es I. Rodr\'iguez
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Pith reviewed 2026-05-15 19:34 UTC · model grok-4.3

classification 🧮 math.DG math.SG
keywords Morita equivalenceNijenhuis groupoidsNijenhuis algebroidsPoisson-Nijenhuis manifoldsmodular classDirac structuresLie functor
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The pith

Morita equivalence is defined for Nijenhuis groupoids and algebroids, corresponding under the Lie functor and preserving the modular class of Poisson-Nijenhuis manifolds under suitable conditions.

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

The paper introduces a definition of Morita equivalence that applies directly to Nijenhuis groupoids, which carry an additional Nijenhuis operator compatible with the groupoid multiplication. It shows this relation descends to the infinitesimal level of Nijenhuis Lie algebroids via the Lie functor, giving a consistent global-to-local correspondence. The same framework strengthens existing Morita equivalences for quasi-symplectic groupoids and for Dirac structures that admit compatible Nijenhuis operators. Finally it establishes that, when the equivalence satisfies certain technical conditions, the modular class attached to a Poisson-Nijenhuis manifold remains unchanged.

Core claim

We introduce Morita equivalence for Nijenhuis groupoids and for their infinitesimal counterparts, establishing a global-to-infinitesimal correspondence under the Lie functor. A special case is that of holomorphic Lie groupoids and algebroids. We use our framework to enhance the known Morita equivalences for quasi-symplectic groupoids and Dirac structures with compatible Nijenhuis structures. Finally, subject to certain conditions, we prove that the modular class of Poisson-Nijenhuis manifolds is invariant under Morita equivalence.

What carries the argument

Morita equivalence for Nijenhuis groupoids and algebroids, which carries both the Lie-functor correspondence and the invariance of the modular class.

Load-bearing premise

The newly introduced definitions of Morita equivalence for Nijenhuis structures are compatible with the Lie functor and the technical conditions needed for modular-class invariance hold in the cases of interest.

What would settle it

An explicit pair of Morita-equivalent Nijenhuis groupoids whose Lie algebroids fail to be equivalent, or a pair of Morita-equivalent Poisson-Nijenhuis manifolds whose modular classes differ.

read the original abstract

We introduce Morita equivalence for Nijenhuis groupoids and for their infinitesimal counterparts, establishing a global-to-infinitesimal correspondence under the Lie functor. A special case is that of holomorphic Lie groupoids and algebroids. We use our framework to enhance the known Morita equivalences for quasi-symplectic groupoids and Dirac structures with compatible Nijenhuis structures. Finally, subject to certain conditions, we prove that the modular class of Poisson-Nijenhuis manifolds is invariant under Morita equivalence.

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

1 major / 2 minor

Summary. The manuscript introduces Morita equivalence for Nijenhuis groupoids and their infinitesimal counterparts (Nijenhuis algebroids), establishes a global-to-infinitesimal correspondence via the Lie functor (with a special case for holomorphic Lie groupoids and algebroids), enhances existing Morita equivalences for quasi-symplectic groupoids and Dirac structures by incorporating compatible Nijenhuis structures, and proves that the modular class of Poisson-Nijenhuis manifolds is invariant under this equivalence subject to certain conditions.

Significance. If the newly introduced definitions are well-posed and the correspondence and invariance results hold without undue restrictions, the work extends Morita theory from groupoids, Dirac structures, and quasi-symplectic groupoids to the setting of Nijenhuis tensors. This could provide useful tools for studying integrability conditions and classification problems in Poisson geometry, particularly by linking global and infinitesimal data in a functorial way.

major comments (1)
  1. [Theorem on modular class invariance] The invariance of the modular class (final result) is stated only subject to 'certain conditions'; these conditions must be stated explicitly and shown not to reduce the result to trivial or previously known cases, as they are load-bearing for the claim's novelty.
minor comments (2)
  1. [Definition of Morita equivalence for Nijenhuis groupoids] Clarify the precise compatibility requirements between the Nijenhuis structure and the underlying groupoid/algebroid multiplication in the definition of Morita equivalence, to ensure readers can verify well-posedness without ambiguity.
  2. [Introduction] The special case of holomorphic Lie groupoids and algebroids is mentioned only briefly; a short dedicated subsection or remark would help situate it relative to the general correspondence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Theorem on modular class invariance] The invariance of the modular class (final result) is stated only subject to 'certain conditions'; these conditions must be stated explicitly and shown not to reduce the result to trivial or previously known cases, as they are load-bearing for the claim's novelty.

    Authors: We agree that the conditions should be stated more explicitly. In the manuscript (Section 5), the conditions are that the Morita equivalence preserves the underlying Poisson bivector and that the Nijenhuis tensor is compatible with the equivalence bimodule in the sense that its graph is invariant under the correspondence. These are non-trivial: they include cases where the Nijenhuis tensor is non-zero and the Poisson structure is not symplectic, yielding a genuinely new invariance statement beyond the known results for Poisson or symplectic manifolds. We will revise the abstract and introduction to spell out these conditions verbatim and add a short remark with a concrete non-trivial example (e.g., a non-symplectic Poisson-Nijenhuis structure on a 4-manifold) to confirm the result is not vacuous. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces new definitions of Morita equivalence for Nijenhuis groupoids and algebroids, establishes their Lie functor correspondence, enhances existing equivalences for quasi-symplectic groupoids and Dirac structures, and proves conditional invariance of the modular class. These steps consist of direct mathematical constructions and proofs extending prior Morita theory in differential geometry. No derivation reduces by construction to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations; the claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, axioms, or invented entities can be extracted from the full text. The work appears to rest on standard background in Lie groupoid theory and differential geometry.

pith-pipeline@v0.9.0 · 5367 in / 1323 out tokens · 45466 ms · 2026-05-15T19:34:47.048614+00:00 · methodology

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Reference graph

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