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arxiv: 2606.19955 · v1 · pith:4LNDW2HJnew · submitted 2026-06-18 · 🧮 math.RA · math.CT· math.RT

Nijenhuis Lie 2-algebras

Apurba Das This is my paper

Pith reviewed 2026-06-26 15:08 UTC · model grok-4.3

classification 🧮 math.RA math.CTmath.RT
keywords Nijenhuis Lie 2-algebras2-term L_infinity-algebras2-representationsrepresentations up to homotopysemidirect productscategory equivalencesLie algebras
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The pith

The category of Nijenhuis Lie 2-algebras is equivalent to the category of 2-term Nijenhuis L_∞-algebras.

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

The paper defines Nijenhuis Lie 2-algebras as the categorification of Nijenhuis Lie algebras. It proves that this category is equivalent to the category of 2-term Nijenhuis L_∞-algebras. For a fixed Nijenhuis Lie algebra, it introduces 2-representations whose semidirect products inherit Nijenhuis Lie 2-algebra structure. It also constructs 2-term Nijenhuis L_∞-algebras from 2-term representations up to homotopy via semidirect product. Finally it shows the two categories of representations are equivalent to each other.

Core claim

We introduce Nijenhuis Lie 2-algebras and prove that their category is equivalent to the category of 2-term Nijenhuis L_∞-algebras. Given a Nijenhuis Lie algebra, we define 2-representations such that the semidirect product is a Nijenhuis Lie 2-algebra and 2-term representations up to homotopy such that the semidirect product is a 2-term Nijenhuis L_∞-algebra; the corresponding categories of these representations are equivalent.

What carries the argument

Nijenhuis Lie 2-algebra, defined via categorification of Nijenhuis Lie algebras, equipped with semidirect product functors that preserve the structure and establish the stated equivalences.

If this is right

  • The semidirect product of any Nijenhuis Lie algebra with one of its 2-representations carries a Nijenhuis Lie 2-algebra structure.
  • The semidirect product of any Nijenhuis Lie algebra with one of its 2-term representations up to homotopy carries a 2-term Nijenhuis L_∞-algebra structure.
  • Any property preserved by the equivalence functors can be transferred between strict 2-representations and representations up to homotopy.
  • Nijenhuis operators on ordinary Lie algebras extend canonically to operators on the associated 2-term structures.

Where Pith is reading between the lines

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

  • The equivalence supplies a single setting in which both strict and homotopy-coherent representations of Nijenhuis Lie algebras can be studied simultaneously.
  • Cohomology or deformation theories already defined on the L_∞ side become available, via the equivalence, for the 2-algebra side.
  • The construction suggests that higher-categorical versions of Nijenhuis operators may appear naturally when lifting ordinary Lie-algebraic structures to 2-term complexes.

Load-bearing premise

The newly introduced definitions of Nijenhuis Lie 2-algebra, 2-representation, and the semidirect product constructions satisfy all required axioms and coherence conditions so that the functors establishing the stated category equivalences are well-defined, essentially surjective, full, and faithful.

What would settle it

A specific Nijenhuis Lie algebra together with a 2-representation whose semidirect product fails to obey the Nijenhuis Lie 2-algebra axioms, or a pair of objects in the two representation categories that the constructed functors map to non-isomorphic objects.

read the original abstract

In this paper, we first introduce Nijenhuis Lie 2-algebras as the categorification of Nijenhuis Lie algebras. We prove that the category of Nijenhuis Lie 2-algebras is equivalent to the category of 2-term Nijenhuis $L_\infty$-algebras. Next, given a Nijenhuis Lie algebra, we introduce the notion of a 2-representation and show that the corresponding semidirect product inherits a Nijenhuis Lie 2-algebra structure. On the other hand, we consider a $2$-term representation up to homotopy of a Nijenhuis Lie algebra and obtain a $2$-term Nijenhuis $L_\infty$-algebra as the semidirect product. Finally, we show that the category of $2$-representations and the category of $2$-term representations up to homotopy of a Nijenhuis Lie algebra are equivalent.

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 / 3 minor

Summary. The paper introduces Nijenhuis Lie 2-algebras as the categorification of Nijenhuis Lie algebras. It proves that the category of Nijenhuis Lie 2-algebras is equivalent to the category of 2-term Nijenhuis L_∞-algebras. Given a Nijenhuis Lie algebra, it defines 2-representations such that the semidirect product inherits a Nijenhuis Lie 2-algebra structure, and likewise obtains a 2-term Nijenhuis L_∞-algebra from a 2-term representation up to homotopy; it then proves that the category of 2-representations is equivalent to the category of 2-term representations up to homotopy.

Significance. If the definitions and functors are well-defined, the two category equivalences supply a consistent higher-categorical framework for Nijenhuis structures, allowing properties to be transferred between the 2-algebra and L_∞ models. The semidirect-product constructions follow the standard pattern used for ordinary Lie 2-algebras and are presented as the main technical contribution.

minor comments (3)
  1. The precise axioms imposed on a Nijenhuis Lie 2-algebra (in particular the compatibility conditions between the Nijenhuis operator and the 2-algebra brackets) should be stated in a numbered definition early in the paper so that subsequent verifications can cite them directly.
  2. In the proof that the semidirect product of a 2-representation carries a Nijenhuis Lie 2-algebra structure, the verification that all higher coherence conditions hold should be expanded; currently the argument appears to rely on the corresponding properties of the underlying Nijenhuis Lie algebra without spelling out the 2-categorical lifts.
  3. Notation for the two distinct notions of “2-representation” (strict versus up to homotopy) should be made visually distinct (e.g., different fonts or subscripts) to prevent confusion when the two categories are compared.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments are provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines Nijenhuis Lie 2-algebras and 2-representations from standard Lie algebra and L∞ axioms via explicit semidirect-product constructions, then proves two category equivalences by verifying the functors are well-defined, essentially surjective, full, and faithful. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the derivations are self-contained against the stated axioms and coherence conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on the new definitions of the structures and the assumption that the category equivalences hold via explicitly constructed functors; no free parameters appear.

axioms (2)
  • standard math Standard axioms for Lie algebras, Lie 2-algebras, and L_∞-algebras
    The paper builds all new structures on these background results from the literature.
  • domain assumption The introduced definitions of Nijenhuis operator on 2-algebras and of 2-representations satisfy the necessary compatibility and coherence axioms
    These are the load-bearing definitions whose verification is required for the equivalences to hold.
invented entities (2)
  • Nijenhuis Lie 2-algebra no independent evidence
    purpose: Categorified version of Nijenhuis Lie algebra
    Newly defined object whose properties are proved in the paper.
  • 2-representation of a Nijenhuis Lie algebra no independent evidence
    purpose: To form semidirect products that inherit the Nijenhuis 2-algebra structure
    New notion introduced to enable the representation theory results.

pith-pipeline@v0.9.1-grok · 5679 in / 1517 out tokens · 35078 ms · 2026-06-26T15:08:06.289285+00:00 · methodology

discussion (0)

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Reference graph

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