Deformations and homotopy theory of Nijenhuis associative algebras

Chao Song; Guodong Zhou; Kai Wang; Yuanyuan Zhang

REVIEW 4 major objections 5 minor 1 cited by

Deformations and homotopy theory of Nijenhuis associative algebras

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper constructs homotopy Nijenhuis associative algebras and proves that the differential graded operad governing them is the minimal model of the operad for ordinary Nijenhuis associative algebras.

desk verdict A real first: minimal model and L∞-deformation theory for Nijenhuis associative algebras, but the proof leans on two delegated computations that a referee should ask to see in full. read the letter →

arxiv 2412.17253 v1 pith:Y3DAVMUM submitted 2024-12-23 math.KT math.ATmath.RAmath.RT

Chao Song , Kai Wang , Yuanyuan Zhang , Guodong Zhou This is my paper · ORCID

classification math.KTmath.ATmath.RAmath.RT MSC 16E4016S8017B3818M6018M6518M70

keywords homotopyNijenhuisassociativealgebraminimalmodelKoszuldualcooperaddeformationcomplexcohomologyL∞-algebraoperad

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Nijenhuis associative algebras are associative algebras equipped with an endomorphism $P$ satisfying the Nijenhuis identity, a structure that appears in quantum bi-Hamiltonian systems and as the associative analogue of the classical Nijenhuis tensor. The paper's goal is to give such algebras a full deformation and homotopy theory. It constructs a differential graded operad $\mathrm{NjA}_\infty$ whose algebras are homotopy Nijenhuis associative algebras, and proves that $\mathrm{NjA}_\infty$ is the minimal model of the operad $\mathrm{NjA}$ of Nijenhuis associative algebras. From this it extracts an $L_\infty$-algebra whose Maurer-Cartan elements are exactly Nijenhuis associative algebra structures, and whose twisting recovers the cohomology and deformation complex defined in Section 2. Deformations of the associative product and of the Nijenhuis operator can therefore be studied simultaneously through a single controlling algebra.

What carries the argument

The load-bearing object is the Koszul dual homotopy cooperad $\mathrm{NjA}^{¡}$, the Hadamard product $S(\mathrm{NjA}^{¡}) \otimes_H S^{-1}$; its underlying generators are $e_n$ and $o_n$, with degrees $n-1$ and $n$, and its cooperadic structure is encoded by the type-(I) and type-(II) trees of Section 3. The differential graded operad $\mathrm{NjA}_\infty$ is the cobar construction $\Omega(\mathrm{NjA}^{¡})$, freely generated by $m_n = s^{-1} e_n$ and $P_n = s^{-1} o_n$, with differential given by Formulas (11)–(12). The differential on the $m_n$ is the standard A-infinity identity; the differential on the $P_n$ is the higher Nijenhuis relation. The proof of Theorem 4.4 introduces a monomial order on tree monomials, with weights $\phi(m_n)=n-1$ and $\phi(P_n)=2n-1$, so that the leading terms of $\partial(m_n)$ and $\partial(P_n)$ coincide with the leading terms in the Rota-Baxter case; the remaining cancellations are then shown to follow by the same argument as in that case. The convolution homotopy operad $\mathrm{Hom}(\mathrm{NjA}^{¡},\mathrm{End}(V))$ gives the controlling $L_\infty$-algebra $C_{\mathrm{NjA}}(V)$.

What would settle it

Work out $\partial^2=0$ directly on the generator $y_4$ using Formulas (9)–(10); the seven groups of terms labelled (I)–(VII) must cancel exactly, and any leftover monomial would destroy the cooperad structure and with it the minimal model. Alternatively, compute $H^1(\mathrm{NjA}_\infty,\partial)$ from Formulas (11)–(12): a nonzero class would contradict the quasi-isomorphism to $\mathrm{NjA}$.

Watch

Extended reading notes

Core claim

A Nijenhuis operator on an associative algebra $(A,m)$ is a linear endomorphism $P$ satisfying $m \circ (P \otimes P) = P \circ (m \circ (P \otimes \mathrm{Id}) + m \circ (\mathrm{Id} \otimes P) - P \circ m)$. The paper defines homotopy Nijenhuis associative algebras by replacing $m$ and $P$ with families of higher operations $m_n$ and $P_n$, obeying the A-infinity identity for the products and a higher Nijenhuis identity for the operators. The central structural result, Theorem 4.4, is that the differential graded operad $\mathrm{NjA}_\infty = \Omega(\mathrm{NjA}^{¡})$ is the minimal model of $\mathrm{NjA}$, so that homotopy Nijenhuis algebras are the up-to-homotopy version of Nijenhuis associative algebras. Proposition 5.4 then realizes a Nijenhuis associative algebra structure on a space $V$ as a Maurer-Cartan element in the $L_\infty$-algebra $C_{\mathrm{NjA}}(V)=\mathrm{Hom}(\mathrm{NjA}^{¡},\mathrm{End}(V))^{\prod}$, which in low degrees combines the classical bracket on associative-algebra cochains with higher brackets that mix cochains and Nijenhuis-operator cochains. Twisting this $L_\infty$-algebra by a Maurer-Cartan element reproduces, up to suspension, the cochain complex of Nijenhuis associative algebras defined in Section 2, justifying that cohomology as the deformation complex. The paper also shows that homotopy relative Rota-Baxter operators of weight $0$ on $A$ with respect to a bimodule $M$ correspond exactly to homotopy Nijenhuis associative algebra structures on the semi-direct product $A \oplus M$.

Load-bearing premise

The proof rests on a lengthy sign-cancellation computation that is only summarized as a table of seven term groups, and on the assumption that an argument from the Rota-Baxter case transfers unchanged once the leading monomials are matched; if either point fails, the minimal model theorem and the $L_\infty$-algebra built from it collapse.

Editorial extensions

If this is right

Every Nijenhuis associative algebra carries a cochain complex, built as a mapping cone of a chain map from associative-algebra cochains to Nijenhuis-operator cochains, whose cohomology controls simultaneous deformations of the product and the Nijenhuis operator.
Maurer-Cartan elements of the controlling $L_\infty$-algebra are in bijection with Nijenhuis associative algebra structures, placing these deformation problems in the standard Maurer-Cartan framework.
Homotopy Nijenhuis associative algebras reduce on homology to ordinary Nijenhuis associative algebras: $m_1$ is a differential, $m_2$ induces an associative product, and $P_1$ induces a Nijenhuis operator on homology, with $P_2$ measuring the homotopy failure of the Nijenhuis relation.
Weight-zero relative Rota-Baxter operators, including their homotopy versions, fit into Nijenhuis theory: a homotopy relative Rota-Baxter operator on $(A,M)$ is equivalent to a homotopy Nijenhuis structure on the semi-direct product $A \oplus M$.
Because the Nijenhuis relation is cubic, the operad $\mathrm{NjA}$ is not Koszul; the paper thereby provides a non-Koszul example where a minimal model and a complete homotopy deformation theory are still obtained.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the minimal model is as small as claimed, the same monomial-order strategy should apply to Nijenhuis Lie algebras and to Nijenhuis operators on bimodules, since the proof only needs the leading terms to match; the announced sequel has a concrete template to test.
The $L_\infty$-algebra $C_{\mathrm{NjA}}(V)$ should support an explicit obstruction theory for extending formal deformations, with degree-2 classes of the twisted complex serving as obstructions; this is not spelled out in the paper.
Because the Nijenhuis-operator differential contains an extra term $-P_M \circ \delta$ beyond the ordinary associative-algebra differential, computing explicit examples of the long exact cohomology sequence would clarify what that extra term measures empirically.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing you should know: this is a genuine extension of the Wang–Zhou Rota-Baxter program to Nijenhuis associative algebras, and the main constructions—the homotopy cooperad NjA¡, the cobar minimal model NjA∞, and the controlling L∞-algebra—are new. The paper is the first operadic treatment of Nijenhuis associative algebras, and since the operad is not Koszul (Remark 1.7 is explicit about this), the minimal model is not coming from standard Koszul duality. That makes it a meaningful test case for the non-Koszul technology.

What the paper does well: it gives a clean cochain complex via a mapping cone in Section 2, then later recovers that complex from the twisted L∞-algebra in Proposition 5.5. That is a satisfying way to justify the cohomology. The authors are also honest about what they are doing: Theorem 4.4 states that, once the leading terms match, the homotopy argument carries verbatim from [43], and Proposition 3.1 lists the cancellations I–VII rather than pretending every sign is displayed.

The soft spots are exactly the two the stress-test note names, and they are real. Proposition 3.1 needs ∂²=0 to hold for the homotopy cooperad structure; the proof asserts cancellations like (I)+(II)=0 without showing the sign conventions or the index bijections. Since a single sign error would destroy the dg structure, this is load-bearing. Similarly, Theorem 4.4 delegates the quasi-isomorphism to [43, Theorem 3.5] after saying the leading terms are the same. The leading terms are displayed, but the identification with the Rota-Baxter case is asserted, not demonstrated, and the induction in [43] is long. Self-citation is not the issue; the issue is that the argument is not reproduced or even sketched. These are verification gaps rather than demonstrated contradictions, and the structure is coherent, so I would bet the cancellations do work out. But "likely correct" is not the same as "proved."

One thing in the paper's favor: Proposition 2.6 is proved directly, so the cochain complex is well-defined without needing the L∞-algebra; Lemmas 2.2 and Proposition 2.4 are deferred but later justified. No circularity.

This paper is for people working on operadic deformation theory of operated algebras and possibly on Nijenhuis geometry via the announced sequel. It deserves a serious referee, not a desk reject, and I would take the referee report.

Recommendation: send to peer review. Ask the authors to expand Proposition 3.1 with the actual ∂² computation, and to write out the carry-over in Theorem 4.4 instead of one sentence. Those are fixable revisions, and the core novelty is worth the effort.

Referee Report

4 major / 5 minor

Summary. The paper develops an operadic deformation theory for Nijenhuis associative algebras. It defines a homotopy cooperad NjA¡ with generators u_n, v_n, constructs the cobar dg operad NjA∞ = Ω(NjA¡), and claims in Theorem 4.4 that NjA∞ is the minimal model of the operad NjA for Nijenhuis associative algebras. It then uses NjA¡ to produce an L∞-algebra CNjA(V) whose Maurer-Cartan elements are claimed to correspond to Nijenhuis associative structures (Proposition 5.4), and shows that twisting this L∞-algebra by a Maurer-Cartan element recovers the cochain complex defined in Section 2 (Proposition 5.5). The final section relates homotopy relative Rota-Baxter associative algebras of weight 0 to homotopy Nijenhuis associative algebras.

Significance. If the main theorems are fully established, this would be a substantial contribution: an explicit minimal model for a non-Koszul operad, a resulting controlling L∞-algebra for deformations, and a homotopy notion of Nijenhuis associative algebras. The paper is constructive: generators, differentials, monomial orders, and Maurer-Cartan equations are written out in detail, and the proposed relationship with homotopy relative Rota-Baxter algebras is concrete. However, the proof of the minimal-model statement relies on two delegated computations — the ∂²=0 verification in Proposition 3.1 and the 'carries verbatim' transfer of the Wang–Zhou argument in Theorem 4.4 — and these are precisely the load-bearing points. The current version is therefore conditional, but the gaps appear fillable.

major comments (4)

[Section 3, Proposition 3.1] The proof of ∂²=0 on the generators y_n is not complete. The expansion of ∂²(y_n) is grouped into seven families I–VII and the text asserts cancellations such as (I)+(II)=0, (III,i=1)+(VII,s≠0)=0, and (IV,i=t,t=p)+(VI)=0, but it does not display the sign factors or the index bijections that realize these cancellations. The formulas (9)–(10) and the definition of Δ_T(v_n) contain several nontrivial signs, including (−1)^t and (−1)^{p(p−1)/2}; a sign error in any of these operations would destroy ∂²=0 and hence the dg cooperad structure. Since NjA∞ is defined as Ω(NjA¡), this verification is essential for Theorem 4.4 and for every subsequent result built on the minimal model. Please supply the complete cancellation argument, or provide a machine-checkable verification of Proposition 3.1.
[Section 4.2, Theorem 4.4] The proof reduces the quasi-isomorphism statement to showing that the leading terms of ∂(m_n) and ∂(P_n) under the monomial order Ξ are respectively m_{n−1}m_2 and P_{n−1}m_2P_1, and then asserts that 'the remaining part of the proof carries verbatim as that of [43, Theorem 3.5]'. The leading-term comparison is not shown. In particular, ∂(P_n) in equation (12) is a sum over many partitions, grafting positions i_h, k_j, and signs α′; one must verify that every other summand is strictly smaller than P_{n−1}m_2P_1 in the order Ξ. Moreover, the phrase 'carries verbatim' is a substantive claim: the homotopy used in [43] must be compatible with the present differential, with the present filtration, and with the new monomial order. This requires proof rather than assertion, because the minimal model theorem and the L∞-algebra of Section 5 depend on it.
[Sections 2.2–2.4] Lemma 2.2 and Proposition 2.4 are not proved where they are introduced. The text states that they 'can be verified through direct inspection' but says the proofs will be deduced later from the L∞-algebra constructed in Section 5, with the general coefficient case following from Subsection 2.4. This makes Definition 2.5 — the deformation complex and cohomology of Nijenhuis associative algebras — logically dependent on Theorem 4.4 and Proposition 5.5. The dependency should be made explicit at the point of Definition 2.5, or the authors should provide direct proofs of Lemma 2.2 and Proposition 2.4 in Section 2. As written, a reader cannot check the cochain complex without first accepting the minimal-model construction.
[Section 5.2, Proposition 5.5] The proof that twisting CNjA(A) by the Maurer-Cartan element corresponding to (m,P) yields the complex sC*NjA(A) is central, but several identifications are merely asserted. For example, the text states that '−[ν,sf]_G corresponds to −(−1)^{n+1}δ^n_Alg(~sf) under the fixed isomorphism (19)' and that certain combinations 'correspond to Φ^n(~sf)' and 'correspond to (−1)^n δ^{n−1}_NjO(ĝ)' without showing the intermediate sign computations. Since the cochain complex of Section 2 was defined before the L∞-algebra, an independent and fully displayed sign verification is necessary; a sign error in these correspondences would change the stated cohomology theory. Please expand the computation of l^α_1 on both components.

minor comments (5)

[Throughout] The text contains many typographical artifacts, including 'n /greaterorequalslant0' in Section 2, 'Provinence' in the affiliation, and 'Iheoret. Phys.' in reference [24]; please proofread the manuscript carefully.
[Section 3, Proposition 3.1] The seven-group expansion of ∂²(y_n) is typeset with very large brace diagrams that are difficult to read. A structured cancellation table with explicit signs and index ranges for each group would greatly improve readability and verifiability.
[Section 4.2] In the proof of Theorem 4.4, the phrase 'It can be easily seen that the differential ∂ satisfies Conditions (i) and (ii) in Definition 4.2' is too terse. In particular, the filtration M(i) required by Condition (ii) is not specified; although a degree filtration is plausible, it should be stated explicitly.
[Section 6, Corollary 6.8] The corollary is described as a direct consequence of Theorem 6.7, but Theorem 6.7 gives a structure-wise construction on A⊕M for each pair (A,M); it does not by itself define a natural operad morphism from NjA∞ to the color-forgotten operad RBArel∞. Please clarify the intended naturality or state a weaker conclusion.
[Section 5.2] Proposition 5.4 is stated to follow from [5, Proposition 3.9]; this is plausible, but the paper should explicitly check that the Maurer-Cartan equation in CNjA(V) is equivalent to equations (17)–(18) with the stated signs, since this equivalence underpins the identification of homotopy Nijenhuis algebras with Maurer-Cartan elements.

Circularity Check

1 steps flagged · score 4.0 of 10

The minimal-model theorem delegates its quasi-isomorphism proof to the authors' own Rota-Baxter paper, but the rest of the derivation chain is self-contained and not a fitted-input prediction.

self citation load bearing [Theorem 4.4, proof (Section 4.2)]
"Once the leading terms are seen to be the same as the case of Rota-Baxter associative operad, the remaining part of the proof carries verbatim as that of [43, Theorem 3.5]."

Reference [43] is K. Wang and G. Zhou, two of the present authors, and its Theorem 3.5 is the minimal-model theorem for the Rota-Baxter associative operad. The present proof reduces the essential quasi-isomorphism verification, namely the vanishing of all positive-degree homology of NjA-infty, to an asserted 'verbatim' carry-over of that theorem. The paper neither displays the transferred homotopy nor verifies the sign and index bijections for the Nijenhuis cooperad; it only asserts that the leading terms agree. Thus the central claim of Theorem 4.4 is load-bearing on a self-citation whose applicability to the Nijenhuis case is asserted rather than demonstrated. This is a genuine reliance on prior work by the same authors, though the surrounding construction is largely self-contained.

full rationale

Most of the paper's derivation chain is not circular. The cohomology theory in Section 2 is defined explicitly as a mapping cone of an explicit chain map, and Section 5 later identifies the twisted L-infinity differential with that same complex, so the cohomology is not a fitted quantity renamed as a prediction. The homotopy cooperad in Section 3 is verified by an internal cancellation computation, and the Maurer-Cartan equivalence in Proposition 5.4 is an application of the general machinery of [5] rather than a restatement of the target result. The main circularity-adjacent point is Theorem 4.4, where the proof of quasi-isomorphism is delegated to the authors' previous Rota-Baxter minimal-model theorem [43] with the phrase 'carries verbatim.' Because [43] is by two of the present authors and concerns a different operad, this is a load-bearing self-citation rather than an independent verification. Still, the present paper supplies a new monomial order, identifies the leading terms, and constructs the full cooperad and L-infinity structure, so the central claim has substantial independent content. The score reflects one load-bearing self-citation in the minimal-model proof, not a wholesale reduction of the results to their inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The paper introduces new algebraic objects (the cooperad NjA¡, the operad NjA∞, the L∞-algebra CNjA(V)) but all are constructed and then verified by theorems. No free parameters are fitted. The main external assumptions are standard operad-theoretic background plus the self-cited result [43].

assumptions (5)

standard math Field of characteristic 0; all vector spaces over k
Assumed throughout the paper (page 3). Used for operadic and L∞-algebra methods.
standard math The minimal model theorem and homotopy argument for Rota-Baxter operads (Wang-Zhou [43, Theorem 3.5])
Invoked in the proof of Theorem 4.4: 'the remaining part of the proof carries verbatim'. This is a self-cited published result by two of the authors.
standard math Cobar construction of a coaugmented homotopy cooperad and its convolution L∞-algebra (Chen-Guo-Wang-Zhou [5, Definition 3.4, Proposition 3.5, Proposition 3.8])
Used in Definitions 4.1 and 5.1 to obtain NjA∞ and CNjA(V).
standard math Pre-Jacobi identity for brace operations (Gerstenhaber-Voronov, Getzler, Chen et al.)
Used in the proof of Proposition 3.1 to verify ∂²=0 on the cobar generators.
domain assumption The operad NjA is not Koszul because relation (6) is cubic (Remark 1.7)
Motivates the use of minimal models, though the proof of Theorem 4.4 does not formally depend on it.

invented entities (3)

Homotopy cooperad NjA¡ independent evidence
purpose: Koszul dual homotopy cooperad of NjA; its cobar construction is the minimal model NjA∞.
Constructed in Section 3. Its correctness as Koszul dual rests on Theorem 4.4, which the paper proves up to carry-over from [43].
dg operad NjA∞ independent evidence
purpose: Governing structure for homotopy Nijenhuis associative algebras.
Defined as Ω(NjA¡). The MC elements of the induced L∞-algebra reproduce Nijenhuis associative structures (Section 5).
L∞-algebra CNjA(V) independent evidence
purpose: Controls simultaneous deformations of product and Nijenhuis operator.
Maurer-Cartan elements correspond to homotopy Nijenhuis associative algebra structures, and its twisted complex equals the cohomology from Section 2.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Deformations and homotopy theory of Nijenhuis associative algebras." pith.science (2026). https://pith.science/paper/Y3DAVMUM

@misc{pith2026241217253,
  author       = {Pith},
  title        = {Pith review of: Deformations and homotopy theory of Nijenhuis associative algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y3DAVMUM}},
  note         = {Machine review of arXiv:2412.17253}
}

abstract

This paper is the first in a series of works devoted to an operadic study of Nijenhuis structures, focusing on Nijenhuis associative algebras. We introduce the concept of homotopy Nijenhuis associative algebras and demonstrate that the differential graded (=dg) operad $\NjAoperad_{\infty}$ governing these structures serves as the minimal model of the operad $\NjAoperad$ for Nijenhuis associative algebras. Additionally, we determine the Koszul dual homotopy cooperad of $\NjAoperad$. We construct an $L_\infty$-algebra that controls the simultaneous deformations of associative products and Nijenhuis operators. The Maurer-Cartan elements of this $L_\infty$-algebra correspond bijectively to Nijenhuis associative algebra structures. From this, we derive a cochain complex (deformation complex) and an associated cohomology theory of Nijenhuis associative algebras. Finally, we explore the connection between homotopy relative Rota-Baxter associative algebras of weight $0$ and homotopy Nijenhuis associative algebras. A sequel to this work will extend the study to Nijenhuis Lie algebras, with applications to Nijenhuis geometry.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Deformations theory and minimal model of operads for Nijenhuis algebras morphisms
math.RA 2025-08 reject novelty 6.0 of 10

A new cohomology theory for Nijenhuis algebra morphisms is introduced, with deformation and operadic minimal model consequences.

Reference graph

Works this paper leans on

44 extracted references · 43 canonical work pages · cited by 1 Pith paper

[43]

Wang and G

K. Wang and G. Zhou, The minimal model of Rota-Baxter operad with arbitrary weig ht, Selecta Math. (N.S.) 30 (2024), no. 5, Paper No. 99. 3, 7, 17, 18, 19 HOMOTOPY NIJENHUIS ALGEBRAS 29

work page 2024
[1]

A. V . Bolsinov, A. Y . Konyaev, and V . S. Matveev, Nijenhuis geometry , Adv. Math. 394 (2022), Paper No. 108001, 52. 2

work page 2022
[2]

M. R. Bremner and V . Dotsenko, Algebraic operads. An algorithmic companion , CRC Press, Boca Raton, FL,

work page
[3]

J. Burke, Koszul duality for representations of an A-inﬁnity algebra deﬁned over a commutative ring , Prelimi- nary version, https://maths-people.anu.edu.au/ ˜burkej/koszul/old-Koszul.pdf 24

work page
[4]

J. F. Cari˜ nena, J. Grabowski, and G. Marmo, Quantum bi-Hamiltonian systems, Internat. J. Modern Phys. A 15 (2000), no. 30, 4797-4810. 2, 4

work page 2000
[5]

J. Chen, L. Guo, K. Wang, and G. Zhou, Koszul duality, minimal model and L ∞-structure for di ﬀerential algebras with weight, Adv. Math. 437 (2024), Paper No. 109438, 41. 3, 7, 11, 16, 20, 21, 22

work page 2024
[6]

Das, Deformations of associative Rota-Baxter operators , J

A. Das, Deformations of associative Rota-Baxter operators , J. Algebra 560 (2020), 144-180. 2, 3, 24

work page 2020
[7]

Das, Cohomology and deformations of twisted Rota-Baxter operat ors and NS-algebras, J

A. Das, Cohomology and deformations of twisted Rota-Baxter operat ors and NS-algebras, J. Homotopy Relat. Struct. 17 (2020), 233-262. 5

work page 2020

Show all 44 references

[8]

Das and S

A. Das and S. K. Mishra, The L∞-deformations of associative Rota-Baxter algebras and hom otopy Rota-Baxter operators, J. Math. Phys. 63 (2022), no. 5, 051703. 24, 25

2022
[9]

Dotsenko and N

V . Dotsenko and N. Poncin, A tale of three homotopies , Appl. Categ. Structures 24 (2016), no. 6, 845-873. 3

2016
[10]

Drummond-Cole and B

G. Drummond-Cole and B. V allette, The minimal model for the Batalin-Vilkovisky operad , Selecta Math. 19 (2013), no. 1, 1-47. 17

2013
[11]

Ebrahimi-Fard, On the associative Nijenhuis relation, Electron

K. Ebrahimi-Fard, On the associative Nijenhuis relation, Electron. J. Combin. 11 (2004), no. 1, Research Paper 38, 13pp. 2

2004
[12]

Gerstenhaber, The cohomology structure of an associative ring , Ann

M. Gerstenhaber, The cohomology structure of an associative ring , Ann. of Math. 78 (1963), 267-288. 11, 21

1963
[13]

Gerstenhaber and A

M. Gerstenhaber and A. V oronov, Homotopy G-algebras and moduli space operad , Int. Math. Res. Notices (1995), 141-153. 11, 21 28 CHAO SONG, KAI W ANG, YUANYUAN ZHANG AND GUODONG ZHOU

1995
[14]

Getzler, Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology, Israel Math

E. Getzler, Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology, Israel Math. Conf. Proc. 7 (1993), 65-78. 11

1993
[15]

Getzler and D

E. Getzler and D. S. J. Jones, Operads, homotopy algebra and iterated integrals for doubl e loop spaces, arXiv: hep-th/9403055 (1994). 2, 5

1994 arXiv
[16]

Ginzburg and M

V . Ginzburg and M. Kapranov, Koszul duality for operads , Duke Math. J. 76 (1994), no. 1, 203-272. 2, 5

1994
[17]

L. Guo, Y . Li, Y . Sheng, and G. Zhou, Cohomologies, extensions and deformations of di ﬀerential algebras with arbitrary weight, Theory Appl. Categ. 8 (2022) 1409-1433. 7

2022
[18]

Helgason, Diﬀerential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, vol

S. Helgason, Diﬀerential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishe rs], New Y ork-London, 1978. xv+628 pp. 2

1978
[19]

Keller, Introduction to A-inﬁnity algebras and modules , Homology Homotopy Appl

B. Keller, Introduction to A-inﬁnity algebras and modules , Homology Homotopy Appl. 3 (2001), no. 1, 1-35. 24

2001
[20]

Keller, A-inﬁnity algebras, modules and functor categories , Contemp

B. Keller, A-inﬁnity algebras, modules and functor categories , Contemp. Math. 406 (2006), 97-94. 24

2006
[21]

Kontsevich and Y

M. Kontsevich and Y . Soibelman, Deformations of algebras over operads and the Deligne conje cture, Conf´ erence Mosh´ e Flato 1999, V ol. I (Dijon), Math. Phys. Stud. 21 (2000), 255-307. 3

2000
[22]

Kosmann-Schwarzbach and F

Y . Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures, Ann. Inst. H. Poincar´ e Phys. Th´ eor.,53 (1990), no. 1, 35–81. 2

1990
[23]

Lada and M

T. Lada and M. Markl, Strongly homotopy Lie algebras, Comm. Algebra 23 (1995), 2147-2161. 3

1995
[24]

Lada and J

T. Lada and J. Stashe ﬀ, Introduction to sh Lie algebras for physicists , Internat. J. Pheoret. Phys. 32 (1993), 1087-1103. 3

1993
[25]

Lei and L

P . Lei and L. Guo, Nijenhuis algebras, NS algebras, and N-dendriform algebra s, Front. Math. China 7 (2012), no. 5, 827-846. 2

2012
[26]

Loday and B

J.-L. Loday and B. V allette, Algebraic operads, Grundlehren der Mathematischen Wissenschaften [Fundame n- tal Principles of Mathematical Sciences], 346. Springer, Heidelberg, 2012. xxiv +634 pp. 3

2012
[27]

Lurie, Derived Algebraic Geometry X: F ormal moduli problems , https://people.math.harvard.edu/ ˜lurie/papers/DAG-X.pdf 2

J. Lurie, Derived Algebraic Geometry X: F ormal moduli problems , https://people.math.harvard.edu/ ˜lurie/papers/DAG-X.pdf 2
[28]

Magri and C

F. Magri and C. Morosi, A geometrical characterization of integrable Hamiltoniansystems through the theory of Poisson-Nijenhuis manifolds, no. S 19, Dipartimento di Matematica dell’Universita’ di M ilano, January 1984. 2

1984
[29]

Markl, Cotangent cohomology of a category and deformations , J

M. Markl, Cotangent cohomology of a category and deformations , J. Pure Appl. Algebra 113 (1996), no. 2, 195-218. 3

1996
[30]

Markl, Operads and PROPs , Handbook of algebra

M. Markl, Operads and PROPs , Handbook of algebra. V ol. 5, 87-140, Handb. Algebr., 5, Els evier/North- Holland, Amsterdam, 2008. 3

2008
[31]

Markl, S

M. Markl, S. Shnider, J. Stashe ﬀ, Operads in algebra, topology and physics. Math. Surveys Monogr., 96 Amer- ican Mathematical Society, Providence, RI, 2002, x +349 pp. ISBN: 0-8218-2134-2. 3

2002
[32]

Merkulov and B

S. Merkulov and B. V allette, Deformation theory of representations of prop(erad)s. I , J. Reine Angew. Math. 634 (2009), 51-106. 3

2009
[33]

Merkulov and B

S. Merkulov and B. V allette, Deformation theory of representations of prop(erad)s. II , J. Reine Angew. Math. 636 (2009), 123-174. 3

2009
[34]

Newlander and L

A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds , Ann. of Math. 65 (1957), 391-404. 2

1957
[35]

Nijenhuis, Xn−1-forming sets of eigenvectors , Indag Math, 13 (1951), 200-212

A. Nijenhuis, Xn−1-forming sets of eigenvectors , Indag Math, 13 (1951), 200-212. 2

1951
[36]

J. P . Pridham, Unifying derived deformation theories , Adv. Math. 224 (2010), no. 3, 772-826. 2

2010
[37]

C. Song, K. Wang, and Y . Y . Zhang, Deformations and cohomology theory of Ω-Rota-Baxter algebras of arbitrary weight, J. Geom. Phys. 201 (2024), Paper No. 105217. 25

2024
[38]

Stashe ﬀ, Homotopy associativity of H-spaces I , Proc

J. Stashe ﬀ, Homotopy associativity of H-spaces I , Proc. Amer. Math. Soc. 108 (1963), 275-292. 19

1963
[39]

Stashe ﬀ, Diﬀerential graded Lie algebras, quasi-Hopf algebras and high er homotopy algebras , Quantum groups (Leningrad, 1990), pp

J. Stashe ﬀ, Diﬀerential graded Lie algebras, quasi-Hopf algebras and high er homotopy algebras , Quantum groups (Leningrad, 1990), pp. 120–137, Lecture Notes in Mat hematics, vol. 1510, Springer, Berlin, (1992). 3

1992
[40]

Tradler, Inﬁnity-inner-products on A-inﬁnity-algebras , J

T. Tradler, Inﬁnity-inner-products on A-inﬁnity-algebras , J. Homotopy Relat. Struct. 3 (2008), no. 1, 245–271. 24

2008
[41]

V an der Laan,Operads up to Homotopy and Deformations of Operad Maps , arXiv:0208041

P . V an der Laan,Operads up to Homotopy and Deformations of Operad Maps , arXiv:0208041. 3
[42]

V an der Laan,Coloured Koszul duality and strongly homotopy operads , arXiv:0312147

P . V an der Laan,Coloured Koszul duality and strongly homotopy operads , arXiv:0312147. 3
[44]

Y au, Colored operads

D. Y au, Colored operads . Grad. Stud. Math., 170 American Mathematical Society, Providence, RI, 2016, xxviii+428 pp. ISBN: 978-1-4704-2723-8. 3 Chao Song, School of MathematicalSciences, East China Normal University, Shanghai 200241, China Email address: 52265500011@stu.ecnu....

2016

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [43]

Wang and G

K. Wang and G. Zhou, The minimal model of Rota-Baxter operad with arbitrary weig ht, Selecta Math. (N.S.) 30 (2024), no. 5, Paper No. 99. 3, 7, 17, 18, 19 HOMOTOPY NIJENHUIS ALGEBRAS 29

work page 2024

[2] [1]

A. V . Bolsinov, A. Y . Konyaev, and V . S. Matveev, Nijenhuis geometry , Adv. Math. 394 (2022), Paper No. 108001, 52. 2

work page 2022

[3] [2]

M. R. Bremner and V . Dotsenko, Algebraic operads. An algorithmic companion , CRC Press, Boca Raton, FL,

work page

[4] [3]

J. Burke, Koszul duality for representations of an A-inﬁnity algebra deﬁned over a commutative ring , Prelimi- nary version, https://maths-people.anu.edu.au/ ˜burkej/koszul/old-Koszul.pdf 24

work page

[5] [4]

J. F. Cari˜ nena, J. Grabowski, and G. Marmo, Quantum bi-Hamiltonian systems, Internat. J. Modern Phys. A 15 (2000), no. 30, 4797-4810. 2, 4

work page 2000

[6] [5]

J. Chen, L. Guo, K. Wang, and G. Zhou, Koszul duality, minimal model and L ∞-structure for di ﬀerential algebras with weight, Adv. Math. 437 (2024), Paper No. 109438, 41. 3, 7, 11, 16, 20, 21, 22

work page 2024

[7] [6]

Das, Deformations of associative Rota-Baxter operators , J

A. Das, Deformations of associative Rota-Baxter operators , J. Algebra 560 (2020), 144-180. 2, 3, 24

work page 2020

[8] [7]

Das, Cohomology and deformations of twisted Rota-Baxter operat ors and NS-algebras, J

A. Das, Cohomology and deformations of twisted Rota-Baxter operat ors and NS-algebras, J. Homotopy Relat. Struct. 17 (2020), 233-262. 5

work page 2020