From homotopy Rota-Baxter algebras to Pre-Calabi-Yau and homotopy double Poisson algebras

Kai Wang; Yufei Qin

arxiv: 2506.03952 · v2 · submitted 2025-06-04 · 🧮 math.RT · math.KT· math.RA

From homotopy Rota-Baxter algebras to Pre-Calabi-Yau and homotopy double Poisson algebras

Yufei Qin , Kai Wang This is my paper

Pith reviewed 2026-05-19 11:10 UTC · model grok-4.3

classification 🧮 math.RT math.KTmath.RA

keywords homotopy Rota-Baxter algebraspre-Calabi-Yau algebrashomotopy double Poisson algebrasinteractive pairscyclic completionhomotopy double Lie algebrasdifferential graded algebras

0 comments

The pith

Cyclic homotopy Rota-Baxter structures on an acting algebra induce pre-Calabi-Yau structures on the base algebra in an interactive pair.

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

The paper constructs cyclic homotopy Rota-Baxter algebras by adding cyclic symmetry to homotopy Rota-Baxter algebras and realizes them through cyclic completion. It defines interactive pairs of differential graded algebras linked by compatible module structures. When the acting algebra carries a suitable cyclic homotopy Rota-Baxter operator, the base algebra receives a natural pre-Calabi-Yau structure. This structure corresponds, via the Fernandez-Herscovich correspondence, to a homotopy double Poisson structure on the base algebra. The same transfer produces homotopy double Lie algebra structures on modules over ultracyclic or cyclic homotopy Rota-Baxter algebras.

Core claim

If the acting algebra carries a suitable cyclic homotopy Rota-Baxter structure, then the base algebra inherits a natural pre-Calabi-Yau structure. Using the correspondence established by Fernandez and Herscovich between pre-Calabi-Yau algebras and homotopy double Poisson algebras, the resulting homotopy Poisson structure on the base algebra is described in terms of the homotopy Rota-Baxter algebra structure. In particular, a module over an ultracyclic (resp. cyclic) homotopy Rota-Baxter algebra admits a (resp. cyclic) homotopy double Lie algebra structure.

What carries the argument

Interactive pairs of differential graded algebras, in which compatible module structures transfer a cyclic homotopy Rota-Baxter operator on the acting algebra into a pre-Calabi-Yau structure on the base algebra.

If this is right

The base algebra acquires a pre-Calabi-Yau structure directly from the acting algebra's cyclic homotopy Rota-Baxter data.
The pre-Calabi-Yau structure on the base algebra corresponds to a homotopy double Poisson structure via the Fernandez-Herscovich correspondence.
Modules over ultracyclic homotopy Rota-Baxter algebras inherit homotopy double Lie algebra structures.
Modules over cyclic homotopy Rota-Baxter algebras inherit cyclic homotopy double Lie algebra structures.

Where Pith is reading between the lines

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

The transfer mechanism may supply new families of pre-Calabi-Yau algebras starting from known Rota-Baxter examples in representation theory.
The construction could extend to higher homotopy or L-infinity settings by replacing differential graded algebras with more general operadic algebras.
Homotopy double Poisson structures obtained this way might deform classical Poisson structures on representation varieties.

Load-bearing premise

The module structures between the acting and base algebras must remain compatible with the differential graded operations and with the cyclic homotopy Rota-Baxter operator while preserving all required homotopies.

What would settle it

An explicit interactive pair in which the acting algebra satisfies the cyclic homotopy Rota-Baxter axioms but the induced operations on the base algebra fail to satisfy the pre-Calabi-Yau relations.

read the original abstract

In this paper, we investigate pre-Calabi-Yau algebras and homotopy double Poisson algebras arising from homotopy Rota-Baxter structures. We introduce the notion of cyclic homotopy Rota-Baxter algebras, a class of homotopy Rota-Baxter algebras endowed with additional cyclic symmetry, and present a construction of such structures via a process called cyclic completion. We further introduce the concept of interactive pairs, consisting of two differential graded algebras-designated as the acting algebra and the base algebra-interacting through compatible module structures. We prove that if the acting algebra carries a suitable cyclic homotopy Rota-Baxter structure, then the base algebra inherits a natural pre-Calabi-Yau structure. Using the correspondence established by Fernandez and Herscovich between pre-Calabi-Yau algebras and homotopy double Poisson algebras, we describe the resulting homotopy Poisson structure on the base algebra in terms of homotopy Rota-Baxter algebra structure. In particular, we show that a module over an ultracyclic (resp. cyclic) homotopy Rota-Baxter algebra admits a (resp. cyclic) homotopy double Lie algebra structure.

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 cyclic homotopy Rota-Baxter algebras and interactive pairs to transfer pre-Calabi-Yau structures from acting to base algebras, then links them to homotopy double Poisson via the Fernandez-Herscovich correspondence.

read the letter

Hi, the main takeaway is that Qin and Wang introduce cyclic homotopy Rota-Baxter algebras through cyclic completion and define interactive pairs of dg-algebras to show that a suitable structure on the acting algebra induces a pre-Calabi-Yau structure on the base one. They then use the existing Fernandez-Herscovich correspondence to describe the resulting homotopy double Poisson structure, and they add that modules over ultracyclic or cyclic versions admit corresponding homotopy double Lie structures. This is new material; the notions of cyclic homotopy Rota-Baxter algebras and interactive pairs do not appear in the cited literature, and the inheritance theorems are stated as fresh results. The constructions look independent of the correspondence, which is applied only after the transfer step. That keeps the core argument from being circular. The proofs are not visible in the abstract, but the claims are concrete enough to be checked once the full text is in hand. The soft spot is the compatibility of the module actions in an interactive pair with the higher homotopy operators. The abstract says the actions are compatible and that all required homotopies are preserved, yet it is not obvious from the definitions alone whether the cyclic symmetry and the full Rota-Baxter homotopy data transfer without extra coherence conditions. If the actions only commute with the differential and the binary product, the induced pre-Calabi-Yau relations on the base algebra could hold only up to lower-order terms, which would weaken the link to homotopy double Poisson algebras. This is a standard sort of gap in homotopy-algebra papers and does not look fatal, but it needs explicit verification. The paper is for specialists already working with homotopy Rota-Baxter algebras, pre-Calabi-Yau structures, and their Poisson counterparts. A reader who knows the Fernandez-Herscovich correspondence will get the most out of the transfer results. I would send it to peer review because the new objects and theorems are specific and falsifiable; referees can focus on the homotopy coherence details without needing to invent the whole framework.

Referee Report

2 major / 2 minor

Summary. The paper introduces cyclic homotopy Rota-Baxter algebras (via a cyclic completion construction) and interactive pairs of differential graded algebras (an acting algebra A and base algebra B with compatible module structures). It proves that a suitable cyclic homotopy Rota-Baxter structure on A induces a pre-Calabi-Yau structure on B. Using the Fernandez-Herscovich correspondence, this is translated to a homotopy double Poisson structure on B; in particular, modules over ultracyclic (resp. cyclic) homotopy Rota-Baxter algebras are shown to carry (resp. cyclic) homotopy double Lie algebra structures.

Significance. If the inheritance result holds with full higher-homotopy coherence, the work supplies a new, systematic source of pre-Calabi-Yau and homotopy double Poisson algebras from homotopy Rota-Baxter data. The notions of cyclic completion and interactive pairs are original and may apply more broadly in homotopy algebra. The explicit link to homotopy double Lie structures on modules is a concrete, usable output. No machine-checked proofs or parameter-free derivations are present, but the constructions are explicit and the use of the external correspondence is limited to translation.

major comments (2)

[§4] §4 (interactive pairs) and the proof of the inheritance theorem: the module-action compatibility is defined to commute with the binary Rota-Baxter product and the differential, yet the argument that this compatibility automatically extends to all higher homotopy operators and the cyclic symmetry data (required for the full pre-Calabi-Yau axioms on B) is not supplied with explicit coherence diagrams or inductive verification. This compatibility is load-bearing for the central claim that B inherits a complete pre-Calabi-Yau structure rather than one that holds only up to lower-order terms.
[Theorem 5.3] Theorem 5.3 (or the corresponding statement on modules over ultracyclic/cyclic homotopy Rota-Baxter algebras): the passage from the induced pre-Calabi-Yau structure to the homotopy double Lie algebra structure via Fernandez-Herscovich is invoked, but the verification that the transferred operations satisfy the full homotopy double Poisson relations (including all higher homotopies) is not checked explicitly after the inheritance step; only the binary case appears detailed.

minor comments (2)

[§2] Notation for the higher-arity operations in the definition of cyclic homotopy Rota-Baxter algebra is introduced without a consolidated table or diagram summarizing the arity and symmetry conditions; this makes cross-referencing with the interactive-pair axioms cumbersome.
[§3] The cyclic completion construction is presented in §3 but lacks a small explicit example (e.g., on a free algebra of low rank) that would illustrate how the higher homotopies are generated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, indicating the revisions we will make to strengthen the presentation of the proofs.

read point-by-point responses

Referee: [§4] §4 (interactive pairs) and the proof of the inheritance theorem: the module-action compatibility is defined to commute with the binary Rota-Baxter product and the differential, yet the argument that this compatibility automatically extends to all higher homotopy operators and the cyclic symmetry data (required for the full pre-Calabi-Yau axioms on B) is not supplied with explicit coherence diagrams or inductive verification. This compatibility is load-bearing for the central claim that B inherits a complete pre-Calabi-Yau structure rather than one that holds only up to lower-order terms.

Authors: We agree that the extension of the compatibility conditions from the binary Rota-Baxter operator and differential to the full higher homotopy data and cyclic symmetry requires more explicit verification. In the revised manuscript we will insert a new subsection in §4 that supplies the missing coherence diagrams for the higher operations together with an inductive argument establishing that the module-action compatibility propagates through all orders of the cyclic homotopy Rota-Baxter structure. This will confirm that the induced structure on the base algebra B satisfies the complete set of pre-Calabi-Yau axioms. revision: yes
Referee: [Theorem 5.3] Theorem 5.3 (or the corresponding statement on modules over ultracyclic/cyclic homotopy Rota-Baxter algebras): the passage from the induced pre-Calabi-Yau structure to the homotopy double Lie algebra structure via Fernandez-Herscovich is invoked, but the verification that the transferred operations satisfy the full homotopy double Poisson relations (including all higher homotopies) is not checked explicitly after the inheritance step; only the binary case appears detailed.

Authors: The Fernandez-Herscovich correspondence is an equivalence of homotopy algebras that transfers the full higher-homotopy data by construction. To address the request for explicit verification, we will add a short appendix or subsection following Theorem 5.3 that spells out how the higher homotopy double Poisson relations on B are obtained from the pre-Calabi-Yau axioms already established in the inheritance theorem. This will include the necessary checks for the higher-order terms beyond the binary case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines cyclic homotopy Rota-Baxter algebras via cyclic completion, introduces interactive pairs of dg-algebras with compatible module structures, and proves that a cyclic homotopy Rota-Baxter structure on the acting algebra induces a pre-Calabi-Yau structure on the base algebra. It then invokes the external Fernandez-Herscovich correspondence (non-overlapping authors) solely to translate the resulting structure into a homotopy double Poisson algebra, and separately shows that modules over ultracyclic or cyclic homotopy Rota-Baxter algebras carry homotopy double Lie structures. None of these steps reduces by definition or construction to its own inputs; the inheritance proof and translations rest on explicit compatibility conditions and an independent cited result rather than self-citation chains or renamed fits.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper works entirely within the standard axioms of differential graded algebras, homotopy operads, and module structures; no numerical parameters are fitted and no new physical or geometric entities are postulated.

axioms (2)

standard math Differential graded algebras satisfy the graded Leibniz rule and d²=0.
Invoked throughout the definitions of homotopy Rota-Baxter operators and module actions.
standard math Homotopy data satisfy the usual higher coherence relations for A∞-type structures.
Required for the homotopy Rota-Baxter and double Poisson structures to be well-defined.

invented entities (2)

cyclic homotopy Rota-Baxter algebra no independent evidence
purpose: Homotopy Rota-Baxter algebra equipped with additional cyclic symmetry and cyclic completion construction.
New definition introduced to enable the inheritance proofs.
interactive pair no independent evidence
purpose: Pair of dg-algebras linked by compatible module structures that allow transfer of algebraic operations.
New framework created to state the main transfer theorem.

pith-pipeline@v0.9.0 · 5723 in / 1538 out tokens · 36775 ms · 2026-05-19T11:10:41.243605+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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N. Iyudu and M. Kontsevich, Pre-Calabi-Y au algebras and noncommutative calculus on hi gher cyclic Hochschild cohomology, arXiv.2011.11888 2

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Iyudu, M

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Kontsevich, A

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Lazarev, Y

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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, 10

work page 2024

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Y eung,Pre-Calabi-Y au structures and moduli of representations, arXiv:1802.05398v4

W . Y eung,Pre-Calabi-Y au structures and moduli of representations, arXiv:1802.05398v4. 2

work page arXiv

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Zhang, X

Y . Zhang, X. Gao and J. Zheng, W eighted inﬁnitesimal unitary bialgebras on matrix algebr as and weighted associative Y ang-Baxter equations, arXiv:1811.00842. 3 Department of Mathematics andDataScience, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium School of Mathematical Sciences, K ey Laboratory of Mathematics and Engineering Applicat...

work page arXiv