{"id":"0497d632-3c70-4d04-b3d2-a83983db8599","arxiv_id":"2509.21232","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors define a completed double Poisson cohomology valid for all double Poisson brackets, and introduce three cohomology theories for double Poisson vertex algebras, with representation functor compatibility.","lead":"This mathematics paper builds new cohomology theories for noncommutative Poisson structures that work without a restrictive technical assumption, and extends them to noncommutative vertex algebras. It proves the new theories are compatible with representation functors, linking noncommutative geometry to ordinary Poisson geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Square-zero of bd in Theorem 4.3 is the load-bearing step; it rests on sign-heavy cancellation identities (Lemmas 1.5–1.6, Steps iv–v) that have not been independently verified.","rationale":"The reader correctly identifies Theorem 4.3 as the fragile structural premise, and I agree with that identification: the removal of the bivector assumption and the Kontsevich–Rosenberg compatibility for ddPH both depend on bd being a square-zero differential. Two observations strengthen the concern. First, the same combinatorial mechanism recurs in the quasi-Poisson chapter and in the dPVA differentials, so a failure would affect the second half of the paper as well as the first. Second, the alternative proof mentioned in Remark 4.8 is not part of the manuscript and cannot serve as independent verification. The paper is otherwise careful, gives detailed cancellations, and computes examples consistent with the claimed theory, but those examples would not necessarily expose a sign error in a high-degree cancellation identity. A symbolic or formal check is cheap and decisive. Since the issue is about verification rather than a demonstrated contradiction, I would move from ACCEPT to CONDITIONAL, requiring that check or an independent referee rederivation of Theorem 4.3 before the headline claims are treated as established. If the check passes, the present evidence supports acceptance.","tokens_in":82311,"tokens_out":9013,"duration_ms":83058,"concrete_test":"Encode the tensor operations of §1.1 (products (1.6)–(1.7), cyclic permutations, extensions (1.9), Lemmas 1.5–1.6) in a computer algebra system or proof assistant, then symbolically verify Theorem 4.3(2) for n = 0, 1, 2, 3 with a generic double bracket (symbolic coefficients) on a free algebra. Specifically, print the residual after cancellation in Steps iv–v; any nonzero residual disproves the complex. This check would also validate the same combinatorial identities used in Proposition 5.11 and in the dPVA differentials.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central existence claim “completed double Poisson cohomology valid in any situation” reduces to Theorem 4.3(2), the proof that bd² = 0. That proof is a long cancellation argument explicitly invoking Lemmas 1.5 and 1.6; Lemma 1.5(d) even contains a small display typo (“t≤h≤n−1”), and Steps iv–v are exactly the places where a sign error would silently break the complex. The same combinatorial core is reused for quasi-Poisson brackets (Proposition 5.11) and, in analogous form, for the basic and variational dPVA differentials (Theorems 9.17 and 10.6). The paper contains no formal verification, and Remark 4.8’s alternative proof via [42] is only cited as unpublished shared work, so it does not currently provide independent support. If any of the cyclic-permutation identities has an unstated restriction or a wrong sign, bd is not a differential and ddPH is not a cohomology theory; the representation-functor maps of Theorem 7.8 would then be maps out of an object that need not exist. This is not an observed inconsistency, but it is the least externally secured premise under the paper’s headline claims.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops cohomology theories for double Poisson algebras and double Poisson vertex algebras. In Part 1, it defines a completed double Poisson cohomology ddPH(A) for an arbitrary double Poisson bracket on A, without requiring the bracket to be defined by a noncommutative bivector, and proves that the associated differential bd squares to zero (Theorem 4.3). It compares ddPH with the Pichereau--Van de Weyer cohomology dPH, extends the construction to double quasi-Poisson and gauged double Poisson brackets, and carries out explicit computations including acyclicity for path algebras of quivers and several one- and two-generator examples. In Part 2, the paper constructs basic, reduced and variational double Poisson vertex algebra cohomologies, proves their compatibility with representation functors, and computes them for constant 2-fold lambda-brackets, obtaining infinite-dimensional cohomology spaces in contrast with the commutative case. Part 3 relates the two theories through noncommutative jet and quotient functors.","tokens_in":82648,"tokens_out":5508,"duration_ms":51115,"significance":"If the central technical claim holds, this is a substantial contribution: it removes the standing assumption in Pichereau--Van de Weyer cohomology that a noncommutative bivector exists, and it provides the first cohomology theories for double Poisson vertex algebras that are compatible with the Kontsevich--Rosenberg principle. The paper is also commendable for its explicit computations, its honest correction of a claim in [35] in Remark 3.15, and its clear statement of an unproved conjecture in Remark 5.18. The dPVA cohomology computations and the quiver acyclicity result are concrete and falsifiable. However, the square-zero property of bd is the load-bearing step, and its proof rests on long, sign-heavy combinatorial identities whose verification is not fully self-contained, as discussed below.","major_comments":[{"comment":"The construction of completed double Poisson cohomology reduces to Theorem 4.3(2), the assertion bd^2 = 0. The proof is an elaborate cancellation argument whose Steps iv--v rely directly on the tensor identities in Lemma 1.5 and Lemma 1.6. Lemma 1.5(d) contains a display typo: the second case is printed as \"t≤h≤n−1\" although t is not defined in the statement; the intended inequality presumably begins with h = i or h = i+1. Since (1.19) is used in the cancellation argument for the terms C_{1,≥2}, the current text does not allow the reader to check the proof independently. I ask that the authors correct this identity and provide a complete proof of Lemmas 1.5--1.6, or else include a machine-checkable verification of the cancellations in Steps iv--v. This is the load-bearing step for all subsequent results, including Theorem 7.8 and the applications in Chapter 6, so it cannot be left in its present state.","section":"§4.1.2, Theorem 4.3; Lemma 1.5(d), Eq. (1.19)"},{"comment":"Remark 4.8 states that the square-zero property of bd and the morphism of complexes can alternatively be proved using the graded Lie bracket constructed in the unpublished work [42]. For a claim of this centrality, an unpublished note shared with the authors is not an independent verification available to the readers. The current manuscript should contain the full proof of bd^2 = 0, or at least state explicitly that Theorem 4.3 is the only published proof. This issue appears again in Remark 7.9, so it should be addressed in both places.","section":"Remark 4.8"},{"comment":"Proposition 5.11 asserts that bd remains a square-zero differential for double quasi-Poisson brackets. The proof reuses the bulk of the proof of Theorem 4.3 and replaces only Steps i and ii by a new computation using (5.11). Consequently the quasi-Poisson analogue inherits any sign error in the shared combinatorial core of Lemmas 1.5--1.6. This dependence should be stated explicitly, and the quasi-Poisson-specific cancellations should be written out in full; otherwise the computations in Section 6.2.3, such as Proposition 6.15, are only as reliable as the unverified identities of Chapter 1.","section":"§5.2, Proposition 5.11"}],"minor_comments":[{"comment":"The proof of Lemma 1.2 is omitted as \"straightforward\". While this is acceptable for a book-length manuscript, given the central role of the related identity (1.19), a one-line proof or a cross-reference to the proof of Lemma 1.5 would help the reader.","section":"Chapter 1, Lemma 1.2"},{"comment":"In the proof of Lemma 6.17, the notation Q∆ for the class of Q modulo im ι∆ is used without being defined in that section; a brief reminder of the definition from Section 5.1 would improve readability.","section":"§6.2.4, Lemma 6.17"},{"comment":"The conjecture that the square-zero property may fail for the completed gauged cohomology in full generality is an honest and useful caveat. I suggest adding a short mention of this limitation in the introduction so that readers do not assume the gauged completed theory is proven in the same cases as the non-completed one.","section":"Remark 5.18"},{"comment":"The memoir is very long and sign-heavy; a short appendix collecting the key identities (1.15)--(1.19), with complete proofs, would substantially improve verifiability and is strongly recommended.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main risk is that the square-zero proof of bd relies on lengthy sign computations that have not been independently checked and that contain at least one clear typo (Lemma 1.5(d)). I would recommend that the editors require the authors to correct the typo, expand the proofs of Lemmas 1.5--1.6, and either remove the reliance on the unpublished [42] or make the alternative proof fully available. The paper is otherwise well-organized and carries significant new results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The two headline results are real: the completed double Poisson cohomology, defined for any double Poisson bracket without requiring a noncommutative bivector, and the three dPVA cohomology theories (basic, reduced, variational). Both are shown to be compatible with representation functors, which is exactly what the Kontsevich–Rosenberg principle asks for. The paper also does the computational work: quiver path algebras come out acyclic, the constant bracket on differential polynomials has infinite-dimensional cohomology, and it even fixes a typo in the Pichereau–Van de Weyer paper. There is no sign of circularity or overclaiming; the authors are explicit about what they have not done (quasi-Poisson and gauged dPVA cohomology, an operadic approach).\n\nThe soft spot is exactly where the stress-test put it. The square-zero property of the differential bd in Theorem 4.3 is the load-bearing step, and the proof is a long cancellation argument built on Lemmas 1.5 and 1.6. Those combinatorial identities are exactly where a sign error or an unstated restriction would silently kill the whole theory. There is a real typo in Lemma 1.5(d)—\"t≤h≤n−1\" should almost certainly read \"i≤h≤n−1\"—which does not help confidence, even though the surrounding text suggests it is a display typo rather than a mathematical error. The alternative proof via [42] is only described as unpublished shared work, so it does not currently provide independent support. This is not an observed inconsistency, but it is the least externally secured premise under the headline claims.\n\nThe later chapters are also long and sign-heavy, and the authors themselves invite careful verification. For a referee, the task is not to re-derive the whole memoir but to check the cancellation steps in Theorem 4.3 and the cyclic-permutation lemmas, ideally on small cases by hand or with a computer algebra system.\n\nWho gets value from this? Anyone working in noncommutative Poisson geometry, representation schemes, or double Poisson vertex algebras and their Hamiltonian PDE applications. It deserves a serious referee: the definitions are natural, the range of examples is convincing, and the central construction removes a restriction that has been in place since Van den Bergh's work. My recommendation is to send it to peer review with an expert referee asked specifically to verify the square-zero proof and the tensor identities in Chapter 1.","headline":"This is a serious, substantial memoir that delivers the central claim—a bivector-free completed double Poisson cohomology plus new dPVA cohomologies—though the square-zero proof at its core is a sign-heavy cancellation argument that deserves independent checking.","tokens_in":83087,"tokens_out":1456,"would_cite":true,"duration_ms":16757,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B63","17B69"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes a completed double Poisson cohomology that needs no noncommutative bivector and induces classical Poisson cohomology on every representation space.","keywords":["double Poisson algebra","double Poisson vertex algebra","completed double Poisson cohomology","Poisson vertex algebra cohomology","representation functor","quiver path algebra","noncommutative multivector fields","variational cohomology"],"falsifier":"Take a double Poisson bracket that is not induced by a bivector, for example the bracket (6.10) on the truncated algebra $k\\langle u,v \\rangle/(u^2,v^2)$, and compute $bd^2$ on a general 2-bracket by writing out every term in the decomposition (4.6). Any nonzero surviving term would disprove Theorem 4.3; a full cancellation in a case where the multivector-to-bracket maps are neither injective nor surjective would confirm that the square-zero property genuinely follows from the Poisson condition alone.","tokens_in":82067,"feed_emoji":"🧮","tokens_out":9422,"duration_ms":80489,"temperature":0.7,"pith_summary":"The paper's central claim is that double Poisson algebras—noncommutative analogues of Poisson algebras, defined by a double bracket that induces ordinary Poisson brackets on every representation space—carry a cohomology theory that needs no noncommutative bivector. This completed double Poisson cohomology is built directly from the double bracket, and its differential squares to zero precisely because the bracket satisfies the double Jacobi identity. The theory has quasi-Poisson and gauged variants, and it obeys the representation-functor compatibility principle: for each dimension $N$ it produces a map to the ordinary Poisson cohomology of the $N$-th representation algebra. A parallel family of basic, reduced, and variational cohomologies is constructed for double Poisson vertex algebras, again compatible with representations. Applications include acyclicity of the completed complex for path algebras of quivers and explicit computations for constant $\\lambda$-brackets on algebras of noncommutative differential polynomials.","feed_headline":"Double Poisson cohomology now works without a bivector","feed_subtitle":"Every double Poisson bracket gets a cochain complex whose cohomology maps to ordinary Poisson cohomology on representation algebras.","key_machinery":"The load-bearing object is the completed double Poisson complex $(dBR(A), bd)$. In degree 0, $dBR(A)$ contains $A^{\\sharp} = A/[A,A]$; in degree $n \\geq 1$ it contains all $n$-brackets, i.e. cyclically skewsymmetric maps $A^{\\otimes n} \\to A^{\\otimes n}$ satisfying a Leibniz rule. The differential $bd$ is defined by summing over cyclic permutations the two ways of inserting the double bracket $\\{\\{-,-\\}\\}$ into an $n$-bracket; the double Poisson identity is exactly the condition that makes $bd^2 = 0$. The same differential can be rewritten as a compact cyclic formula that makes the comparison with double Lie-Rinehart cohomology and the proof of compatibility with representation functors straightforward.","core_discovery":"The central discovery is that the obstruction to defining double Poisson cohomology without a bivector can be removed. Given any double Poisson bracket $\\{\\{-,-\\}\\}$ on an algebra $A$, the graded space $dBR(A)$ of $n$-brackets carries a square-zero differential $bd$ defined by cyclically inserting the double bracket into an $n$-bracket; the paper proves $bd^2=0$ and calls the resulting cohomology the completed double Poisson cohomology $ddPH(A)$. When the bracket does come from a noncommutative bivector, the new differential is compatible with the older bivector differential via a morphism of complexes, so the two cohomologies agree whenever the natural map from multivector fields to brackets is an isomorphism. The same construction is adapted to double quasi-Poisson and gauged double Poisson brackets, and a further theorem shows that each representation functor sends $ddPH(A)$ into the ordinary Poisson cohomology $H_{CE}(A_N)$ of the representation algebra. On the vertex side, the paper defines basic, reduced, and variational double Poisson vertex algebra cohomologies and proves the analogous representation-functor compatibility for the variational theory.","pith_inferences":["If the construction is sound, deformation theory for double Poisson brackets can be run without knowing a bivector, and the second cohomology group already yields a criterion for compatible pairs of double Poisson brackets on arbitrary, including singular, algebras.","The representation-functor compatibility suggests a tool for bi-Hamiltonian integrability on all representation spaces simultaneously: a noncommutative 2-cocycle that is itself Poisson should induce compatible Poisson brackets on every representation algebra, giving a testable route to families of integrable systems.","The contrast between infinite-dimensional dPVA cohomology and the finite-dimensional commutative PVA cohomology suggests that noncommutative integrable systems have many more deformation directions; one could test this by computing first-order deformations of constant brackets and asking whether the new brackets remain compatible.","One might conjecture that the comparison map from bivector-based double Poisson cohomology to the completed theory is an isomorphism not only when the map from multivector fields to brackets is an isomorphism, but also for a wider class of quasi-free algebras; this could be checked on free algebras and path algebras."],"forward_implications":["Completed double Poisson cohomology is now defined for every double Poisson algebra, including truncated polynomial algebras whose brackets do not come from noncommutative bivectors.","Each representation functor produces a linear map $ddPH(A) \\to H_{CE}(A_N)$ and, for vertex algebras, $dPvH(V) \\to PvH(V_N)$, extending the representation-functor compatibility principle to cohomology.","The completed double Poisson cohomology of a path algebra of a quiver with a non-degenerate constant double Poisson bracket vanishes in positive degree, with degree-zero cohomology of dimension equal to the number of vertices.","For a constant 2-fold $\\lambda$-bracket on noncommutative differential polynomials, the basic and reduced/variational dPVA cohomologies are infinite-dimensional for $M \\geq 1$, unlike their commutative counterparts.","The jet and quotient functors induce maps from double Poisson algebra cohomology to variational dPVA cohomology and back, relating the two theories together with their representation-algebra counterparts."],"supporting_citations":[{"why":"introduces double brackets, double Poisson algebras, and the representation-functor mechanism that the paper's cohomologies must be compatible with.","marker":"[38]"},{"why":"defines the earlier bivector-dependent double Poisson cohomology that the new completed theory generalizes and compares with.","marker":"[35, 40]"},{"why":"introduces double Poisson vertex algebras and n-fold $\\lambda$-brackets, the objects whose cohomology the second part of the paper develops.","marker":"[21]"},{"why":"supplies the basic, reduced and variational Poisson vertex algebra cohomologies that the dPVA versions are modelled on and computed against.","marker":"[19]"},{"why":"formulates the principle that noncommutative structures should induce standard structures on all representation algebras.","marker":"[28]"},{"why":"provides the double Lie-Rinehart differential whose cyclic expression is used as a compact formula for the new differential.","marker":"[14]"},{"why":"introduces gauged noncommutative Poisson cohomology with gauge elements, which the paper adapts into completed gauged double Poisson cohomology.","marker":"[1]"}],"fun_headline_variants":["No bivector, no problem: double Poisson cohomology defined","Quiver path algebras: acyclic double Poisson cohomology","New cohomology for double Poisson vertex algebras","Double Poisson cohomology without a bivector"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction collapses if the long permutation-and-sign identities behind $bd^2 = 0$ fail: the proof that the differential squares to zero is a direct computation that relies on the tensor identities of Lemmas 1.5 and 1.6, and any hidden restriction there—say on how cyclic permutations interact with the bimodule actions—would leave a differential that is not square-zero.","fun_headline_variants_meta":{"raw":{"variants":["No bivector, no problem: double Poisson cohomology defined","Quiver path algebras: acyclic double Poisson cohomology","New cohomology for double Poisson vertex algebras","Double Poisson cohomology without a bivector"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00178,"raw_usage":{"total_tokens":7108,"prompt_tokens":1123,"completion_tokens":5985,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":739,"completion_tokens_details":{"reasoning_tokens":5917}},"tokens_in":739,"tokens_out":5985,"duration_ms":41754,"temperature":1.0,"reasoning_tokens":5917,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:46:19.839851+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a double Poisson bracket that is not induced by a bivector, for example the bracket (6.10) on the truncated algebra $k\\langle u,v \\rangle/(u^2,v^2)$, and compute $bd^2$ on a general 2-bracket by writing out every term in the decomposition (4.6). Any nonzero surviving term would disprove Theorem 4.3; a full cancellation in a case where the multivector-to-bracket maps are neither injective nor surjective would confirm that the square-zero property genuinely follows from the Poisson condition alone.","supporting_citations":[{"cited_title":"The variational Poisson cohomology","cited_arxiv_id":"1106.0082","evidence_quote":"supplies the basic, reduced and variational Poisson vertex algebra cohomologies that the dPVA versions are modelled on and computed against."},{"cited_title":"In: The Gelfand Mathe- matical Seminars, 1996–1999 , Birkh¨ auser, Boston, pp","cited_arxiv_id":null,"evidence_quote":"formulates the principle that noncommutative structures should induce standard structures on all representation algebras."},{"cited_title":"Goldman-Turaev formality implies Kashiwara-Vergne","cited_arxiv_id":"1812.01159","evidence_quote":"introduces gauged noncommutative Poisson cohomology with gauge elements, which the paper adapts into completed gauged double Poisson cohomology."}],"review_version":2}