{"id":"b994d129-5f5e-4abc-b4c3-0801f48b3235","arxiv_id":"2608.06736","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Extended O-operators on Poisson algebras are defined and shown to be equivalent to solutions of the new extended Poisson Yang-Baxter equation, generalizing the classical O-operator characterization of Poisson Yang-Baxter solutions.","lead":"This paper introduces extended O-operators on Poisson algebras and the associated extended Poisson Yang-Baxter equation, then proves they correspond to each other in several settings. It unifies known results on O-operators and Poisson bialgebras under a single framework with adjustable mass parameters.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the balanced-extension conditions are explicit and are exactly supplied by the (ad,L)-invariance hypothesis in the central theorem.","rationale":"The reader's verdict is ACCEPT with MODERATE confidence, and the weakest assumption is identified as the balanced A-module homomorphism condition on the extension S. I examined whether this imposes an unstated gap in the central claim, Theorem 3.17. It does not: the hypothesis that Theta is (ad,L)-invariant is precisely the condition that makes Theta_+ balanced and A-invariant (Lemma 3.8), and in the mass (kappa,0) case the further equivalence condition (24) is vacuous because mu=0. The proof of Theorem 3.17 is a direct tensor computation, and the delicate cancellations of mixed Lambda_+-Theta_+ terms are justified by the balanced identity and by A-invariance; no step divides by kappa or mu, so degenerate mass parameters do not create hidden singularities. The apparent factor 1/4 is consistent: r13+r31=2Theta13 and r23+r32=2Theta23, so the EPYBE right-hand side contributes (kappa+1)[Theta13,Theta23], aligning with the kappa[Theta,Theta] defect term in the extended O-operator equation. The correspondence is explicitly restricted to solutions whose symmetric part is (ad,L)-invariant, which narrows the scope but is stated in the theorem and abstract rather than being a hidden flaw. Therefore the reader's concern is a legitimate stress point but not a reason to change the ACCEPT verdict; independent symbolic verification on small examples would further raise confidence.","tokens_in":33483,"tokens_out":16545,"duration_ms":151998,"concrete_test":"Use a computer algebra system to instantiate Theorem 3.17 on the 3-dimensional Poisson algebra of Example 2.19: choose a generic symmetric Theta satisfying Eqs. (44)-(45) and a generic Lambda, and verify for several values of kappa that the EPYBE of mass (kappa+1)/4 is equivalent to Eqs. (25)-(26) for Lambda_+ with extension Theta_+ of mass (kappa,0). Separately symbolically verify Lemma 2.26(i)-(iii), especially Eq. (27), on the same example to confirm the balanced-condition cancellations used in Theorem 2.28.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The natural stress point is the heavy use of the balanced, A-invariant and equivalence conditions on the extension S in Theorem 2.28, since the central correspondence in Theorem 3.17 passes through that engine. On inspection, these hypotheses are not smuggled in: Theorem 3.17 assumes that the symmetric part Theta is (ad,L)-invariant, and Lemma 3.8 identifies that assumption with Theta_+ being balanced and A-invariant of mass 1. For the mass (kappa,0) case, the equivalence condition (24) is vacuous because mu=0, so Theorem 2.28 only needs the balanced and A-invariant conditions, which are automatic for Theta_+ under the theorem's hypothesis. Tracing the proof of Theorem 3.17, the mixed Theta_+- and Lambda_+-terms cancel exactly using the balanced identity rho(S(u))v=-rho(S(v))u and the A-invariance identity [a,S(x*)]=S(-ad*(a)x*); the remaining computation is the tensor identity pairing C(r)-((kappa+1)/4)[r13+r31,r23+r32] with the extended-O-operator defect. The only caveat is that the correspondence is not unconditional: it characterizes exactly those EPYBE solutions whose symmetric part is (ad,L)-invariant, a restriction the authors state clearly. Thus I do not find a load-bearing gap in the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces extended O-operators and generalized/extended Poisson Yang-Baxter equations (GPYBE/EPYBE) for Poisson algebras. It proves that ordinary O-operators of weight λ produce post-Poisson algebra structures (Prop. 2.17), that extended O-operators, under the balanced, A-invariant, and equivalence conditions (22)-(24), induce Poisson algebra structures on the module space (Thm. 2.28), and that equivalent characterizations hold through the symmetrizer-antisymmetrizer decomposition (Thm. 2.33). The central result, Theorem 3.17, states that for r=Θ+Λ with Θ (ad,L)-invariant, r solves the EPYBE of mass (κ+1)/4 if and only if Λ_+ is an extended O-operator of weight 0 with extension Θ_+ of mass (κ,0). This is applied to quadratic and semi-direct product Poisson algebras, recovering known O-operator/PYBE characterizations in the skew-symmetric case.","tokens_in":33790,"tokens_out":13658,"duration_ms":116858,"significance":"If correct, the paper provides a useful unifying framework: it extends the O-operator method from Lie and associative algebras to Poisson algebras, connects post-Poisson algebras with O-operators, and gives a tensor-form correspondence between extended O-operators and solutions of the EPYBE. The main theorems are proven by explicit computations and reduce in special cases (κ=-1 or S=0) to known characterizations of the PYBE, which is a good consistency check. The paper also contains concrete worked examples. The central correspondence is explicitly conditional on the (ad,L)-invariance of the symmetric part of r; this is a stated scope restriction rather than a hidden assumption. I found no load-bearing technical gap in the main derivation.","major_comments":[],"minor_comments":[{"comment":"The statement contains a typo: “P_S = S I_B : P*→P” should read “P_S = S I_B : A*→A” (and similarly in the following line).","section":"§3.3, Proposition 3.26"},{"comment":"There is a duplicated plus sign: “ζ(T([w,v]_V))u+ +λζ(T([w,u]_V))v” should have a single plus.","section":"§3.4, Proposition 3.11, Eq. (61)"},{"comment":"For μ=0 the “equivalent of mass μ” condition (24) is vacuous, and for κ=0 the “A-invariant of mass κ” condition (23) is vacuous; since the degenerate cases μ=0 and κ=0 are used in Theorem 3.17, a short remark making this explicit would prevent misreading.","section":"§2.3, Definition 2.22"},{"comment":"In the displayed computation, the bracket “[v,ζ(S(u))w]” is the bracket on V, not on A; using “[,]_V” consistently throughout the proof would remove ambiguity.","section":"§2.1, Lemma 2.26(i), Eq. (27)"},{"comment":"The remark proves the equivalence of the three forms of the bracket identity but only asserts the corresponding multiplication identity; adding the one-line computation for A(r) would make the remark self-contained.","section":"§3.2, Remark 3.16"},{"comment":"The phrase “the 2-tensor form of r± = T I_B ± S I_B” is imprecise because T I_B and S I_B are linear maps, while r± denotes tensors; it would be clearer to say “the 2-tensor forms of T I_B±S I_B”.","section":"§3.3, Corollary 3.27"},{"comment":"The notation r_id is used before it is defined; it should be introduced explicitly, for example as r_id = Σ_i e_i⊗e_i for a basis (e_i) of A.","section":"§3.4, Corollary 3.32(ii)"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript relies on the authors' earlier work [16,17] for Lemma 3.8 and Proposition 2.15; this is appropriate but the dependence could be noted more visibly in the introduction. The paper is within the scope of math.RA and would be a solid contribution after the typographical issues are fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it says: it transplants extended O-operators and the extended Yang-Baxter equation from Lie and associative algebras to Poisson algebras, and proves a genuine correspondence (Theorem 3.17) between EPYBE solutions with (ad,L)-invariant symmetric part and extended O-operators of weight 0. The symmetrizer-antisymmetrizer decomposition is a clean organizing device, and the quadratic and semi-direct product sections show the framework connects to known PYBE results without forcing them. The post-Poisson algebra material is mostly recalled, but the new constructions via O-operators and the invertible-operator characterization are a reasonable addition.\n\nThe main theorems are long direct computations. I did not machine-check them, but I traced the key steps and found no hidden load-bearing assumption. The balanced, A-invariant, and equivalence conditions on the extension S look restrictive, but they are stated in Definition 2.22 and used transparently; in Theorem 3.17 they are exactly supplied by the (ad,L)-invariance of the symmetric part. The stress-test note confirms this, and I agree. The definitions of extended O-operators and EPYBE are admittedly engineered so the equivalences work, but the equivalences themselves are not tautological and they reduce correctly to known characterizations (e.g., mass -1 gives the PYBE/O-operator correspondence).\n\nSoft spots, in proportion: the paper is conceptually incremental, following the template of Bai-Guo-Ni almost line by line. The novelty is in the Poisson setting, not in new ideas. There are minor typos (e.g., 'P*\\to P' in Proposition 3.26, some garbled symbols in Theorem 3.10's proof) and some proofs are only sketched for the 'other case' with 'analogous'. The self-citation to [16,17] is heavy but the cited results are genuinely used, not padded. The abstract promises operad statements (trisuccessor) that are only cited, not proved; that is acceptable but should be labeled more clearly.\n\nFor whom? Specialists in O-operator and Yang-Baxter theory for Poisson algebras, and people working on post-Poisson algebras. A serious referee will find the claims checkable and likely correct. I would send it to peer review; if I worked in this narrow area I would cite it. It needs minor cleanup, not restructuring.\n\nRecommendation: send to a specialist referee. Likely accept after minor revisions.","headline":"Workmanlike extension of O-operator/Yang-Baxter machinery to Poisson algebras; the central equivalence is real, the hypotheses are explicit, and the paper deserves refereeing.","tokens_in":34295,"tokens_out":1407,"would_cite":true,"duration_ms":17079,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16T25","17A30","17A60","17B38","17B62","17B63"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes a one-to-one correspondence between solutions of an extended Poisson Yang-Baxter equation with invariant symmetric part and extended O-operators, unifying the classical O-operator description of skew-symmetric…","keywords":["Poisson algebras","post-Poisson algebras","extended O-operators","extended Poisson Yang-Baxter equation","Poisson bialgebras","Rota-Baxter operators","O-operators","Poisson Yang-Baxter equation"],"falsifier":"Take the paper's 3-dimensional example, choose an extension map $S$ that is balanced and $A$-invariant but fails the equivalence condition with $\\mu=\\lambda$, and compute whether the operations (15)--(16) satisfy the Jacobi identity and Leibniz rule; any counterexample would show the matching condition is necessary. To test Theorem 3.17 directly, search in a quadratic Poisson algebra for an $r=\\Theta+\\Lambda$ with invariant $\\Theta$ that satisfies the EPYBE of mass $(\\kappa+1)/4$ while $\\Lambda_+$ fails the two extended operator identities (25)--(26).","tokens_in":33269,"feed_emoji":"🔗","tokens_out":11726,"duration_ms":91124,"temperature":0.7,"pith_summary":"This paper establishes that the Poisson Yang-Baxter equation, which normally only sees skew-symmetric tensors, can be extended so that solutions with a nonzero $(\\mathrm{ad},L)$-invariant symmetric part are governed by a new kind of operator, the extended $\\mathcal{O}$-operator. The central equivalence says that a tensor $r=\\Theta+\\Lambda$ solves the extended Poisson Yang-Baxter equation of mass $(\\kappa+1)/4$ exactly when its skew-symmetric part $\\Lambda_+$ is an extended $\\mathcal{O}$-operator of weight $0$ with symmetric extension $\\Theta_+$ of mass $(\\kappa,0)$. This turns a nonlinear tensor equation into a linear-algebraic operator condition and folds the classical $\\mathcal{O}$-operator characterization of skew-symmetric solutions into a wider framework. The paper also shows that ordinary $\\mathcal{O}$-operators of weight $\\lambda$ build post-Poisson algebras, and that extended $\\mathcal{O}$-operators induce new Poisson algebra structures on module spaces under symmetry conditions on the extension map.","feed_headline":"Symmetric Yang-Baxter solutions equal extended O-operators","feed_subtitle":"Extended O-operators and symmetric Yang-Baxter solutions are now the same data, one-to-one.","key_machinery":"The paper's load-bearing object is the extended $\\mathcal{O}$-operator: a pair $(T,S)$ of linear maps $V\\to A$, where $S$ is called the extension and is required to be balanced, $A$-invariant, and equivalence-compatible, while $T$ satisfies the deformed $\\mathcal{O}$-operator identities with mass parameters $(\\kappa,\\mu)$. The companion object is the extended Poisson Yang-Baxter equation (EPYBE) of mass $\\epsilon$, defined by modifying the classical $C(r)=A(r)=0$ equations with a term $\\epsilon[r_{13}+r_{31},r_{23}+r_{32}]$ and $\\epsilon(r_{13}+r_{31})\\cdot(r_{23}+r_{32})$. The argument is carried by the symmetrizer-antisymmetrizer decomposition: writing $r=\\Theta+\\Lambda$, the operator $T$ corresponds to the skew-symmetric part and $S$ to the symmetric part, and the identities for an extended $\\mathcal{O}$-operator are exactly the tensor identities of the EPYBE after this decomposition. Under the hypotheses that $S$ is balanced and $A$-invariant and that the equivalence condition holds with $\\mu=\\lambda$, the intermediate Lemma 2.26 and Theorem 2.28 convert the extension terms into cancellations that make $(V,\\{\\cdot,\\cdot\\}_T,\\circ_T)$ a Poisson algebra.","core_discovery":"The central discovery is a correspondence at the level of tensors and operators on Poisson algebras. For a Poisson algebra $A$, decompose an element $r\\in A\\otimes A$ into its symmetric and skew-symmetric parts, $r=\\Theta+\\Lambda$, and suppose $\\Theta$ is $(\\mathrm{ad},L)$-invariant. Then $r$ solves the extended Poisson Yang-Baxter equation of mass $(\\kappa+1)/4$, namely $C(r)=\\frac{\\kappa+1}{4}[r_{13}+r_{31},r_{23}+r_{32}]$ and $A(r)=\\frac{\\kappa+1}{4}(r_{13}+r_{31})\\cdot(r_{23}+r_{32})$, if and only if the induced map $\\Lambda_+:A^*\\to A$ is an extended $\\mathcal{O}$-operator of weight $0$ with extension $\\Theta_+$ of mass $(\\kappa,0)$. This generalizes the classical result that skew-symmetric solutions of the Poisson Yang-Baxter equation correspond to ordinary $\\mathcal{O}$-operators; the symmetric part is no longer required to vanish. On the structural side, the paper proves that when the extension map $S$ is balanced, $A$-invariant, and satisfies an equivalence condition with $\\mu=\\lambda$, any extended $\\mathcal{O}$-operator $T$ with extension $S$ makes $(V,\\{\\cdot,\\cdot\\}_T,\\circ_T)$ into a Poisson algebra.","pith_inferences":["If the $(\\mathrm{ad},L)$-invariance assumption on $\\Theta$ could be removed, the correspondence would cover solutions whose symmetric part is arbitrary; the paper only treats the invariant case, so finding a counterexample without it would clarify how load-bearing that hypothesis is.","The mass parameter $(\\kappa+1)/4$ interpolates between the classical PYBE and deformed equations, so varying $\\kappa$ suggests a family of Yang-Baxter-type equations whose classical limit is the standard one; quantifying this deformation on concrete examples is a natural next step.","Since the extension $S$ is no longer required to vanish, the theory gives a way to measure the failure of a map $T$ to be an ordinary $\\mathcal{O}$-operator; one could try to associate this 'defect' with geometric or integrability data on Poisson manifolds.","The same symmetrizer-antisymmetrizer mechanism likely transfers to other algebras with both a Lie-type and an associative-type operation, such as pre-Lie or Novikov settings, where extended operators and extended Yang-Baxter equations already exist; establishing the analogous correspondence there is a testable extension."],"forward_implications":["Solutions of the EPYBE with $(\\mathrm{ad},L)$-invariant symmetric part are reduced from quadratic tensor equations to one operator condition on $\\Lambda_+$.","The classical $\\mathcal{O}$-operator characterization of skew-symmetric solutions of the PYBE is recovered as the special case $\\kappa=-1$; the new theorem is a strict generalization.","On quadratic Poisson algebras, the nondegenerate bilinear form converts extended $\\mathcal{O}$-operators on the adjoint module into solutions of the EPYBE, and taking $S=0$ recovers the Rota-Baxter description of skew-symmetric PYBE solutions.","In semi-direct product Poisson algebras, extended $\\mathcal{O}$-operators on a module correspond to EPYBE solutions in the larger algebra, and Rota-Baxter operators of nonzero weight produce pairs of PYBE solutions.","Because ordinary $\\mathcal{O}$-operators of weight $\\lambda$ build post-Poisson algebras, the paper connects the extended operator/EPYBE correspondence to the operadic trisuccessor picture of post-Poisson algebras."],"supporting_citations":[{"why":"introduces extended O-operators on Lie algebras, the template this paper adapts to Poisson algebras.","marker":"[3]"},{"why":"introduces extended O-operators on associative algebras and the associative Yang-Baxter equation, the second template.","marker":"[5]"},{"why":"defines Poisson bialgebras, the Poisson Yang-Baxter equation, and the O-operator-to-post-Poisson construction that the paper extends.","marker":"[21]"},{"why":"establishes the (ad,L)-invariant symmetric-part setting and the quasi-triangular Poisson bialgebra results that Theorem 3.17 generalizes.","marker":"[17]"},{"why":"identifies post-Poisson algebras as the trisuccessor of the Poisson operad, motivating the post-Poisson framework.","marker":"[2]"},{"why":"supplies the post-Lie algebra axioms used in the definition of post-Poisson algebras.","marker":"[27]"},{"why":"supplies the commutative dendriform trialgebra axioms used in the post-Poisson definition and in Proposition 2.13.","marker":"[19]"},{"why":"gives the known O-operator characterization of skew-symmetric solutions of the Poisson Yang-Baxter equation that the paper recovers.","marker":"[16]"},{"why":"also gives the skew-symmetric O-operator characterization cited in Remark 3.20(b) as the special case of the extended theory.","marker":"[18]"}],"fun_headline_variants":["Symmetric solutions = extended O-operators","Symmetric Yang-Baxter solutions match extended O-operators","Extended O-operators are symmetric Yang-Baxter solutions","One-to-one: symmetric solutions and extended O-operators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the correction map $S$ is highly symmetric: swapping its two inputs must flip the sign in the bracket part and keep the product part unchanged, it must commute with the algebra's actions as a module homomorphism, and an equivalence condition tied to the weight must hold exactly; without these, the paper does not prove that the induced operations form a Poisson algebra or that the Yang-Baxter correspondence holds.","fun_headline_variants_meta":{"raw":{"variants":["Symmetric solutions = extended O-operators","Symmetric Yang-Baxter solutions match extended O-operators","Extended O-operators are symmetric Yang-Baxter solutions","One-to-one: symmetric solutions and extended O-operators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001244,"raw_usage":{"total_tokens":5171,"prompt_tokens":1083,"completion_tokens":4088,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":699,"completion_tokens_details":{"reasoning_tokens":4023}},"tokens_in":699,"tokens_out":4088,"duration_ms":27766,"temperature":1.0,"reasoning_tokens":4023,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:37:20.391378+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the paper's 3-dimensional example, choose an extension map $S$ that is balanced and $A$-invariant but fails the equivalence condition with $\\mu=\\lambda$, and compute whether the operations (15)--(16) satisfy the Jacobi identity and Leibniz rule; any counterexample would show the matching condition is necessary. To test Theorem 3.17 directly, search in a quadratic Poisson algebra for an $r=\\Theta+\\Lambda$ with invariant $\\Theta$ that satisfies the EPYBE of mass $(\\kappa+1)/4$ while $\\Lambda_+$ fails the two extended operator identities (25)--(26).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces extended O-operators on Lie algebras, the template this paper adapts to Poisson algebras."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces extended O-operators on associative algebras and the associative Yang-Baxter equation, the second template."},{"cited_title":"Ni and C","cited_arxiv_id":null,"evidence_quote":"defines Poisson bialgebras, the Poisson Yang-Baxter equation, and the O-operator-to-post-Poisson construction that the paper extends."},{"cited_title":"Lin and D","cited_arxiv_id":null,"evidence_quote":"establishes the (ad,L)-invariant symmetric-part setting and the quasi-triangular Poisson bialgebra results that Theorem 3.17 generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"identifies post-Poisson algebras as the trisuccessor of the Poisson operad, motivating the post-Poisson framework."},{"cited_title":"Vallette, Homology of generalized partition posets, J","cited_arxiv_id":null,"evidence_quote":"supplies the post-Lie algebra axioms used in the definition of post-Poisson algebras."},{"cited_title":"Loday, On the algebra of quasi-shuffles, Manuscripta Math","cited_arxiv_id":null,"evidence_quote":"supplies the commutative dendriform trialgebra axioms used in the post-Poisson definition and in Proposition 2.13."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the known O-operator characterization of skew-symmetric solutions of the Poisson Yang-Baxter equation that the paper recovers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"also gives the skew-symmetric O-operator characterization cited in Remark 3.20(b) as the special case of the extended theory."}],"review_version":1}