{"id":"511e6f49-2b50-4013-ae51-1fb52d2298cd","arxiv_id":"2505.20735","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Extended O-operators generate new Novikov algebra structures and correspond, under an invariance condition on the symmetric part, to solutions of extended and generalized Novikov Yang-Baxter equations.","lead":"Extended O-operators and post-Novikov algebras are introduced for Novikov algebras, the algebraic structures behind some integrable systems and conformal field theory. The paper proves that these operators generate new Novikov algebra structures and correspond to solutions of generalized Yang-Baxter equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The advertised 'general' NYBE-to-extended-O-operator equivalence is false as stated: in Example 2.28, r=e2⊗e2 solves the NYBE, but its symmetric part is not invariant, so Corollary 3.8 cannot be applied and its conclusion fails.","rationale":"I read the paper as a structural contribution: extended O-operators, post-Novikov algebras, ENYBE, and GNYBES are natural definitions, and the main chain of implications is plausible when the stated invariant-symmetric-part hypothesis is in force. My independent check of the proof of Theorem 2.24(i) found the key identities used correctly, and Theorem 3.5(ii) does reduce the ENYBE condition to the extended O-operator equation under Eq. (51). The reader's weakest assumption is exactly the place where the advertised 'general' claim overreaches. The counterexample r=e2⊗e2 in the paper's own Example 2.28 shows the invariant condition is not automatically satisfied by NYBE solutions: the symmetric part is non-invariant, yet r solves the NYBE, and the corresponding α=0 is not an extended O-operator with extension β. Therefore the Introduction's and Abstract's statements that a general solution of the NYBE is equivalent to some extended O-operator are false without the missing hypothesis. This does not undermine the conditional theorems, but it does mean the paper's headline contribution is stated more strongly than what is proved. The appropriate verdict is the reader's CONDITIONAL: accept provided the claims are restricted to the invariant-symmetric-part setting and the overstatements are corrected. I found no additional internal inconsistency in the central computations I chased, so I do not recommend moving the verdict further.","tokens_in":34201,"tokens_out":18697,"duration_ms":182387,"concrete_test":"In the algebra of Example 2.28, compute the NYBE for r=e2⊗e2: each of r13∘r23, r12⋆r23, and r13∘r12 vanishes, so Eq. (46) holds. Then verify directly that Eq. (51) fails for s=e2⊗e2 and that Eq. (29) fails for β=rhat, x=e1, u=e2*. This settles that Corollary 3.8's invariant-symmetric-part hypothesis is not automatic and that the unconditional 'general solution' equivalence in the Introduction is false. To salvage the claim, either prove the equivalence without Eq. (51) or amend the Introduction and abstract to state the invariant-symmetric-part hypothesis explicitly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption is correct and, moreover, the missing hypothesis is not merely a technical convenience: it is necessary. In the two-dimensional Novikov algebra of Example 2.28 (e1∘e1=e1, e1∘e2=e2, e2∘e1=e2, e2∘e2=0), take r=e2⊗e2. Since all products involving e2∘e2 and e2⋆e2 vanish, every term in Eq. (46) is zero, so r is a solution of the NYBE. But the symmetric part β=rhat is not invariant: Eq. (51) applied to s=e2⊗e2 gives (L_A(e1)⊗id+id⊗L_{A,⋆}(e1))(e2⊗e2)=3e2⊗e2≠0. Equivalently, β is not balanced. If the unconditional equivalence claimed in the Introduction were valid, Corollary 3.8(c) would make α=0 an extended O-operator of weight 0 with extension β of mass (−1,0). Definition 2.19 requires β to be A-invariant of mass −1, but this fails: with x=e1 and u=e2*, e1∘β(e2*)=e1∘e2=e2, whereas β(L*_{A,⋆}(e1)e2*)=−2e2≠e2. Thus r is a NYBE solution whose skew-symmetric part is not an extended O-operator with the symmetric part as extension. The paper's actual theorems are conditional on Eq. (51), and those conditional statements are coherent; the problem is that the Abstract and Introduction repeatedly advertise the equivalence for general solutions of the NYBE without this hypothesis. This is load-bearing for the central advertised claim, not an incidental wording issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces extended O-operators on Novikov algebras, post-Novikov algebras, and an extended Novikov Yang-Baxter equation, and investigates their interrelations. The central results are: an extended O-operator produces a new Novikov algebra via the operation u*v = l_A(α(u))v + r_A(α(v))u + λu·v (Theorem 2.24(i)); O-operators of weight λ produce post-Novikov algebras (Theorem 2.16); under an invariance condition on the symmetric part of a tensor r, solutions of the NYBE correspond to extended O-operators (Theorem 3.5, Corollary 3.8); and extended O-operators are related to generalized Novikov Yang-Baxter equations (Theorem 4.8). The Introduction further claims that this correspondence holds for 'general' solutions of the NYBE, but the theorems in the text require Eq. (51).","tokens_in":34581,"tokens_out":17384,"duration_ms":156046,"significance":"If the conditional results are correct, the paper provides a coherent tensor and operator framework for non-skew-symmetric NYBE solutions with invariant symmetric part, and it extends to Novikov algebras the extended O-operator formalism from Lie and associative algebras. The proof of Proposition 2.21 is detailed, and Example 2.28 gives an explicit construction; the paper also makes precise links to the ENYBE and GNYBES. However, the advertised 'general solution' claim is not supported and is in fact false as stated, so the significance of the paper is narrower than the Introduction suggests.","major_comments":[{"comment":"The Introduction states that 'a general solution of the NYBE is equivalent to some extended O-operator' and cites Corollaries 3.8, 3.14, and 3.17. The corresponding results, however, all carry an additional hypothesis: Theorem 3.5(ii) and Corollary 3.8 assume that the symmetric part β of r is invariant, i.e. Eq. (51); Corollary 3.14 repeats this assumption; Corollary 3.17 assumes β is a balanced A-bimodule homomorphism. The hypothesis is essential. In Example 2.28, r=e2⊗e2 satisfies the NYBE (Eq. (46)), but its symmetric part is not invariant: applying Eq. (51) with x=e1 gives 3e2⊗e2≠0. Consequently β is not A-invariant of mass −1 in the sense of Definition 2.19(ii), and α=0 is not an extended O-operator with extension β of mass (−1,0); Corollary 3.8(c) cannot be applied. The unconditional claim in the Introduction is therefore false as stated, and the paper needs either to state the invariance hypothesis explicitly in the main claims or to present this counterexample and restrict the advertised conclusion.","section":"Introduction; Theorem 3.5; Corollary 3.8"},{"comment":"Lemma 2.15 and Theorem 2.16 are used to establish the fundamental link between O-operators of weight λ and post-Novikov algebras, and Corollary 2.17 depends on them. Both are dismissed with 'It is straightforward', as is Proposition 2.14. Since the post-Novikov definition involves eight compatibility identities, the omitted verifications are not obvious from the text; the authors should supply the computations or at least a representative sample and state which defining identities are used. The same applies to Corollary 2.18, which is said to follow directly from Theorem 2.16.","section":"Lemma 2.15; Theorem 2.16"},{"comment":"The proof of the equivalence between P− being a skew-symmetric solution of the GNYBES and the system (68)-(73) is compressed at several load-bearing points: the reduction of the displayed equality to (75)-(79) is asserted as 'easy to see', the equivalences (75)⇔(68), (76)⇔(70), and (77)⇔(71) are described as 'similar' or 'one can check', and the final step showing (∗∗) is equivalent to (72)-(73) is not shown. Since Theorem 4.8 underpins Corollary 4.9 and the GNYBES connection, these steps should be written out.","section":"Section 4, Theorem 4.8"}],"minor_comments":[{"comment":"The hypothesis says β is a balanced A-bimodule homomorphism from (V*,l*_{A,⋆},−r*_A), but β∈Hom(V,A); it should be from (V,l_A,r_A), or the statement should be phrased in terms of Q_+.","section":"Corollary 3.17"},{"comment":"The definition of B_α(u,w) uses v in the right-hand side; it should be B_α(u,v).","section":"Eq. (74)"},{"comment":"The quantification in (56) should be over a*,b*∈A*, not A.","section":"Corollary 3.8(d)"},{"comment":"The displayed computation contains notation slips (e.g. r(α(v))u instead of r_A(α(v))u) and should be proofread.","section":"Proposition 2.21"},{"comment":"The notation (T±id)^{∓2} is used without defining what a negative power of an endomorphism means; this should be stated explicitly.","section":"Eq. (45)"},{"comment":"The A-invariance condition multiplies both sides by the same parameter κ, so for κ=0 the condition is vacuous and for nonzero κ the numerical value is irrelevant; the terminology 'mass κ' should be clarified or the condition should be written without the redundant factor.","section":"Definition 2.19(ii)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope. The main gap is the mismatch between the advertised 'general NYBE' claim and the actual invariant-symmetric-part assumptions; this is fixable by revision. I would ask the authors to add the counterexample from Example 2.28 as a warning and to supply the missing proofs in Theorems 2.16 and 4.8 before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main thing to know: this paper introduces extended O-operators, post-Novikov algebras, extended and generalized Novikov Yang-Baxter equations for Novikov algebras, and proves a web of equivalences. The definitions are natural and the main theorems (2.24, 3.5) are backed by detailed computations. If you work with Novikov bialgebras or O-operator theory, this is worth reading.\n\nWhat is actually new: the notion of an extended O-operator with an extension of mass (κ, μ) genuinely generalizes O-operators in the Novikov setting. The post-Novikov algebra structure and its connection to O-operators of weight λ is a coherent addition. Theorem 2.24 shows new Novikov algebra structures from extended O-operators, and Theorem 3.5 plus Corollary 3.8 tie NYBE solutions to extended O-operators. But that tie is conditional on the symmetric part of r being invariant (Eq. 51), and the hypothesis is not a technical convenience. The stress-test example is conclusive: in the two-dimensional algebra of Example 2.28, r = e2⊗e2 solves the NYBE, but its symmetric part is not invariant, and the conclusion of Corollary 3.8 fails. So the Introduction's phrase \"a general solution of the NYBE is equivalent to some extended O-operator\" is stronger than anything proved. Because the generality of that correspondence is the paper's main advertised result, this is a load-bearing overstatement, not a wording slip.\n\nThe rest of the paper is on firmer ground. The conditional theorems are precisely stated in most places, and the computations in Proposition 2.21 and Theorem 3.5 are substantial. A few lemmas (2.14, 2.15, 2.16) are dismissed as \"straightforward,\" which makes independent verification more laborious but not impossible. The citation pattern is fine; the paper builds on [10] and credits it appropriately.\n\nWho this is for: researchers in Novikov algebras, pre-Lie algebras, and bialgebra theory. It deserves a serious referee, but the referee should require the authors to add the invariance hypothesis to every \"general solution\" claim and to include a remark or example showing the hypothesis is necessary. I would not cite the unconditional claim, but I would cite the conditional theorems and the new definitions.\n\nRecommendation: send to peer review, with major revision required. The mathematics is mostly solid; the advertised generality needs correction.","headline":"Useful structural paper on Novikov algebras whose main advertised equivalence is overbroad: it holds only under an invariant symmetric-part condition, and the paper's own example shows that condition is necessary.","tokens_in":35139,"tokens_out":3296,"would_cite":true,"duration_ms":32571,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17A30","17B38","17A60","17D25"],"pacs":[],"model":"deepseek-v4-flash","headline":"General solutions of the Novikov Yang-Baxter equation are extended O-operators when the symmetric part is invariant.","keywords":["Novikov algebra","Novikov bialgebra","Novikov Yang-Baxter equation","extended O-operator","post-Novikov algebra","extended Novikov Yang-Baxter equation","generalized Novikov Yang-Baxter equation","Rota-Baxter operator"],"falsifier":"In a low-dimensional Novikov algebra, for instance the two-dimensional algebra of Example 2.28, enumerate all $r\\in A\\otimes A$ solving the NYBE and select one whose symmetric part fails the invariance condition. If its skew-symmetric part does not satisfy the extended $\\mathcal{O}$-operator equation with $\\beta$ equal to the symmetric part, then the claimed equivalence fails precisely when the invariance condition is dropped.","tokens_in":33962,"feed_emoji":"🧮","tokens_out":8152,"duration_ms":71191,"temperature":0.7,"pith_summary":"This paper introduces extended $\\mathcal{O}$-operators on Novikov algebras and proves that they are the operator form of general, not necessarily skew-symmetric, solutions of the Novikov Yang-Baxter equation (NYBE). An extended $\\mathcal{O}$-operator relaxes the defining equation of an $\\mathcal{O}$-operator by allowing a defect term built from an auxiliary extension map, and it still produces a new Novikov algebra structure on the bimodule. The paper also defines post-Novikov algebras, extended Novikov Yang-Baxter equations (ENYBE), and generalized Novikov Yang-Baxter equations, and proves a network of equivalences among these objects. If the main claim is right, the classical correspondence between skew-symmetric NYBE solutions and $\\mathcal{O}$-operators extends to all NYBE solutions whose symmetric part satisfies an invariance condition, bringing Novikov bialgebra theory in line with the extended-operator treatment of the classical Yang-Baxter equation.","feed_headline":"General NYBE solutions are extended O-operators","feed_subtitle":"New operator forms cover non-skew-symmetric r and tie post-Novikov algebras to Yang-Baxter equations.","key_machinery":"The central object is the extended $\\mathcal{O}$-operator: a pair $(\\alpha,\\beta)$ of linear maps from a bimodule $M$ into a Novikov algebra $A$, where $\\beta$ satisfies three compatibility conditions (balanced, $A$-invariant of mass $\\kappa$, equivalent of mass $\\mu$) and $\\alpha$ nearly solves the $\\mathcal{O}$-operator equation, with the defect measured by $\\kappa\\beta(u)\\circ\\beta(v)+\\mu\\beta(u\\cdot v)$. The argument uses symmetrization: from two maps $\\delta_\\pm$ it sets $\\alpha=(\\delta_+ + \\delta_-)/2$ and $\\beta=(\\delta_+ - \\delta_-)/2$, so the difference between candidate operators is controlled by $\\beta$. On the tensor side, $r\\in A\\otimes A$ is identified with a linear map $\\hat{r}:A^*\\to A$; the symmetric part of $r$ plays the role of $\\beta$ and the skew-symmetric part plays the role of $\\alpha$. The invariance condition on the symmetric part, $(L_A(x)\\otimes \\mathrm{id}+\\mathrm{id}\\otimes L_{A,\\star}(x))s=0$, is exactly what makes that symmetric part a balanced $A$-bimodule homomorphism, which Corollary 3.8 needs.","core_discovery":"Let $(A,\\circ)$ be a Novikov algebra and $(M,\\cdot,l_A,r_A)$ an $A$-bimodule Novikov algebra. A linear map $\\alpha$ is an extended $\\mathcal{O}$-operator of weight $\\lambda$ with extension $\\beta$ of mass $(\\kappa,\\mu)$ when $\\alpha(u)\\circ\\alpha(v)-\\alpha(l_A(\\alpha(u))v+r_A(\\alpha(v))u+\\lambda u\\cdot v)=\\kappa\\beta(u)\\circ\\beta(v)+\\mu\\beta(u\\cdot v)$, with $\\beta$ balanced, $A$-invariant of mass $\\kappa$, and equivalent of mass $\\mu$. The paper proves that any such $\\alpha$ endows $M$ with a new Novikov algebra product $u*v=l_A(\\alpha(u))v+r_A(\\alpha(v))u+\\lambda u\\cdot v$, and that when the symmetric part of a tensor $r\\in A\\otimes A$ is invariant, $r$ solves the NYBE exactly when its skew-symmetric part is an extended $\\mathcal{O}$-operator whose extension is its symmetric part. This is the non-skew-symmetric extension of the older statement that skew-symmetric NYBE solutions are precisely $\\mathcal{O}$-operators.","pith_inferences":["The paper leaves open whether the invariance of the symmetric part is also necessary; a low-dimensional search for a NYBE solution with non-invariant symmetric part that is not an extended $\\mathcal{O}$-operator would settle the sharpness of the statement.","Following the Lie-algebra analogue, extended $\\mathcal{O}$-operators on Novikov algebras may correspond to nonabelian Lax pairs or double Novikov algebra structures; this direction is not explored in the paper.","The mass parameter in the ENYBE suggests an interpolation between different Novikov bialgebra structures on the same underlying algebra; one could test whether varying $\\varepsilon$ deforms the associated bialgebra in a controlled way.","A computational enumeration of NYBE solutions in low-dimensional Novikov algebras, checking which satisfy the invariance condition, would give a concrete measure of how much of the solution space the extended-operator description actually covers."],"forward_implications":["A solution of the NYBE with invariant symmetric part yields an extended $\\mathcal{O}$-operator, and therefore a new Novikov algebra structure on $A^*$ by Theorem 2.24.","Every such solution also gives a post-Novikov algebra structure on $A^*$ (Corollary 3.10), and if $\\hat{r}$ is invertible, a compatible post-Novikov structure on $A$ itself.","The extended Novikov Yang-Baxter equation of mass $\\varepsilon$ interpolates between the NYBE ($\\varepsilon=0$) and the equation whose solutions are equivalent to extended $\\mathcal{O}$-operators, so the parameter $\\varepsilon$ controls how far a tensor is from being an $\\mathcal{O}$-operator.","Solutions of the ENYBE with invariant symmetric part are automatically solutions of the generalized Novikov Yang-Baxter equations (Corollary 4.6), and thus feed into the construction of Novikov bialgebras."],"supporting_citations":[{"why":"Defines O-operators, Novikov Yang-Baxter equations, and Novikov bialgebras; the skew-symmetric correspondence that this paper extends.","marker":"[10]"},{"why":"Introduced extended O-operators for Lie algebras and the equivalence with general classical Yang-Baxter solutions, the template for this paper.","marker":"[5]"},{"why":"Developed the associative-algebra analogue of O-operators and Yang-Baxter equations, supplying the parallel framework.","marker":"[3]"},{"why":"Gives the generalized classical Yang-Baxter equations and tensor-form dictionary used in Proposition 3.2.","marker":"[4]"},{"why":"Origin of Novikov algebras as Hamiltonian operators; supplies the defining identities.","marker":"[13]"},{"why":"Defines bimodules of Novikov algebras, the module notion used throughout.","marker":"[22]"},{"why":"Provides the semidirect-product criterion used in Proposition 2.5 to characterize A-bimodule Novikov algebras.","marker":"[16]"},{"why":"Supplies the two-dimensional Novikov algebra used in Example 2.28.","marker":"[2]"}],"fun_headline_variants":["Extended O-operators capture non-skew-symmetric NYBE","Non-skew NYBE solutions are extended O-operators","Extended O-operators yield new Novikov algebras","NYBE's non-skew solutions: extended O-operators","Extended O-operators link post-Novikov to YBE"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The equivalence between NYBE solutions and extended $\\mathcal{O}$-operators rests on the assumption that the symmetric part $s$ of the solution is invariant, $(L_A(x)\\otimes \\mathrm{id}+\\mathrm{id}\\otimes L_{A,\\star}(x))s=0$ for all $x$; without this condition the paper does not show that a general NYBE solution corresponds to an extended $\\mathcal{O}$-operator.","fun_headline_variants_meta":{"raw":{"variants":["Extended O-operators capture non-skew-symmetric NYBE","Non-skew NYBE solutions are extended O-operators","Extended O-operators yield new Novikov algebras","NYBE's non-skew solutions: extended O-operators","Extended O-operators link post-Novikov to YBE"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000314,"raw_usage":{"total_tokens":1800,"prompt_tokens":984,"completion_tokens":816,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":735}},"tokens_in":600,"tokens_out":816,"duration_ms":7170,"temperature":1.0,"reasoning_tokens":735,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:48:28.482355+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In a low-dimensional Novikov algebra, for instance the two-dimensional algebra of Example 2.28, enumerate all $r\\in A\\otimes A$ solving the NYBE and select one whose symmetric part fails the invariance condition. If its skew-symmetric part does not satisfy the extended $\\mathcal{O}$-operator equation with $\\beta$ equal to the symmetric part, then the claimed equivalence fails precisely when the invariance condition is dropped.","supporting_citations":[{"cited_title":"Hong, C.M","cited_arxiv_id":null,"evidence_quote":"Defines O-operators, Novikov Yang-Baxter equations, and Novikov bialgebras; the skew-symmetric correspondence that this paper extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced extended O-operators for Lie algebras and the equivalence with general classical Yang-Baxter solutions, the template for this paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Developed the associative-algebra analogue of O-operators and Yang-Baxter equations, supplying the parallel framework."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the generalized classical Yang-Baxter equations and tensor-form dictionary used in Proposition 3.2."},{"cited_title":"Gelfand and I","cited_arxiv_id":null,"evidence_quote":"Origin of Novikov algebras as Hamiltonian operators; supplies the defining identities."},{"cited_title":"Osborn, Modules for Novikov algebras, Contemp","cited_arxiv_id":null,"evidence_quote":"Defines bimodules of Novikov algebras, the module notion used throughout."},{"cited_title":"Hong, Extending structures and classifying complements for left-symmetric algebras, Results Math","cited_arxiv_id":null,"evidence_quote":"Provides the semidirect-product criterion used in Proposition 2.5 to characterize A-bimodule Novikov algebras."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two-dimensional Novikov algebra used in Example 2.28."}],"review_version":1}