{"id":"a9aff443-38f1-4064-8e66-fa0d1cd53aaf","arxiv_id":"2505.20867","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Nijenhuis Lie conformal algebras are given a cohomology, a 2-term homotopy theory, a classification of non-abelian extensions, and a Wells-type obstruction to automorphism inducibility.","lead":"This paper builds a cohomology theory for Nijenhuis Lie conformal algebras and uses it to classify extensions and study automorphism inducibility through a Wells-type map. A generalist might read it as a compact example of how deformation and extension results are adapted to a new class of conformal algebras.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Wells map subtracts non-abelian 2-cocycles into a pointed set; the difference is never proved to be a cocycle, so Theorems 6.5 and 6.6 are unsupported.","rationale":"","tokens_in":32878,"tokens_out":11823,"duration_ms":131620,"concrete_test":"Analytic check: subtract the cocycle identity (29) for (chi,rho,Phi) from the identity for (chi^(alpha,beta),rho^(alpha,beta),Phi^(alpha,beta)) and isolate the terms bilinear in rho and rho^(alpha,beta). If those terms are not identically equal to [chi^(alpha,beta)-chi, h], the difference triple is not a 2-cocycle; run this on a split extension with H non-abelian and alpha nontrivial on the action to make the failure explicit. If the cross terms do vanish for structural reasons, the check will instead validate Definition 6.3.","verdict_should_be":"REJECT","load_bearing_attack":"Definition 5.6 defines H^2_nab as a quotient set of non-abelian 2-cocycles, so it carries no group structure (the abstract's 'group' is already stronger than the definition). Definition 6.3 then sets W(alpha,beta) = [(chi^(alpha,beta), rho^(alpha,beta), Phi^(alpha,beta)) - (chi,rho,Phi)]. This is not a well-defined element: subtracting two cocycles gives a triple whose status as a cocycle is never checked. The obstacle is visible in Eq. (29): if the identity for the original cocycle and for the twisted cocycle are subtracted, the left-hand side contains a cross term rho^(alpha,beta)(p)_lambda(rho(q)_mu h) - rho(p)_lambda(rho^(alpha,beta)(q)_mu h), which does not cancel and which nothing forces to be of the form [some element, h]. Lemma 6.2 proves only that the twisted triple is itself a cocycle, while Theorems 6.5 and 6.6 need the stronger, unproven statement that the two cocycles are equivalent exactly when the undefined difference is 'zero.' Since no basepoint in H^2_nab is specified and no zero cocycle is identified, the condition W(alpha,beta)=0 cannot be checked as written. The classification in Theorem 5.7 does not depend on this subtraction, but the inducibility theorem and the Wells exact sequence do.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a cohomological and homotopical framework for Nijenhuis Lie conformal algebras, i.e., Lie conformal algebras (L, [·_λ·]) equipped with a Nijenhuis operator N. Section 3 defines a cochain complex for such algebras (a total complex combining the Chevalley–Eilenberg-type complex of L with the Nijenhuis-operator complex), connects it to the formal deformation theory of N, and upgrades it to arbitrary Nijenhuis representations. Section 4 introduces 2-term Nijenhuis L∞-conformal algebras and claims that skeletal ones are classified by third cohomology classes of Nijenhuis Lie conformal algebras, while strict ones are in bijection with crossed modules of Nijenhuis Lie conformal algebras. Section 5 defines non-abelian 2-cocycles (χ, ρ, Φ) attached to extensions 0 → HQ → ER → LN → 0 and states (Theorem 5.7) a bijection between equivalence classes of non-abelian extensions and the set H^2_nab(LN, HQ); the abelian case is treated as a special case (Theorem 5.8). Section 6 studies the inducibility of pairs (α, β) ∈ Aut(HQ) × Aut(LN), introduces a Wells-type map W : Aut(HQ) × Aut(LN) → H^2_nab(LN, HQ), and claims that (α, β) is inducible if and only if W(α, β) vanishes, together with a Wells exact sequence (Theorems 6.5 and 6.6).","tokens_in":33119,"tokens_out":35500,"duration_ms":319979,"significance":"If the main theorems were correct, Nijenhuis Lie conformal algebras would inherit the full standard apparatus: deformation obstructions (Theorem 3.9), homotopical classifications (Theorems 4.7 and 4.10), non-abelian extension classification (Theorem 5.7), and the automorphism inducibility criterion with an associated exact sequence (Theorems 6.5 and 6.6). The paper is largely a systematic adaptation of known constructions (Frégier's non-abelian cohomology [13]; Sahoo–Das and Baez–Lauda for 2-term L∞-conformal algebras; Wells for automorphisms), and its main virtue is explicitness: concrete cochain formulas, detailed cocycle identities in Lemma 5.4, and a direct construction of the extension from a cocycle in the proof of Theorem 5.7. I find Sections 3 and 4 plausible, with the non-abelian classification of Theorem 5.7 recoverable after a small repair to Definition 5.5. The load-bearing problem is concentrated in Section 6: the stress-test concern about subtracting non-abelian 2-cocycles is confirmed by direct inspection of Definition 6.3 and Eq.","major_comments":[{"comment":"Definition 6.3 defines the Wells map by W(α, β) = [(χ^{(α,β)}, ρ^{(α,β)}, Φ^{(α,β)}) − (χ, ρ, Φ)] as an element of H^2_nab(LN, HQ). This is not well-defined. H^2_nab is introduced in Definition 5.6 as the set of equivalence classes of non-abelian 2-cocycles; the paper never equips this set with a group structure, a subtraction, or a zero element. Furthermore, the triple obtained by subtracting two non-abelian 2-cocycles is never shown to be a non-abelian 2-cocycle: Lemma 6.2 proves only that the twisted triple (χ^{(α,β)}, ρ^{(α,β)}, Φ^{(α,β)}) is itself a cocycle. A direct check of Eq. (29) shows the obstruction: subtracting the identity (29) for the twisted cocycle from the identity for the original cocycle leaves cross terms such as ρ^{(α,β)}(p)_λ(ρ(q)_μ h) − ρ(p)_λ(ρ^{(α,β)}(q)_μ h), which do not cancel and are not forced to be of the form [x_{λ+μ} h]_H. Consequently 'W(α, β) = 0' has no defined meaning as written. The proof of Theorem 6.5 silently replaces 'W(α, β) = 0' with the different statement that (χ^{(α,β)}, ρ^{(α,β)}, Φ^{(α,β)}) and (χ, ρ, Φ) are equivalent in the sense of Definition 5.6, and Proposition 6.4 repeats the same subtraction for cocycles coming from two different sections. Since the Wells map and its vanishing are the advertised content of Section 6, Theorems 6.5 and 6.6 are unsupported as stated. The paper's own argument suggests the local fix: regard H^2_nab as a pointed set with basepoint the class of (χ, ρ, Φ), set W(α, β) = [(χ^{(α,β)}, ρ^{(α,β)}, Φ^{(α,β)})], and read 'W(α, β) = 0' as equality with that basepoint; the exact sequence of Theorem 6.6 can then be formulated with 'kernel' interpreted as the preimage of the basepoint. Note that the classification in Theorem 5.7 does not use this subtraction and is not affected by this particular flaw. The abstract's phrase 'second non-abelian cohomology group' should also be corrected to 'set'.","section":"Section 6, Definition 6.3"},{"comment":"Definition 5.5 declares a triple (χ_λ, ρ, Φ) to be a non-abelian 2-cocycle if it satisfies Eqs. (29)–(32), but it omits the conformal sesquilinearity and conformal skew-symmetry of χ (e.g., χ_{−∂−λ}(q, p) = −χ_λ(p, q)) and the analogous sesquilinearity of ρ. Cocycles obtained from a section of an extension via (26)–(28) automatically satisfy these identities, but the converse direction of Theorem 5.7 constructs the bracket [(p, h)_λ(q, k)]_E := ([p_λ q]_L, ρ(p)_λ k − ρ(q)_{−∂−λ} h + χ_λ(p, q) + [h_λ k]_H) on E = L ⊕ H and asserts, without verification, that this is a conformal skew-symmetric λ-bracket; for an arbitrary triple satisfying only (29)–(32), skew-symmetry can fail. The fix is local: add the conformal conditions to Definition 5.5; the converse construction then goes through.","section":"Section 5, Definition 5.5 / Theorem 5.7"},{"comment":"Theorem 5.8 is stated without proof ('Similar to Theorem 5.7 ...'), and the assertion Ext_ab(LN, MQ) ≅ H^2(LN, MQ) requires identifying the equivalence classes of cocycles satisfying (30)–(32) (with trivial bracket on M) with the cohomology of the total complex defined in Section 3.5. This identification is not established: in the abelian case ρ is fixed data of the representation whereas in Section 5 it is part of the cocycle, and the equivalence relations (36)–(37) are not shown to coincide with the coboundary of the total cochain complex, all the more so because the cocycle conditions in Section 5.1 omit the conformal skew-symmetry built into the cochains of Section 3.1. Please supply the proof or a comparison of the two cohomology theories; as it stands, the abelian classification does not follow from Theorem 5.7 as written.","section":"Section 5.1, Theorem 5.8"}],"minor_comments":[{"comment":"The homomorphism condition is stated inconsistently: Definition 2.3 requires ψ ∘ N = N′ ∘ ψ, while the paragraph before Definition 3.11 writes ψ ∘ N′ = N ∘ ψ. The two conditions are incompatible in general, and the former is the one used in the rest of the paper.","section":"Definition 2.3 vs. Section 3.5"},{"comment":"Equation (31) has an unbalanced parenthesis ('− Q((ρ(p)_λ h)') and the placement of the Q-terms should be rewritten to match the calculation in the proof; as printed, the identity is hard to parse and verify.","section":"Lemma 5.4, Eq. (31)"},{"comment":"The equivalence of deformations in Definition 3.5 fixes the first-order term of ψ_t to be t[p_λ −] for a single p ∈ L, whereas the standard notion allows an arbitrary first-order term ψ_1 ∈ C^1(L, L). With this restricted notion, Theorem 3.6's claim that the cohomology class of N_1 depends only on the equivalence class of the deformation is established only for the restricted family.","section":"Definition 3.5, Theorem 3.6"},{"comment":"In the displayed abelian version of Eq. (32) in Section 5.1, the term 'Q[p_λ q]_L' should evidently be 'Φ([p_λ q]_L)': as printed, Q (a map on M) is applied to the L-element [p_λ q]_L.","section":"Section 5.1, abelian Eq. (32)"},{"comment":"The Wells exact sequence is stated without proof; even after the pointed-set reformulation of Major Comment 1, the exactness assertions need at least a brief argument.","section":"Section 6, Theorem 6.6"},{"comment":"Reference [27] contains a duplicated fragment: 'Mathematical Society, 108(2), pp.293-312.' appears twice at the end of the entry.","section":"References"},{"comment":"The symbols H^2_NL, H^2(LN, MQ), and H^2_nab denote three different objects and should be distinguished by notation; the abstract also calls H^2_nab a 'group' although Definition 5.6 defines only a set.","section":"Abstract and notation"}],"recommendation":"major_revision","confidential_remarks":"The stress test supplied to me is confirmed by direct reading: Definition 6.3's subtraction of non-abelian 2-cocycles is ill-defined, and the flaw propagates into Proposition 6.4 and Theorems 6.5–6.6. I nevertheless regard this as a major but local defect: a pointed-set reformulation of the Wells map (basepoint = class of the original cocycle; W(α, β) = class of the twisted cocycle) makes Theorem 6.5 provable along the paper's own lines, and Definition 5.5 and Theorem 5.8 admit small local repairs. I therefore recommend major revision rather than rejection. Editorial notes for the editor: the bibliography includes five self-citations ([1]–[6]), and the closest prior work, Das–Sen [12] on 2-term averaging L∞-algebras and non-abelian extensions of Nijenhuis Lie algebras, is cited but not discussed in the introduction; the author should delimit the novelty relative to [12] explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead the Asif paper on Nijenhuis Lie conformal algebras. The main thing to know: Sections 3–5 contain a decent amount of honest, useful translation work, but Section 6's central construction is not well-defined, and the Wells-type results collapse as stated.\n\nWhat's actually new: the cohomology complex C^*_{NL}, the skeletal and strict 2-term Nijenhuis L∞-conformal algebra correspondences, and the non-abelian extension classification via H^2_nab. The paper does a thorough job of following the established templates from Lie conformal algebras and Nijenhuis Lie algebras. The deformation theory in Section 3 is coherent, and the crossed-module correspondence in Section 4 is spelled out. Theorem 5.7 (extension classification) is the strongest part; it does not rely on the broken Wells map.\n\nThe problem is Definition 6.3. The Wells map is declared as W(α,β) = [(χ^(α,β),ρ^(α,β),Φ^(α,β)) - (χ,ρ,Φ)]. But H^2_nab is a quotient set of non-abelian 2-cocycles, not an abelian group, so there is no subtraction operation defined. Lemma 6.2 only proves the twisted triple is itself a cocycle; it never proves the difference is a cocycle. Without that, the condition W(α,β)=0 in Theorem 6.5 is meaningless, and the exact sequence in Theorem 6.6 is not established. This is not a minor lacuna; it is the load-bearing step for the paper's advertised automorphism results.\n\nAdditional slips are minor by comparison: the abstract calls H^2_nab a group, which is stronger than the definition; Eq. (14) has a degree mismatch in the short exact sequence; and the abelian cocycle equation drops a Φ([pλq]L) term that appears in the non-abelian form.\n\nWho gets value: specialists in conformal algebras and Nijenhuis-type operators. If you are working in that area, Sections 3–5 give a useful catalog of results and a template for your own constructions. Section 6 needs a real repair. The author's self-citations are not the issue; the missing derivation is.\n\nMy advice: send it to a knowledgeable referee, but with a clear instruction that Section 6 must be fixed — either the Wells map has to be redefined in a way that makes sense in a pointed set (e.g., via a genuine difference in a cochain complex), or Theorems 6.5 and 6.6 should be reworked. As it stands, the paper is not acceptable in its current form.","headline":"Sections 3-5 are a solid translation exercise, but the Wells map in Section 6 subtracts non-abelian cocycles in a pointed set, so the automorphism results are unsupported.","tokens_in":33713,"tokens_out":3088,"would_cite":false,"duration_ms":28836,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B65","17B10","17B38","17B69","14F35","18N10","18N40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that Nijenhuis Lie conformal algebras support a complete cohomological toolkit: extension classes, deformation obstructions, and automorphism lifts all live in cohomology.","keywords":["Nijenhuis Lie conformal algebra","Nijenhuis L∞-conformal algebra","cohomology","homotopy","non-abelian extension","automorphism inducibility","crossed module"],"falsifier":"Take the non-abelian extension built from a cocycle $(\\chi_\\lambda,\\rho,\\Phi)$ and compute $W(\\alpha,\\beta)$ for one fixed pair $(\\alpha,\\beta)$ using two different sections; if the two resulting classes differ, the map is not well-defined and Theorem 6.5 fails as stated, while if they always agree the section-independence claim is confirmed.","tokens_in":32586,"feed_emoji":"🧩","tokens_out":13727,"duration_ms":120317,"temperature":0.7,"pith_summary":"Lie conformal algebras are algebraic models of operator product expansions, and a Nijenhuis operator on such an algebra is a linear map that manufactures new bracket laws while preserving the structure. The paper tries to establish that Nijenhuis Lie conformal algebras support the full cohomological package familiar from Lie algebras: a cohomology theory that controls formal deformations, a 2-term homotopy theory matching crossed modules and 3-cocycles, a classification of non-abelian extensions by the second non-abelian cohomology set $H^2_{\\mathrm{nab}}(L_N,H_Q)$, and an obstruction map that decides when a pair of automorphisms lifts to an automorphism of an extension. If correct, extension-theoretic and automorphism-theoretic questions about these deformed conformal algebras reduce to cohomological computations, just as they do for ordinary Lie algebras and groups.","feed_headline":"Extensions of Nijenhuis conformal algebras classified by cohomology","feed_subtitle":"A single obstruction decides which automorphism pairs lift, tying deformations, extensions, and homotopy together.","key_machinery":"The load-bearing objects are four. First, the total cochain complex $\\{C^*_{NL}(L,M),d_{NL}\\}$ combines the Lie conformal coboundary $\\delta$ with the Nijenhuis coboundary $d_N$ through a comparison map $\\xi$; it carries the deformation and obstruction classes. Second, a non-abelian 2-cocycle is a triple $(\\chi_\\lambda,\\rho,\\Phi)$ encoding how a chosen section of an extension fails to respect the bracket and the Nijenhuis operator; it is the datum from which the extension is reconstructed. Third, the obstruction map $W:\\mathrm{Aut}(H_Q)\\times \\mathrm{Aut}(L_N)\\to H^2_{\\mathrm{nab}}(L_N,H_Q)$ twists a cocycle by a pair of automorphisms and measures the difference from the original cocycle; its vanishing is the inducibility criterion. Fourth, a homotopy Nijenhuis operator $(N_0,N_1,N_2)$ on a 2-term $\\mathcal{L}_\\infty$-conformal algebra connects the skeletal case to third cohomology classes and the strict case to crossed modules.","core_discovery":"The central claim is that the category of Nijenhuis Lie conformal algebras is rich enough to carry the classical apparatus of extensions and automorphisms. Concretely, the paper proves that equivalence classes of non-abelian extensions $0\\to H_Q\\to E_R\\to L_N\\to 0$ are in bijection with the second non-abelian cohomology set $H^2_{\\mathrm{nab}}(L_N,H_Q)$ (Theorem 5.7), and that a pair $(\\alpha,\\beta)\\in \\mathrm{Aut}(H_Q)\\times \\mathrm{Aut}(L_N)$ is inducible if and only if the obstruction map $W(\\alpha,\\beta)$ vanishes (Theorem 6.5), yielding the exact sequence of Theorem 6.6. Along the way it builds a cohomology $H^*_{NL}$ for these algebras, shows that low-degree cohomology classes obstruct and control deformations of the Nijenhuis operator, and introduces 2-term Nijenhuis $\\mathcal{L}_\\infty$-conformal algebras whose skeletal representatives are classified by 3-cocycles and whose strict representatives correspond to crossed modules.","pith_inferences":["If the pointed-set issue in the definition of $W$ is repaired, the natural reading of Theorem 6.5 is that $W(\\alpha,\\beta)$ equals the class of the original cocycle, and the exact sequence should be stated with $H^2_{\\mathrm{nab}}$ treated as a pointed set.","The strict/skeletal dichotomy suggests that higher-dimensional Nijenhuis conformal structures would be governed by higher non-abelian cohomology; the crossed-module extension of Example 5.3 is a concrete place to test this.","The deformation results give a practical criterion: for a Nijenhuis operator with $H^2_N(L,L)=0$, every finite-order deformation extends, a statement that could be checked on explicit free conformal algebras."],"forward_implications":["Equivalence classes of non-abelian extensions of $L_N$ by $H_Q$ correspond bijectively to elements of $H^2_{\\mathrm{nab}}(L_N,H_Q)$, so classifying extensions reduces to solving the four cocycle identities that define a non-abelian 2-cocycle.","A pair of automorphisms $(\\alpha,\\beta)$ lifts to an automorphism of the extension exactly when $W(\\alpha,\\beta)$ is trivial, and Theorem 6.6 gives the exact sequence linking automorphism groups with $H^2_{\\mathrm{nab}}$.","The cohomology of the Nijenhuis operator controls deformation theory: each finite-order deformation has an obstruction class in $H^2_N(L,L)$, and the deformation extends exactly when that class vanishes.","Skeletal 2-term Nijenhuis $\\mathcal{L}_\\infty$-conformal algebras are classified by triples $(L_N,M_Q,(g,\\tau))$ with $(g,\\tau)\\in H^3_{NL}(L_N,M_Q)$.","Strict 2-term Nijenhuis $\\mathcal{L}_\\infty$-conformal algebras are in one-to-one correspondence with crossed modules of Nijenhuis Lie conformal algebras, and each such crossed module produces a non-abelian extension."],"supporting_citations":[{"why":"It supplies the non-abelian cocycle identities and the reconstruction of an extension from a cocycle used in Section 5.","marker":"[13]"},{"why":"It gives the correspondence between strict 2-term structures and crossed modules that Theorem 4.10 adapts to the conformal setting.","marker":"[8]"},{"why":"It provides the model for 2-term Nijenhuis $\\mathcal{L}_\\infty$-algebras and their non-abelian extensions that Section 4 follows.","marker":"[12]"},{"why":"It supplies the automorphism-inducibility obstruction and the exact sequence idea behind the map $W$ and Theorem 6.6.","marker":"[29]"},{"why":"It provides the abelian version of the inducibility criterion and the section-change argument used in Proposition 6.4.","marker":"[24]"},{"why":"It supplies the cohomology-of-Nijenhuis-operators pattern, including the comparison map $\\xi$, that Section 3 adapts to conformal algebras.","marker":"[9]"},{"why":"It supplies the deformation and obstruction framework underlying Section 3.3.","marker":"[14]"},{"why":"It supplies the graded Lie bracket and Maurer-Cartan perspective through which Nijenhuis operators are characterized.","marker":"[30]"}],"fun_headline_variants":["Obstruction map decides when automorphisms lift in Nijenhuis algebras","Non-abelian extensions classified by second cohomology set","Cohomology ties deformations, extensions, and homotopy of Nijenhuis algebras","2-term Nijenhuis L∞-algebras correspond to crossed modules and 3-cocycles","Wells-type map yields exact sequence for automorphism inducibility"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction assumes that subtracting one non-abelian 2-cocycle from another yields a well-defined element of the second non-abelian cohomology set, even though that set is not an abelian group and has no built-in zero.","fun_headline_variants_meta":{"raw":{"variants":["Obstruction map decides when automorphisms lift in Nijenhuis algebras","Non-abelian extensions classified by second cohomology set","Cohomology ties deformations, extensions, and homotopy of Nijenhuis algebras","2-term Nijenhuis L∞-algebras correspond to crossed modules and 3-cocycles","Wells-type map yields exact sequence for automorphism inducibility"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000267,"raw_usage":{"total_tokens":1613,"prompt_tokens":945,"completion_tokens":668,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":563}},"tokens_in":561,"tokens_out":668,"duration_ms":6419,"temperature":1.0,"reasoning_tokens":563,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:47:19.078405+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the non-abelian extension built from a cocycle $(\\chi_\\lambda,\\rho,\\Phi)$ and compute $W(\\alpha,\\beta)$ for one fixed pair $(\\alpha,\\beta)$ using two different sections; if the two resulting classes differ, the map is not well-defined and Theorem 6.5 fails as stated, while if they always agree the section-independence claim is confirmed.","supporting_citations":[{"cited_title":"Fr´ egier, Non-abelian cohomology of extensions of Lie algebras as Deligne groupoid, J","cited_arxiv_id":null,"evidence_quote":"It supplies the non-abelian cocycle identities and the reconstruction of an extension from a cocycle used in Section 5."},{"cited_title":"and Lauda, A.D., 2004","cited_arxiv_id":null,"evidence_quote":"It gives the correspondence between strict 2-term structures and crossed modules that Theorem 4.10 adapts to the conformal setting."},{"cited_title":"and Sen, S., 2024","cited_arxiv_id":null,"evidence_quote":"It provides the model for 2-term Nijenhuis $\\mathcal{L}_\\infty$-algebras and their non-abelian extensions that Section 4 follows."},{"cited_title":"Automorphisms of group extensions","cited_arxiv_id":null,"evidence_quote":"It supplies the automorphism-inducibility obstruction and the exact sequence idea behind the map $W$ and Theorem 6.6."},{"cited_title":"and Yadav, M.K., 2010","cited_arxiv_id":null,"evidence_quote":"It provides the abelian version of the inducibility criterion and the section-change argument used in Proposition 6.4."},{"cited_title":"On the deformation of rings and algebras","cited_arxiv_id":null,"evidence_quote":"It supplies the deformation and obstruction framework underlying Section 3.3."},{"cited_title":"and Richardson, R.W., 1966","cited_arxiv_id":null,"evidence_quote":"It supplies the graded Lie bracket and Maurer-Cartan perspective through which Nijenhuis operators are characterized."}],"review_version":1}