{"id":"3a3d8397-6730-4d95-b87f-8fc6a3638b76","arxiv_id":"2509.09884","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Averaging commutative and cocommutative infinitesimal bialgebras induce special apre-perm bialgebras via a new splitting of perm algebras.","lead":"The paper builds a new bridge between averaging operators on commutative algebras and bialgebras, a class of structures with both multiplication and comultiplication. It defines a new splitting of perm algebras and shows that averaging commutative bialgebras induce special apre-perm bialgebras.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest_assumption identifies the external dualization lemma as the most load-bearing unproved input. I checked this lemma directly from Definition 3.6: the original identities force the dual identities, so the lemma is sound. The remaining potential soft spot is the abbreviated proof of Proposition 4.13, where six of the seven defining equations of a special apre-perm bialgebra are asserted by 'similarly'. This is a genuine exposition gap, but it is not load-bearing because the same conclusion follows from the paper's structural route: Theorem 2.16 gives a double construction of averaging Frobenius commutative algebras, Proposition 4.3 produces the corresponding Manin triple of special apre-perm algebras, and Corollary 4.12 translates that back into the special apre-perm bialgebra with exactly the ϑ, θ of (102). I also spot-checked several of the omitted equations and found them consistent. No ad hominem, no manufactured concern; the central claim appears sound.","tokens_in":31593,"tokens_out":28405,"duration_ms":288758,"concrete_test":"Independently expand equations (85)–(90) in Proposition 4.13 using η=(Q⊗id)∆, θ=−∆P, x◦y=P(x)·_A y, x◁y=−Q(x·_A y), and identities (17)–(19); confirm all signs and tensor factors. Separately, re-derive the perm-algebra dualization lemma from Definition 3.6 in coordinates to confirm the dual representation is (l*, l*−r*).","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the main chain, I find no load-bearing flaw. The reader's flagged external input—the perm-algebra dualization lemma (l,r,V) ↔ (l*, l*−r*, V*)—is correct: from the representation identities l(xy)=l(x)l(y)=l(y)l(x) and r(xy)=r(y)r(x)=r(y)l(x)=l(x)r(y), dualizing gives exactly the required identities for (l*, l*−r*). The most abbreviated point is Proposition 4.13, where only (84) is verified and (85)–(90) are dismissed with 'similarly'; however the same statement follows structurally from Theorem 2.16 → Proposition 4.3 → Corollary 4.12, as Remark 4.14 notes. Spot-checking (85), (86) and (89) using η=(Q⊗id)∆, θ=−∆P, x◦y=P(x)y, x◁y=−Q(xy), together with (17)–(19), yields the required equalities. Thus the abbreviated verification is an exposition gap, not a correctness risk.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper lifts the classical construction that an averaging operator on a commutative associative algebra induces a perm algebra (Proposition 1.1) to the bialgebra level. Section 2 introduces representations of averaging commutative algebras, admissible averaging operators, double constructions of averaging Frobenius commutative algebras, and averaging commutative and cocommutative infinitesimal bialgebras; Theorem 2.16 establishes their equivalence. Section 3 introduces a new two-part splitting of perm algebras, the notion of an apre-perm algebra (Definition 3.8), and its special case in which the second multiplication is commutative. It shows that perm algebras with nondegenerate symmetric left-invariant bilinear forms correspond to quadratic special apre-perm algebras (Proposition 3.29), and that admissible averaging commutative algebras induce special apre-perm algebras (Proposition 3.22). Section 4 defines Manin triples for these structures and special apre-perm bialgebras, proves their equivalences (Theorem 4.11, Corollary 4.12), and shows that every averaging commutative and cocommutative infinitesimal bialgebra gives rise to a special apre-perm bialgebra (Proposition 4.13).","tokens_in":31862,"tokens_out":2377,"duration_ms":31605,"significance":"If correct, the paper provides a coherent bialgebra counterpart of the averaging-to-perm induction, using a genuinely new splitting of perm algebras rather than the pre-perm splitting. The main structural results are stated with explicit formulas and are connected in a useful diagram (103), relating averaging bialgebras to Manin triples of special apre-perm algebras and then to Manin triples of Lie algebras with commutative 2-cocycles. The paper is largely self-contained, and the central chain Theorem 2.16 → Proposition 4.3 → Proposition 4.13 is checkable; the computations in Sections 3 and 4 are detailed. The main external input, the representation dualization lemma for perm algebras cited from [27,35], is standard and is correctly applicable in the stated setting, as I verified.","major_comments":[],"minor_comments":[{"comment":"The proof of Proposition 3.12 states that the converse direction follows by 'a similar argument'. Since this equivalent characterization is used later (e.g., in Corollary 3.21 and Proposition 3.37), the omitted verification would be helpful. At minimum, indicate explicitly how Lemma 3.10 supplies the converse.","section":"§3.2, Proposition 3.12"},{"comment":"In the proof of Proposition 4.13, only equation (84) is verified; equations (85)–(90) are dismissed with 'Similarly'. The structural route via Remark 4.14 assures the reader that the claim is sound, but for a self-contained proof the remaining verifications should be included or the correspondence with the ten compatibility equations in Lemma 4.10 should be tabulated explicitly.","section":"§4.2, Proposition 4.13"},{"comment":"The term 'invariant bilinear form' for special apre-perm algebras is defined by (57). This is a different use of 'invariant' from the usual left-invariance (24) on perm algebras. To avoid ambiguity, consider calling it '◁-invariant' or explicitly contrasting the two notions where they are first used.","section":"§3.3, Definition 3.28 and Proposition 3.29"},{"comment":"There are a few typographical and wording slips, e.g., 'a averaging commutative and cocommutative infinitesimal bialgebra' in the abstract of Section 4 and 'cocomutative' in the introduction. These are harmless but should be corrected in the final version.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The paper's novelty is concentrated in the new splitting of perm algebras and the induced bialgebra theorem. The authors draw heavily on their own earlier work on Rota-Baxter and perm bialgebras, but the citations are appropriate and the present construction is not a routine translation: the switch from antisymmetric to symmetric left-invariant forms is a substantive change. I see no correctness issue that would warrant major revision or rejection; the remaining concern is purely expository completeness in the proofs of Proposition 3.12 and Proposition 4.13."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a solid, careful paper. The genuinely new object is the apre-perm algebra—a different splitting of perm algebras than the pre-perm splitting—and the paper shows it is the right underlying structure for perm algebras with nondegenerate symmetric left-invariant forms. The chain from averaging Frobenius algebras to special apre-perm bialgebras is coherent, and the main equivalences are stated with enough detail to be checked.\n\nThe new content: the definition of apre-perm algebras via the representation (L*_•, -R*_◁), the induction theorem from averaging commutative and cocommutative infinitesimal bialgebras to special apre-perm bialgebras (Prop. 4.13), and the Manin-triple characterization. The paper supplies explicit formulas throughout, and the examples in Sections 2 and 3 are genuinely helpful. The structural results are not just abstract: the computations in Section 3 and 4 are detailed enough to follow, and the equivalences in Theorems 2.16, 4.4, and 4.11 are plausible and mostly verified.\n\nThe soft spots are mostly exposition. Several verifications are dismissed as \"straightforward\" or \"similarly\"—most notably parts of Proposition 3.12 and equations (85)-(90) in Proposition 4.13. I spot-checked (85), (86), and (89) and they work out; the missing checks are an exposition gap, not a correctness risk. The representation dualization lemma from [27,35] is load-bearing, but it is a standard published result and the stress test confirms it is correct. The paper also leans heavily on the same group's earlier framework for Manin triples and perm algebra representations; that is a citation-pattern choice, not a flaw, since those results are published and the new contribution sits cleanly on top of them.\n\nThis paper is for people working in operated algebras, bialgebra theory, and Manin-triple classifications. It will not change the field, but it gives a complete and checkable structural result that connects averaging Frobenius algebras to perm algebras with left-invariant forms. I would send it to a serious referee; it deserves a careful reading and will likely survive with minor revisions.\n\nRecommendation: engage with it, and let it through peer review.","headline":"Sound structural paper: new apre-perm splitting and Manin-triple equivalence; main arguments check out, with only exposition gaps in verifications.","tokens_in":32331,"tokens_out":1424,"would_cite":true,"duration_ms":15946,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17A36","17A40","17B10","17D25","18M70"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that every averaging commutative and cocommutative infinitesimal bialgebra gives rise to a special apre-perm bialgebra, lifting the classical construction of a perm algebra from an averaging operator to the bialgebra","keywords":["averaging operator","perm algebra","infinitesimal bialgebra","apre-perm algebra","Manin triple","Frobenius algebra","O-operator","embedding tensor"],"falsifier":"Find a finite-dimensional perm algebra and a triple (l, r, V) for which (l*, l* − r*, V*) is not a representation, or construct an averaging commutative and cocommutative infinitesimal bialgebra whose images under formulas (63) and (102) fail to satisfy one of equations (84)-(90); either would refute the central induction theorem.","tokens_in":31514,"feed_emoji":"🧮","tokens_out":3953,"duration_ms":33050,"temperature":0.7,"pith_summary":"The paper lifts the classical fact that an averaging operator on a commutative associative algebra produces a perm algebra to the level of bialgebras. It introduces averaging commutative and cocommutative infinitesimal bialgebras as the bialgebra counterpart of averaging commutative associative algebras, and shows they are equivalent to double constructions of averaging Frobenius commutative algebras. To capture the induced bialgebra structure, the paper defines apre-perm algebras via a new splitting of perm multiplications into two operations, and special apre-perm algebras when the second operation is commutative. The main result is that an averaging commutative and cocommutative infinitesimal bialgebra induces a special apre-perm bialgebra, with explicit formulas for the new operations and comultiplications. If correct, this gives a coherent bialgebra-level analogue of the averaging-to-perm correspondence, connecting Frobenius algebras, Manin triples, and perm algebras.","feed_headline":"New splitting turns averaging bialgebras into apre-perm bialgebras","feed_subtitle":"The familiar averaging-to-perm algebra construction lifts to bialgebras, linking Frobenius algebras and Manin triples.","key_machinery":"The paper introduces apre-perm algebras: a two-operation splitting (▷, ◁) of a perm algebra whose sum is the perm multiplication and whose dual operators (L*_▷ + R*_◁, −R*_◁) form a representation of the perm algebra on the dual space. A special apre-perm algebra requires ◁ to be commutative. The carrying mechanism is the equivalence between representations of a perm algebra and representations on its dual, used to relate the adjoint representation to (L*_▷ + R*_◁, −R*_◁) and hence to define the splitting. The paper also uses dual a-O-operators and strong special dual a-O-operators to connect compatible apre-perm structures to bilinear forms and Manin triples.","core_discovery":"The central claim is Proposition 4.13: given an averaging commutative and cocommutative infinitesimal bialgebra (A, ·_A, ∆, P, Q), the multiplications defined by x ▷_A y = P(x)·_A y + Q(x·_A y) and x ◁_A y = −Q(x·_A y), together with comultiplications ϑ(x) = (Q⊗id)∆(x) + ∆(Px) and θ(x) = −∆(Px), form a special apre-perm bialgebra. This is proved by showing that the double construction of an averaging Frobenius commutative algebra yields a Manin triple of special apre-perm algebras, and that this Manin triple is equivalent to the bialgebra structure. In particular, the paper shows that perm algebras equipped with nondegenerate symmetric left-invariant bilinear forms correspond exactly to quad","pith_inferences":["The paper's mechanism suggests that other induced bialgebra structures from averaging operators might be obtained by analogous dual-representation splittings; for instance, averaging Lie algebras with symmetric invariant forms could induce special variants of pre-Lie bialgebras.","The explicit formulas (63) and (102) provide a testable recipe for constructing examples: any concrete averaging commutative algebra with a compatible infinitesimal bialgebra structure can be checked directly against equations (84)-(90).","The one-to-one correspondence between quadratic special apre-perm algebras and perm algebras with symmetric left-invariant forms may offer a route to classify symmetric left-invariant perm algebra structures in low dimensions.","The paper leaves open whether the induced special apre-perm bialgebra is functorial in the input averaging bialgebra."],"forward_implications":["Every averaging commutative and cocommutative infinitesimal bialgebra carries a special apre-perm bialgebra structure given explicitly by formulas (63) and (102).","Double constructions of averaging Frobenius commutative algebras are equivalent to Manin triples of special apre-perm algebras, which in turn correspond to Manin triples of perm algebras with nondegenerate symmetric left-invariant bilinear forms.","Perm algebras with nondegenerate symmetric left-invariant bilinear forms are in one-to-one correspondence with quadratic special apre-perm algebras.","Special apre-perm algebras give rise to both a pre-Lie algebra and an anti-pre-Lie algebra under compatible operations, with the anti-pre-Lie structure matching the commutative 2-cocycle of the sub-adjacent Lie algebra.","Manin triples of perm algebras with symmetric left-invariant forms produce Manin triples of Lie algebras with commutative 2-cocycles via the sub-adjacent Lie algebra construction."],"fun_headline_variants":["Averaging bialgebras spawn apre-perm bialgebras","New splitting turns averaging bialgebras into apre-perm","Apre-perm bialgebras emerge from averaging construction","Averaging Frobenius algebras produce apre-perm bialgebras","Manin triples link averaging bialgebras to apre-perm"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The results rely on a cited lemma, not proved in the paper, that a triple (l, r, V) is a representation of a perm algebra if and only if (l*, l* − r*, V*) is a representation; if this duality fails in the setting of finite-dimensional perm algebras, the definition of apre-perm algebras and the induced bialgebra theorem collapse.","fun_headline_variants_meta":{"raw":{"variants":["Averaging bialgebras spawn apre-perm bialgebras","New splitting turns averaging bialgebras into apre-perm","Apre-perm bialgebras emerge from averaging construction","Averaging Frobenius algebras produce apre-perm bialgebras","Manin triples link averaging bialgebras to apre-perm"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000195,"raw_usage":{"total_tokens":1242,"prompt_tokens":838,"completion_tokens":404,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":582,"completion_tokens_details":{"reasoning_tokens":316}},"tokens_in":582,"tokens_out":404,"duration_ms":5016,"temperature":1.0,"reasoning_tokens":316,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T18:30:55.324379+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a finite-dimensional perm algebra and a triple (l, r, V) for which (l*, l* − r*, V*) is not a representation, or construct an averaging commutative and cocommutative infinitesimal bialgebra whose images under formulas (63) and (102) fail to satisfy one of equations (84)-(90); either would refute the central induction theorem.","supporting_citations":[],"review_version":1}