{"id":"b248a066-7758-46c5-8a01-666b51fda8b1","arxiv_id":"1908.09643","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors build Hopf algebroids A_sigma with arbitrary base rings L from coefficients sigma satisfying a dynamical Yang-Baxter condition, and provide quasigroup examples that are not weak Hopf algebras.","lead":"This paper constructs Hopf algebroids, a generalization of Hopf algebras, starting from an arbitrary base algebra and a collection of coefficients that satisfy certain conditions. The construction widens the known class of examples and gives Hopf algebroids that are not weak Hopf algebras even when the underlying index set is finite.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The antipode construction depends on Claim 4.3, whose proof omits generators (1)-(3),(5) and asserts the key generator-(4) conclusion without computation; Theorem 5.2 is then deferred to [32].","rationale":"The paper presents a coherent generalization of the known construction, and the bialgebroid theorems (Theorem 2.2 and Theorem 3.2) have plausible proofs, though with several omitted verifications (e.g., the claim that Δ(Iσ)⊂I2 is not proved). The most load-bearing point is the existence of the antipode S, because without it Theorem 5.2 has nothing to apply. That existence is exactly Claim 4.3, and the proof of Claim 4.3 is visibly incomplete: only one generator type is treated, and the final step for that type is asserted rather than demonstrated. The reader's weakest_assumption focused on the existence of rigid σ and the quasigroup example; I agree that this is also a gap, but the specific unverified step in Claim 4.3 is even closer to the central claim, since it is the only in-paper argument connecting rigidity to the antipode. Because the gap may be fillable by a straightforward computation and there is no indication of an actual falsehood, the conditional verdict is appropriate rather than rejection. The concrete test above would settle whether the omitted computation goes through.","tokens_in":11979,"tokens_out":16208,"duration_ms":138970,"concrete_test":"Verify Claim 4.3 for the QG5 quasigroup example of Section 7 by computing, in the quotient Aσ with the x_ab,y_ab from Theorem 6.1, the image S(a) of each generator a of Iσ of types (1)-(5); for generator (4), carry out the reduction from S(a) to zero using only the rigidity identities of Definition 4.1 and record the intermediate step that bridges the two displayed forms. If all images vanish, the antipode exists in this case; the same check on a random small algebra L would reveal whether the omitted computation hides an assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Claim 4.3 (S(Iσ)={0}) is the load-bearing step that turns the bialgebroid Aσ into a Hopf algebroid via Proposition 4.2(2). The proof only says 'We give the proof only for the case that a is the generator (4)' and does not verify generators (1)-(3) and (5). For generator (4), after deriving the identity from (4.2), the paper jumps to 'Therefore, by virtue of (4.1), S(...)=0.' The expression for S of generator (4) displayed in the text does not visibly follow from that identity: the former is ∑(L^{-1})_{xb}(L^{-1})_{yd}(1⊗σ^{xy}_{ac}) minus ∑(L^{-1})_{ax}(L^{-1})_{cy}(σ^{bd}_{xy}⊗1), while the derived identity involves ∑(L^{-1})_{p a}(L^{-1})_{q c}(σ^{xy}_{ac}⊗1) minus ∑(L^{-1})_{b x}(L^{-1})_{d y}(1⊗σ^{bd}_{pq}). Bridging these two forms requires additional rigidity identities that are not stated. Moreover, Theorem 5.2's proof of the Hopf algebroid axioms (5.1), (5.2), and the compatibility equations is deferred to [32, Theorem 3.9] with no verification in this setting. If Claim 4.3 fails, S does not exist and Aσ is only a bialgebroid.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for an arbitrary k-algebra L equipped with automorphisms T_alpha indexed by a group G, a quotient algebra A_sigma generated by L \\otimes_k L^op and symbols L_ab, (L^{-1})_ab, with relations determined by elements sigma^{ab}_{cd} \\in L. It states that under condition (2.11) the algebra is a left bialgebroid, under condition (3.8) it is a right bialgebroid and also a left bialgebroid, and that if sigma is rigid and satisfies (3.8), then A_sigma is a Hopf algebroid with antipode S. The paper proposes sufficient algebraic conditions (1)-(5) in Theorem 6.1 for rigidity, and Section 7 gives quasigroup-based examples intended to yield Hopf algebroids that are not weak Hopf algebras.","tokens_in":12287,"tokens_out":10319,"duration_ms":88681,"significance":"If the deferred proofs can be supplied, the paper would provide a broad and explicit construction of Hopf algebroids with arbitrary base algebra L, generalizing earlier constructions based on M_H(R). The computations that are actually shown, such as Proposition 3.3, Proposition 2.4, and the partial verification in Proposition 2.3, are internally consistent, and the construction is not circular: rigidity is additional data beyond the bialgebroid presentation, and the conditions in Theorem 6.1 are explicit invertibility conditions. The main weakness is that several load-bearing results are either deferred to the authors' previous paper [32] or left with incomplete proofs, so the central claims are not yet established in the submitted manuscript.","major_comments":[{"comment":"The proof of (1) implies (2) in Proposition 4.2 is incomplete at two load-bearing points. First, Claim 4.3 is not proved: the proof treats only generator (4), with no verification for generators (1)-(3) and (5), and for generator (4) the final equality to 0_{A_sigma} does not follow from the displayed identity derived from (4.2), because the tensor factors and index labels in the two expressions are not matched by any stated rigidity identity. Second, even if Claim 4.3 were true, the induced anti-homomorphism S : A_sigma -> A_sigma would still not be shown to be bijective, although Proposition 4.2(2) requires an anti-automorphism; surjectivity and injectivity are not addressed. Since the existence of S is the bridge between rigidity and Hopf algebroid structure, this gap is central.","section":"§4, Proposition 4.2 and Claim 4.3"},{"comment":"The proof of Theorem 5.2 is deferred with the sentence \"The proof of this theorem is similar to that of Theorem 3.9 in [32]\". The axioms (5.1), (5.2), the two compatibility equations preceding Definition 5.1, and the invertibility of the map S_{A \\otimes_{L'} A} are not checked for the present algebra A_sigma. A Hopf algebroid structure is the central claim of the paper, so this deferral is not sufficient for a self-contained proof.","section":"§5, Theorem 5.2"},{"comment":"The proof of Theorem 6.1 is likewise deferred to [32, Theorem 4.1] with the phrase \"proved in much the same way\", and the formulas for x_ab and y_ab are asserted without derivation. Because Theorem 6.1 is the only result in the paper that produces rigid sigma from checkable conditions, the construction of the antipode for the examples depends on an unproved theorem.","section":"§6, Theorem 6.1"},{"comment":"The quasigroup example is the only concrete source of rigid sigma, but Theorem 7.3 states that the verification is \"straightforward\" and refers to [22] and [32, Section 4] rather than demonstrating (3.8) and conditions (1)-(5). The formula defining sigma^{ab}_{cd} is intricate, and without at least an outline of the verification the existence claim for Hopf algebroids that are not weak Hopf algebras is not independently checkable from the manuscript.","section":"§7, Theorem 7.3 and Corollary 7.4"}],"minor_comments":[{"comment":"In the proof of Proposition 2.3, the case v = L_ab is omitted with the remark that the proof is easy; the computation should either be included or the omitted cases listed explicitly.","section":"§2, Proposition 2.3"},{"comment":"In Claim 4.3, the letters a, b, c, d are used both for the free indices of generator (4) and as summation indices in equation (4.2); this clash makes the derivation hard to follow and should be fixed.","section":"§4, Claim 4.3"},{"comment":"Section 5 introduces a ring L' \"isomorphic to the opposite ring L^op\", duplicating the notation L' = L^op used in Section 3; the two uses should be distinguished.","section":"§5, Definition 5.1"},{"comment":"The proof of Proposition 2.4 states that epsilon(a) = 0 for generator (4) \"by virtue of (2.11)\" without displaying the computation; a one-line verification would improve readability.","section":"§2, Proposition 2.4"},{"comment":"The abstract and introduction promise \"for arbitrary algebras L\", but the construction requires L to carry automorphisms T_alpha satisfying (2.9); the scope should be stated more precisely.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The authors' previous works [32] and [22] carry a substantial part of the proofs, and [22] is a master's thesis in Japanese. If these results are not reproduced or at least stated with complete proofs in an appendix, the paper is not self-contained. The editor may wish to check that [32] is readily accessible and that [22] is not essential for the main construction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new part here is the move from base algebras of functions on finite sets to an arbitrary k-algebra L, plus the explicit conditions (2.11) and (3.8) that make A_sigma a left and right bialgebroid, the sufficient rigidity conditions in Theorem 6.1, and the finite quasigroup examples in Corollary 7.4 that are not weak Hopf algebras. That is a real extension of Shibukawa's earlier construction, and if it holds together it gives people a broader toolkit.\n\nThe bialgebroid half is clean. The definitions are consistent, the epsilon and epsilon-prime computations check out, and Proposition 3.3 is a correct and useful observation that (3.8) implies (2.11). The paper is also honest about what it is not proving.\n\nThe soft spots are all on the Hopf side, and they are substantial. Claim 4.3 is the load-bearing step: it defines the antipode, and without it A_sigma is only a bialgebroid. The proof explicitly covers only generator (4) and skips generators (1), (2), (3), and (5). Even for generator (4), the displayed S-expression does not visibly follow from the preceding identity; bridging the two forms requires extra rigidity identities that are not stated. The stress-test note is right about this gap. Then Theorem 5.2, the main Hopf algebroid statement, is deferred to [32] with \"the proof is similar,\" and Theorem 6.1 is also deferred to [32]. Theorem 7.3, the quasigroup example, is asserted as \"straightforward\" with no actual verification. None of this is nonsense, and the construction is probably correct, but as submitted the paper is not self-contained where it matters most.\n\nThe citation pattern is worth noting: the paper leans on [32], written by one of the authors, for three major theorems. That is not automatically a flaw, especially since [32] is published, but it makes refereeing hard when the new paper's contribution depends on unstated details from that earlier paper.\n\nWho is this for? People working on dynamical Yang-Baxter maps, face algebras, or Hopf algebroids. It deserves a serious referee, because the construction is meaningful and the examples are concrete.\n\nMy recommendation: send it to review, and require as major revision that Claim 4.3 be proved in full, that the author state precisely which results in [32] are being invoked and why they apply, and that the quasigroup verification be shown rather than asserted. With those changes I would be comfortable citing it.","headline":"A plausible and useful generalization of the earlier finite-H construction, but the Hopf-algebroid proofs are deferred to [32] and the key antipode claim has a real gap; worth refereeing with major revision.","tokens_in":12807,"tokens_out":3255,"would_cite":true,"duration_ms":104044,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16T20","20G42","81R50","20N05","20N10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Rigid families of algebra elements σ satisfying condition (3.8) turn the quotient algebra A_σ into a Hopf algebroid with base L.","keywords":["Hopf algebroids","bialgebroids","rigid sigma","antipode","quasigroups","dynamical Yang-Baxter maps","weak Hopf algebras","FRT construction"],"falsifier":"For the quasigroup QG5, write out $σ^{{ab}}$_{cd} explicitly using the formula in Section 7, compute the matrices Q, Q′, Q′′, Q′′′ of Theorem 6.1, and verify the four rigidity equalities of Definition 4.1; if any of those equalities fails, the rigidity theorem is false. As a broader test, search for any σ satisfying (3.8) such that the anti-automorphism S fails to satisfy the Hopf algebroid axiom (5.2); such a σ would disprove Theorem 5.2.","tokens_in":11800,"feed_emoji":"🧮","tokens_out":8197,"duration_ms":215150,"temperature":0.7,"pith_summary":"The paper presents a general construction of Hopf algebroids whose base ring is an arbitrary k-algebra L, using elements $σ^{{ab}}$_{cd} in L that satisfy a short list of algebraic conditions. The main theorem shows that when σ is rigid and satisfies condition (3.8), the quotient algebra A_σ carries an anti-automorphism S and is a Hopf algebroid. The paper also proves that condition (3.8) alone makes A_σ both a left and a right bialgebroid, so rigidity is the only extra ingredient needed for the antipode. Concrete rigid σ are produced from finite quasigroups, and for non-separable coefficient algebras the resulting Hopf algebroids are not weak Hopf algebras. This matters because it extends the known construction beyond base rings of function-type and separates Hopf algebroids from weak Hopf algebras.","feed_headline":"Rigid sigma turns an algebra into a Hopf algebroid","feed_subtitle":"A short list of conditions on σ supplies the antipode, and quasigroup examples escape weak Hopf algebras.","key_machinery":"The machinery is the algebra A_σ, defined as the quotient of the free algebra generated by symbols L_{ab}, ($L^{{-1}}$)_{ab}, and L ⊗_k L^op by relations encoding σ. The load-bearing notion is rigidity of σ: for each a, b there exist elements x_{ab}, y_{ab} in A_σ satisfying the four equalities in Definition 4.1. Rigidity is equivalent to the existence of the antipode S (Proposition 4.2), and the paper's sufficient conditions (Theorem 6.1) turn rigidity into the invertibility of certain matrices Q, Q′, Q′′, Q′′′ built from σ, providing explicit formulas for x_{ab} and y_{ab}.","core_discovery":"The paper's central claim is that a family σ = ($σ^{{ab}}$_{cd}) in an arbitrary algebra L satisfying the identity (3.8) and a rigidity condition yields a Hopf algebroid A_σ over L. Rigidity is exactly the existence of elements x_{ab}, y_{ab} in A_σ that behave like inverses of the generators L_{ab}, and Proposition 4.2 shows this is equivalent to having a k-algebra anti-automorphism S on A_σ, the antipode. Theorem 5.2 then states that A_σ together with S is a Hopf algebroid, and Theorem 6.1 reduces rigidity to five explicit invertibility conditions on matrices built from σ, with explicit formulas for x_{ab} and y_{ab}. Section 7 provides a quasigroup-based σ that satisfies all these conditions, and Corollary 7.4 concludes that the resulting Hopf algebroids are not weak Hopf algebras when the base is not separable.","pith_inferences":["The five conditions in Theorem 6.1 resemble dynamical analogues of the Yang-Baxter equation; if σ solves such an equation, rigidity may be equivalent to the existence of a dynamical R-matrix inverse, linking this construction to known integrable models.","The explicit formulas suggest a concrete algorithm: compute the matrices Q, Q′, Q′′, Q′′′ for any candidate σ and test invertibility; this could be used to search for new examples beyond quasigroups, for instance from arbitrary Latin squares or from group action data.","Because the quasigroup verification is only sketched as 'straightforward', a fully written check of the five conditions for QG5 would independently confirm the existence of rigid σ over non-separable bases; if that check failed, the paper's main supply of examples would be in doubt."],"forward_implications":["The construction works for any algebra L, so the previous restriction to base rings of the form M_H(R) is lifted.","Whenever σ is rigid and satisfies (3.8), the antipode S exists and the full Hopf algebroid axioms hold, so rigidity is the complete obstruction to upgrading the bialgebroid.","The quasigroup examples give Hopf algebroids with finite-dimensional function-type bases that are not weak Hopf algebras, so the Hopf algebroid notion is genuinely broader.","The explicit formulas for x_{ab} and y_{ab} in Theorem 6.1 mean that, given a candidate σ, checking the five invertibility conditions is enough to construct the antipode directly."],"supporting_citations":[{"why":"Provides the earlier construction of Hopf algebroids from dynamical Yang-Baxter maps and the proof template for Theorems 5.2 and 6.1.","marker":"[32]"},{"why":"Source of the quasigroup examples in Section 7 and of the assertion that the constructed σ is rigid.","marker":"[22]"},{"why":"Supplies the theorem used to conclude that A_σ is not a weak Hopf algebra when the base ring is not separable.","marker":"[27]"},{"why":"Provides the definition of Hopf algebroid with bijective antipode and the axioms used in Section 5.","marker":"[5]"},{"why":"The notion of rigid σ (cf. Section 4.5) is drawn from this work on quantum dynamical Yang-Baxter equations.","marker":"[7]"}],"fun_headline_variants":["Rigid sigma turns any algebra into a Hopf algebroid","Five matrix conditions build Hopf algebroids","Quasigroup sigma escapes weak Hopf algebras","Antipode via rigidity: new Hopf algebroids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction collapses unless there exists at least one rigid σ satisfying (3.8); the paper's only concrete example is the quasigroup construction whose verification is asserted as 'straightforward' rather than demonstrated.","fun_headline_variants_meta":{"raw":{"variants":["Rigid sigma turns any algebra into a Hopf algebroid","Five matrix conditions build Hopf algebroids","Quasigroup sigma escapes weak Hopf algebras","Antipode via rigidity: new Hopf algebroids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000276,"raw_usage":{"total_tokens":1552,"prompt_tokens":759,"completion_tokens":793,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":375,"completion_tokens_details":{"reasoning_tokens":724}},"tokens_in":375,"tokens_out":793,"duration_ms":8258,"temperature":1.0,"reasoning_tokens":724,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:58:41.279380+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the quasigroup QG5, write out $σ^{{ab}}$_{cd} explicitly using the formula in Section 7, compute the matrices Q, Q′, Q′′, Q′′′ of Theorem 6.1, and verify the four rigidity equalities of Definition 4.1; if any of those equalities fails, the rigidity theorem is false. As a broader test, search for any σ satisfying (3.8) such that the anti-automorphism S fails to satisfy the Hopf algebroid axiom (5.2); such a σ would disprove Theorem 5.2.","supporting_citations":[{"cited_title":"Shibukawa: Hopf algebroids and rigid tensor categor ies associated with dynamical Yang-Baxter maps","cited_arxiv_id":null,"evidence_quote":"Provides the earlier construction of Hopf algebroids from dynamical Yang-Baxter maps and the proof template for Theorems 5.2 and 6.1."},{"cited_title":"Otsuto: Left bialgebroid, rigidity, and Hopf algebr oid","cited_arxiv_id":null,"evidence_quote":"Source of the quasigroup examples in Section 7 and of the assertion that the constructed σ is rigid."},{"cited_title":"Schauenburg: Weak Hopf algebras and quantum groupoi ds","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem used to conclude that A_σ is not a weak Hopf algebra when the base ring is not separable."},{"cited_title":"B¨ ohm and K","cited_arxiv_id":null,"evidence_quote":"Provides the definition of Hopf algebroid with bijective antipode and the axioms used in Section 5."},{"cited_title":"Etingof and A","cited_arxiv_id":null,"evidence_quote":"The notion of rigid σ (cf. Section 4.5) is drawn from this work on quantum dynamical Yang-Baxter equations."}],"review_version":1}