Construction of Hopf algebroids

Youichi Shibukawa (Hokkaido University); Yudai Otsuto (Hokkaido University)

REVIEW 4 major objections 5 minor 38 references

Construction of Hopf algebroids

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Rigid families of algebra elements σ satisfying condition (3.8) turn the quotient algebra A_σ into a Hopf algebroid with base L.

desk verdict 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. read the letter →

arxiv 1908.09643 v1 pith:XX2TVLEZ submitted 2019-08-21 math.RA math.QA

Yudai Otsuto (Hokkaido University) , Youichi Shibukawa (Hokkaido University) This is my paper · ORCID

classification math.RAmath.QA MSC 16T2020G4281R5020N0520N10

keywords HopfalgebroidsbialgebroidsrigidsigmaantipodequasigroupsdynamicalYang-BaxtermapsweakalgebrasFRTconstruction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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}.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

The 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.

Who 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.

My 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.

Referee Report

4 major / 5 minor

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.

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 (4)

[§4, Proposition 4.2 and Claim 4.3] 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.
[§5, Theorem 5.2] 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.
[§6, Theorem 6.1] 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.
[§7, Theorem 7.3 and Corollary 7.4] 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.

minor comments (5)

[§2, Proposition 2.3] 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.
[§4, Claim 4.3] 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.
[§5, Definition 5.1] 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.
[§2, Proposition 2.4] 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.
[Abstract and Introduction] 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.

Circularity Check

1 steps flagged · score 2.0 of 10

No reduction-by-construction; the generalized A_sigma construction is independent, but the key Hopf-algebroid verification is deferred to the same author's prior paper.

other [Theorem 5.2, Section 5; cf. Theorem 6.1, Section 6]
"Theorem 5.2. The algebra A_sigma with the k-algebra anti-automorphism S in Proposition 4.2 (2) is a Hopf algebroid for a rigid sigma satisfying (3.8). The proof of this theorem is similar to that of Theorem 3.9 in [32]."

The final Hopf-algebroid verification for the new arbitrary-base-L construction is not carried out in this paper; the reader is referred to Theorem 3.9 of [32], a paper by the same second author. Since the paper's stated novelty is the generalization from M_H(K) to arbitrary L, citing the earlier theorem without adapting the proof leaves part of the main claim resting on a self-citation. This is not a reduction by construction: A_sigma is explicitly defined by generators and relations, and the bialgebroid and antipode existence are proved here. The step is therefore a minor self-citation/omitted-proof issue rather than a circular derivation.

full rationale

The derivation of A_sigma as a left/right bialgebroid is self-contained: Theorem 2.2 proves that condition (2.11) makes the explicit quotient (2.10) into a left bialgebroid, Theorem 3.2 proves condition (3.8) gives a right bialgebroid, and Proposition 3.3 derives (2.11) from (3.8). The antipode is obtained from the rigidity hypothesis: Proposition 4.2 proves the equivalence between rigidity and the existence of the anti-automorphism S, and Theorem 5.2 then checks the further Hopf-algebroid axioms. No fitted quantity is renamed as a prediction, no uniqueness theorem is imported to force a choice, and no known result is merely relabelled. The only concern is the reliance on [32] for the proof of Theorem 5.2, Theorem 6.1, and the 'straightforward' verification of the quasigroup example in Theorem 7.3; these are self-citations and omitted details, not circular reductions. The proof of Claim 4.3 also checks only generator (4) and asserts the final cancellation, but again this is a gap in presentation rather than circularity. Overall, the central construction is independent of its inputs, so the circularity score is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

There are no fitted numerical parameters; the free data are the algebra L, the automorphisms T_deg, the coefficient family sigma, and, for the antipode, the rigid structure. The main axioms are the two dynamical Yang-Baxter-like conditions and the five rigidity conditions. The formal symbols L_ab and (L^-1)_ab are generators of a free algebra with explicit defining relations, not independent postulates.

assumptions (5)

standard math Associative unital rings, free k-algebra quotient by an ideal, Sweedler notation, and tensor products over L behave as standard background.
The construction in Section 2 builds A_sigma as a quotient of a free k-algebra, and later sections use tensor products over L and Sweedler notation without proof.
domain assumption T_alpha are k-algebra automorphisms of L satisfying T_alpha composed with T_alpha-inverse equals the identity.
Equation (2.9) is assumed for the degree map and is used throughout the commutation relations in the ideal I_sigma.
domain assumption The coefficient family sigma satisfies condition (2.11) for the left bialgebroid and condition (3.8) for the right bialgebroid.
These are the theorem assumptions in Theorem 2.2 and Theorem 3.2; they are not derived from any deeper principle.
domain assumption The rigidity conditions (1)-(5) in Theorem 6.1 hold, including the existence of i*(sigma) and invertible Q matrices.
These conditions guarantee the antipode via the x_ab and y_ab elements. Their existence is assumed and the only concrete verification is deferred for the quasigroup example.
domain assumption If R is not separable with an idempotent Frobenius system, then L isomorphic to R^{|H|} is also not separable, and non-separability of the base implies A_sigma is not a weak Hopf algebra.
Used in Corollary 7.4; relies on [27, Theorem 5.1] and stability of the property under finite products, which is not proved in the paper.

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Pith. "Pith review of Construction of Hopf algebroids." pith.science (2026). https://pith.science/paper/XX2TVLEZ

@misc{pith2026190809643,
  author       = {Pith},
  title        = {Pith review of: Construction of Hopf algebroids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XX2TVLEZ}},
  note         = {Machine review of arXiv:1908.09643}
}

abstract

For arbitrary algebras $L$, we construct Hopf algebroids $A_\sigma$ with base rings $L$ by means of $\sigma^{ab}_{cd}\in L$ satisfying suitable properties.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [32]

Shibukawa: Hopf algebroids and rigid tensor categor ies associated with dynamical Yang-Baxter maps

Y. Shibukawa: Hopf algebroids and rigid tensor categor ies associated with dynamical Yang-Baxter maps. J. Algebra 449 (2016), 408–445

work page 2016

[2] [22]

Otsuto: Left bialgebroid, rigidity, and Hopf algebr oid

Y. Otsuto: Left bialgebroid, rigidity, and Hopf algebr oid. Master thesis (2018), Hokkaido University (in Japanese)

work page 2018

[3] [1]

Abe: Hopf algebras

E. Abe: Hopf algebras. Translated from the Japanese by Hi sae Ki- noshita and Hiroko Tanaka. Cambridge Tracts in Mathematics , 74. Cambridge University Press, Cambridge-New York, 1980. 14

work page 1980

[4] [2]

R. J. Baxter: Partition function of the eight-vertex lat tice model. Ann. Physics 70 (1972), 193–228

work page 1972

[5] [3]

R. J. Baxter: Solvable eight-vertex model on an arbitrar y planar lattice. Philos. Trans. Roy. Soc. London Ser. A 289 (1978), no. 1359, 315–346

work page 1978

[6] [4]

B¨ ohm: An alternative notion of Hopf algebroid

G. B¨ ohm: An alternative notion of Hopf algebroid. Hopf a lgebras in noncommutative geometry and physics, 31–53, Lecture Notes in Pure and Appl. Math., 239, Dekker, New York, 2005

work page 2005

[7] [5]

B¨ ohm and K

G. B¨ ohm and K. Szlach´ anyi: Hopf algebroids with bijective antipodes: axioms, integrals, and duals. J. Algebra 274 (2004), no. 2, 708–750

work page 2004

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V. G. Drinfel’d: Quantum groups. Proceedings of the Inte rnational Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 19 86), 798– 820, Amer. Math. Soc., Providence, RI, 1987

work page 1987