Nondegenerate module categories

Chelsea Walton; Harshit Yadav

REVIEW 3 major objections 4 minor 7 references

Nondegenerate module categories

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that a braided module category is nondegenerate if and only if it is factorizable, extending Shimizu's theorem to module categories.

desk verdict A genuine extension of Shimizu's nondegeneracy framework to braided module categories, with a checkable Hopf-level payoff, but the main equivalence currently rests on two deferred proofs and an asserted monadicity step. read the letter →

arxiv 2411.18453 v2 pith:TQL6FFNN submitted 2024-11-27 math.QA math.CTmath.RT

Chelsea Walton , Harshit Yadav This is my paper · ORCID

classification math.QAmath.CTmath.RT MSC 18M1516T05

keywords braidedmodulecategorynondegeneratefactorizablemonadicityquasitriangularcomodulealgebrareflectivecenter

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 proves that, for braided module categories over a braided finite tensor category, two separate-looking conditions coincide: nondegeneracy (an explicit morphism θ_M built from a universal copairing is an isomorphism) and factorizability (a functor G_M comparing the module category to its reflective center is an equivalence). This is the module-category analogue of Shimizu's theorem that several nondegeneracy conditions are equivalent for braided finite tensor categories. In the Hopf setting, the equivalence becomes a concrete algebra-level criterion: a quasitriangular left H-comodule algebra is factorizable precisely when its category of finite-dimensional representations is nondegenerate as a module category over H-modules. The proof rests on a new monadicity result for module categories.

What carries the argument

The load-bearing machinery is the monadic adjunction ρ ⊣ ρ_ra (Proposition 4.12), where ρ(X) = X ⋉ (−) sends an object of C to the endofunctor of M it induces, and ρ_ra(F) = ∫_M Hom(M, F(M)) is its right adjoint. Although the codomain Fun(M,M) is not a rigid category, the paper shows the adjunction is monadic and yields equivalences H_M: M ⊠ Fun_C|(M,M) → E_M-Mod(M) and H_C: E_C(M) → C_C-Mod(M). These equivalences turn the question of whether G_M is an equivalence into the question of whether the restriction functor Res_{θ_M} is an equivalence, which holds exactly when θ_M is an isomorphism. The universal copairing ω_M and the end algebras E_M and E_C are the objects that enter the nondegeneracy condition.

What would settle it

Find an exact indecomposable braided left C-module category M over a braided finite tensor category C (for instance, a module category of B-FdMod for a non-semisimple H-comodule algebra B) for which θ_M is an isomorphism but G_M is not an equivalence, or vice versa; the theorem predicts this never happens. A direct check of the deferred proof of Proposition 4.17 would also settle the matter: if the claimed equivalence between the reflective center E_C(M) and the Eilenberg-Moore category C_C-Mod(M) fails for any such M, the commuting-diagram argument breaks.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 5.12: under the standing hypotheses that M is nonzero, exact, indecomposable, and finite, M is factorizable as a left C-module category if and only if it is nondegenerate. Factorizability means the functor G_M: M ⊠ Fun_C|(M,M) → E_C(M) is an equivalence of braided module categories; nondegeneracy means the morphism θ_M: E_C^* → E_M, defined via the universal copairing ω_M, is an isomorphism in C. The proof commutes a diagram whose vertices are M ⊠ Fun_C|(M,M), E_C(M), E_M-Mod(M), and E_C^*-Mod(M), using the equivalences H_M and H_C from the new monadicity theorem. In the Hopf case (Theorem 6.19), this reproduces an explicit linear-algebra condition: B-FdMod is nondegenerate over H-FdMod if and only if the map θ_B: H^* → E(H,B), f ↦ [h ↦ ⟨f, S(S(h_(1)))K_i h_(2)⟩] K^i, is an isomorphism, i.e., B is factorizable as a quasitriangular comodule algebra.

Load-bearing premise

The argument assumes that the Hopf-monadic machinery imported from Shimizu's work still functions when the category of endofunctors Fun(M,M) is not rigid, and that two auxiliary equivalences (one with proof deferred to an appendix) and a technical lemma are valid; if any of these fails, the commutative-diagram proof equating factorizability with nondegeneracy collapses.

Editorial extensions

If this is right

Nondegeneracy and factorizability can be used interchangeably for braided finite module categories, so results proved for one condition transfer to the other.
Every reflective center E_C(M) of an exact indecomposable module category is nondegenerate if and only if it is factorizable, giving a large class of examples.
In the Hopf case, factorizability of a quasitriangular comodule algebra is detected by an explicit injectivity (or bijectivity) of the map θ_B, which is checkable by dimension count since dim_k E(H,B) = dim_k H.
The theorem opens the door to a module-category version of Shimizu's full equivalence: weak factorizability and trivial symmetric center are one-way implications here, and Question 5.23 asks when the converses hold.

Reading between the lines

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

The monadicity result may apply to other settings where the endofunctor category is not rigid, suggesting that Hopf-monad techniques could be used to prove analogous nondegeneracy–factorizability equivalences for other enrichments.
The explicit formula for θ_B in the Hopf case gives a practical computational criterion: to test whether a comodule algebra is factorizable, one only needs to check injectivity of a single map, and the paper's dimension equality reduces that to testing that the map is nonzero on all basis elements.
A possible testable extension: verify whether the one-way implications in Lemma 5.16 and Theorem 5.21 become equivalences under a modified definition of trivial symmetric center, since the paper notes that Definition 5.18 may need adjustment.

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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 paper does something genuinely new: it defines nondegenerate and factorizable braided module categories, proves they are equivalent (Theorem 5.12), and then gives a concrete Hopf-algebra incarnation where a quasitriangular comodule algebra is factorizable iff its representation category is nondegenerate (Theorem 6.19). The definitions are anchored by examples: for the regular module category they reduce to Shimizu's conditions, and the Hopf map theta_B is explicit enough to check by hand. The paper is also honest about its scope—one-directional implications, open questions, and the caveat that Definition 5.18 may need modification.

The soft spots are exactly where the reader's stress-test lands. Proposition 4.17, the equivalence between the reflective center E_C(M) and the Eilenberg-Moore category C_C-Mod(M), is stated with "we leave the proof to the reader" and deferred to the v1 appendix. Lemma 5.13, the technical identity used in the proof of Theorem 5.12, is also left to the reader. Those two results are not decorative: the diagram chase in Theorem 5.12 identifies the factorizability functor G_M with pH_C circle Res_thetaM circle H_M, and that identification uses both. If either fails, the theorem reduces to a statement about an EM-condition, not factorizability. Proposition 4.12 also invokes [BV07, Theorem 3.14] for a Hopf monad on a non-rigid functor category; the hypotheses are asserted rather than demonstrated in the text.

That said, I do not think the paper is circular or sloppy in its citation pattern. The reliance on [LWY23] for reflective centers is real, but [LWY23] supplies constructions, not the main equivalence. And the regular-case reductions and the Hopf-level theorem are independent evidence that the framework is the right one. The missing proofs are likely routine but lengthy—the v1 appendix exists—and a referee can verify them. The structural argument is coherent.

This paper deserves a serious referee, but the referee should insist that the deferred proofs be included or at least explicitly and verifiably referenced. I would send it to peer review with that condition. A reader working on braided module categories, quantum symmetric pairs, or non-semisimple TQFTs will want this on hand.

Referee Report

3 major / 4 minor

Summary. The paper proposes module-theoretic analogues of nondegeneracy and factorizability for braided finite tensor categories. For a braided left C-module category M (Hypothesis 5.1: nonzero, exact, indecomposable, finite), the authors define nondegeneracy as the isomorphism property of θ_M : E_C^* → E_M (Definition 5.9) and factorizability as the equivalence property of G_M : M ⊠ Fun_C|(M,M) → E_C(M) (Definition 5.5). The main theorem, Theorem 5.12, asserts that these two properties are equivalent. The proof combines a monadicity result (Proposition 4.12), category equivalences (Propositions 4.16 and 4.17), a restriction-functor criterion (Lemma 4.11), and a claimed commutation diagram. The paper also studies weak factorizability and trivial symmetric center (Definitions 5.14 and 5.18), proving nondegeneracy ⇒ weak factorizability ⇒ trivial symmetric center (Lemma 5.16, Theorem 5.21). In the Hopf setting, the authors introduce factorizable quasitriangular left H-comodule algebras (Definition 6.15) and prove that B-FdMod is nondegenerate over H-FdMod if and only if (B,K) is factorizable (Theorem 6.19), with explicit descriptions of the copairing and θ_M (Propositions 6.20 and 6.21).

Significance. If the central results hold, the paper gives a meaningful extension of Shimizu's theorem to braided module categories, with clear relevance to quantum symmetric pairs, reflective centers, and applications to braided module categories in quantum character variety theory. The Hopf-algebraic part is particularly valuable: Theorem 6.19 and Proposition 6.20 give explicit, checkable formulas, and the examples (including reflective algebras in Example 6.25) provide concrete tests. The framework also suggests a module-theoretic route toward understanding nondegeneracy for module categories, and the new monadicity statement for non-rigid endofunctor categories is potentially useful beyond this paper. However, the significance is conditional because the proof of Theorem 5.12 rests on two deferred or sketched technical results (Propositions 4.17 and 5.13), and the claimed application of Hopf-monad theory to the non-rigid category Fun(M,M) is not fully verified in the text.

major comments (3)

[§4.5, Proposition 4.17] Proposition 4.17 is a load-bearing ingredient: in the proof of Theorem 5.12, the identification of the composite pHC ∘ ResθM ∘ HM with GM requires H_C to be an equivalence. The proof is omitted, with the text saying 'We leave the proof to the reader; detailed computations can be found in the appendix of the ArXiv version 1.' A journal submission cannot defer a central equivalence to an appendix of an earlier preprint version. The authors must include a complete proof or a precise pointer to a published, accessible source; otherwise Theorem 5.12 is unsupported.
[§5.3, Lemma 5.13] Lemma 5.13 is stated without proof ('the proof is left to the reader') yet it supplies the crucial interchange identity that converts the expression involving ξ_{F(M)} into s_{X,M} F(e_{X,M}) s^{-1}_{X,M}. This is exactly where the object-level equality e^{F(M)}_X = s_{X,M} F(e_{X,M}) s^{-1}_{X,M} is established. A proof or a full derivation should be included; as written, the main theorem depends on an unverified equality.
[§4.3, Proposition 4.12] Proposition 4.12(b) invokes [BV07, Theorem 3.14] to conclude that the adjunction ρ ⊣ ρ_ra is Hopf monoidal, even though the codomain Fun(M,M) is not rigid. The text asserts that the conditions of [BV07, Theorem 3.14] are satisfied, but it does not verify the specific hypotheses (e.g., the relevant rigidity or Hopf-monad axioms) for this non-rigid situation. Since this is the paper's declared novelty, the verification should be spelled out or replaced by a reference that explicitly covers non-rigid codomains. Without this, Proposition 4.12 and hence the equivalences built on it are not fully justified.

minor comments (4)

[Title/Abstract] The title contains spurious spacing: 'NONDEGENERA TE MODULE CA TEGORIES' should read 'NONDEGENERATE MODULE CATEGORIES'. The same OCR-like artifact appears in the abstract. Please correct these.
[Throughout] The notation 'eX,M ⊠ pF,sq :="eX,M ⊠ idpF,sq" in Proposition 5.6(a) is confusing; the braiding on the Deligne product should be defined more explicitly, because a braided module category structure on a Deligne product is not immediate.
[§5.3, proof of Theorem 5.12] The proof relies on a long chain of equalities in which several steps are labeled only 'level ex.' or use unstated naturality. This is acceptable as a computation sketch, but for a journal version the authors should either expand the derivation or provide a diagram that makes the level-exchange steps precise.
[References] The reference [Shi19b] is cited for the statement that H-FdMod is nondegenerate iff (H,R) is factorizable, but the bibliography lists only the conference proceedings; please provide the full publication data if the version is published, or cite the relevant section of [KL01] or [Rad12] more precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the main equivalences are proved from external monadicity results and direct computations, not from their own conclusions.

full rationale

The central equivalence Theorem 5.12 does not reduce by construction to a fitted parameter, a self-citation chain, or a definitional identity. The proof assembles the commuting diagram via the equivalences H_M (Proposition 4.16) and H_C (Proposition 4.17), both built on the monadic adjunction ρ ⊣ ρ_ra (Proposition 4.12). The key exactness and faithfulness of ρ_ra are imported from Shimizu's external work [Shi20, Theorem 3.4], and the Hopf-monad step cites Bruguieres–Virelizier [BV07, Theorem 3.14]; neither is a result of the present authors, and neither assumes the target theorem. The final reduction uses Lemma 4.11, proved in the text, that an algebra map φ is an isomorphism iff Res_φ is an equivalence; this is a standard categorical fact, not a renamed version of nondegeneracy or factorizability. The equality p H_C ∘ Res_{θ_M} ∘ H_M = G_M is obtained by a diagram chase, with G_M defined independently in (5.4); it is not a definitional tautology. In the Hopf case, Theorem 6.19 is also a direct computation: Proposition 6.20 gives the copairing ω_B explicitly, Proposition 6.21 shows θ_{B-FdMod} = θ_B ∘ S_{H^*}, and since S_{H^*} is bijective the two isomorphism conditions coincide. This is a translation between two independently defined conditions, not a prediction obtained from a fitted input. The self-citation [LWY23] (with author overlap Walton) supplies the reflective-center construction and the Hopf-level equivalence E_{H-FdMod}(A-FdMod) ≅ R_H(A)-FdMod; these are established tools used in examples and applications rather than unverified assumptions of the main equivalence. Two proof obligations are explicitly deferred: Proposition 4.17 says 'We leave the proof to the reader; detailed computations can be found in the appendix of the ArXiv version 1 of this work,' and Lemma 5.13 says 'the proof is left to the reader.' These are completeness or correctness risks, not circularity, because no step is equivalent to its own input by definition.

Assumptions & free parameters 0 free parameters · 9 assumptions · 5 invented entities

This is a pure mathematics paper: there are no fitted parameters, no ad hoc numerical constants, and no data, so the free_parameters list is empty. The axioms are the standard external theorems the paper builds on (Shimizu's theorem, Beck's monadicity, results of Shimizu, of Bruguieres-Lack-Virelizier, of Bruguieres-Virelizier, and of Davydov-Nikshych) plus the paper's own standing hypotheses (Hypothesis 5.1) and conventions (strictness, algebraically closed field). The invented entities are the paper's new definitions; each is anchored by independent evidence, either through reduction to Shimizu's regular case or through explicit computable formulas, so the risk of postulating an entity with no external handle is low.

assumptions (9)

standard math Shimizu's Theorem [Shi19a, Theorem 1.1]: nondegeneracy, weak factorizability, factorizability, and trivial symmetric center are equivalent for braided finite tensor categories
Quoted in §2.6 and used as the external benchmark: the regular module category cases (Examples 5.7, 5.10, 5.15) check that the new definitions reduce to Shimizu's conditions.
domain assumption Hypothesis 5.1: C is a braided finite tensor category and M is a nonzero, exact, indecomposable, braided finite left C-module category
Standing hypothesis for the main theorem and for §6 via Lemma 6.7; exactness and indecomposability are used in Proposition 4.12 through [Shi20, Theorem 3.4].
standard math [Shi20, Theorem 3.4]: for M exact and indecomposable, the adjunction ρ ⊣ ρ_ra exists with ρ_ra exact and faithful
Imported in §4.3 to invoke Beck's monadicity theorem in Proposition 4.12(a).
standard math Beck's monadicity theorem [ML98, §VI.7]
Used in Proposition 4.12(a) to conclude that the adjunction ρ ⊣ ρ_ra is monadic.
standard math [BV07, Theorem 3.14] and [BLV11, Theorem 6.6]: conditions under which the bimonad ρ_ra ∘ ρ is a Hopf monad and E_M = ρ_ra(id_M) is a commutative algebra in Z(C)
Used in the proof of Proposition 4.12(b); the paper's claimed new twist is applying this machinery to a non-rigid codomain Fun(M,M).
standard math [DN13, Lemma 3.2]: Fun_C|(Mod-A(C), M) is equivalent to A-Mod(M)
Used as step (iii) in the proof of Proposition 4.16.
domain assumption The reflective center E_C(M) construction and the reflective algebra theorem [LWY23, Theorem 6.6, Corollary 6.8]
External preprint co-authored by this paper's first author; the reflective center is the target of the factorizability functor G_M (Definition 5.5) and reflective algebras appear in Proposition 6.24 and Example 6.25. The present paper's target theorem is not assumed from [LWY23].
standard math MacLane's strictness theorem (Hypotheses 2.1 and 3.1)
Conventions used throughout the paper to simplify notation.
domain assumption All linear structures are over an algebraically closed field k
Standing convention; required for finiteness and Deligne product statements in §2.1.

invented entities (5)

Nondegenerate braided module category (Definition 5.9) independent evidence
purpose: Module-categorical analogue of nondegeneracy for braided finite tensor categories; defined by θ_M : E_C* → E_M being an isomorphism
Anchored externally: for the regular module category, nondegeneracy reduces to Shimizu's nondegeneracy of C (Example 5.10), and Theorem 6.19 gives a computable Hopf-level test.
Factorizable braided module category (Definition 5.5) independent evidence
purpose: Module-categorical analogue of factorizability; defined by G_M : M ⊠ Fun_C|(M,M) → E_C(M) being an equivalence
Reduces to factorizability of C in the regular case (Example 5.7), connecting to the Drinfeld center Z(C).
Universal copairing ω_M (Definition 5.8) independent evidence
purpose: Morphism 1 → E_C ⊗ E_M defined by universal properties; used to define θ_M and weak factorizability
Explicitly computed in the Hopf case (Proposition 6.20), giving a concrete formula that supports the later theorems.
Weakly factorizable module category (Definition 5.14) and symmetric center Z_2(M) (Definition 5.18) independent evidence
purpose: Partial extension of Shimizu's weak factorizability and trivial symmetric center conditions to module categories
One-directional implications are proved (Lemma 5.16, Theorem 5.21); the converse directions are explicitly left open (Questions 5.17, 5.23), so the partial status is honest and checkable.
Factorizable quasitriangular comodule algebra (Definition 6.15) independent evidence
purpose: Algebraic criterion for B-FdMod to be nondegenerate; θ_B : H* → E(H,B) is an explicit map built from the K-matrix
Concrete and computable: the group algebra case forces G trivial (Example 6.22) and reflective algebras from Drinfeld doubles are factorizable (Proposition 6.24, Example 6.25).

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Cite this review

Pith. "Pith review of Nondegenerate module categories." pith.science (2026). https://pith.science/paper/TQL6FFNN

@misc{pith2026241118453,
  author       = {Pith},
  title        = {Pith review of: Nondegenerate module categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQL6FFNN}},
  note         = {Machine review of arXiv:2411.18453}
}

read the original abstract

Due to the work of Shimizu (2019), various nondegeneracy conditions for braided finite tensor categories are equivalent. This theory is partially extended to braided module categories here. We introduce when a braided module category is "nondegenerate" and "factorizable", and establish that these properties are equivalent. The proof involves a new monadicity result for module categories. Lastly, we examine the Hopf case, using Kolb's (2020) notion of a quasitriangular comodule algebra to introduce "factorizable" comodule algebras. We then show that the representation category of a quasitriangular comodule algebra is nondegenerate in our sense precisely when the comodule algebra is factorizable. Several examples are provided.

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Works this paper leans on

7 extracted references · 6 canonical work pages

[1]

Andruskiewitsch and J

[AM07] N. Andruskiewitsch and J. M. Mombelli, On module categories over ﬁnite-dimensional Hopf algebras , J. Algebra 314 (2007), no. 1, 383–418. [BK01] B. Bakalov and A. Kirillov Jr., Lectures on tensor categories and modular functors , University Lecture Series, vol. 21, American Mathematical Society, Providenc e, RI,

work page 2007
[4]

Kolb, Braided module categories via quantum symmetric pairs , Proc

NONDEGENERATE MODULE CATEGORIES 37 [Kol20] S. Kolb, Braided module categories via quantum symmetric pairs , Proc. Lond. Math. Soc. (3) 121 (2020), no. 1, 1–31. [L WY23] R. Laugwitz, C. Walton, and M. Yakimov, Reﬂective centers of module categories and quantum K-matri ces, arXiv preprint arXiv:2307.14764 (2023). [M¨ ug03] M. M¨ uger,From subfactors to cate...

arXiv 2020
[36]

Shimizu, Recent developments of the categorical Verlinde formula , Proceedings of the Meeting for Study of Number Theory, Hopf Algebras and Related Topics, 20 19, pp

[Shi19b] K. Shimizu, Recent developments of the categorical Verlinde formula , Proceedings of the Meeting for Study of Number Theory, Hopf Algebras and Related Topics, 20 19, pp. 197–222. [Shi20] K. Shimizu, Further results on the structure of (co)ends in ﬁnite tensor categories, Appl. Categ. Structures 28 (2020), no. 2, 237–286. [Shi23] K. Shimizu, Relat...

work page 2020
[2001]

Balagovi´ c and S

[BK19] M. Balagovi´ c and S. Kolb, Universal K-matrix for quantum symmetric pairs , J. Reine Angew. Math. 747 (2019), 299–353. [BL V11] A. Brugui` eres, S. Lack, and A. Virelizier, Hopf monads on monoidal categories , Adv. Math. 227 (2011), no. 2, 745–800. [BM21a] N. Bortolussi and M. Mombelli, The character algebra for module categories over Hopf algeb r...

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[2010]

Wen, Colloquium: zoo of quantum-topological phases of matter , Rev

[Wen17] X.-G. Wen, Colloquium: zoo of quantum-topological phases of matter , Rev. Modern Phys. 89 (2017), no. 4, 041004,

work page 2017
[2012]

[Rad94] D. E. Radford, On Kauﬀman ’s knot invariants arising from ﬁnite-dimensiona l Hopf algebras , Advances in Hopf algebras (Chicago, IL, 1992), 1994, pp. 205–266. [RSTS88] N. Yu. Reshetikhin and M. A. Semenov-Tian-Shansky , Quantum R-matrices and factorization problems , J. Geom. Phys. 5 (1988), no. 4, 533–550 (1989). [Seg89] G. Segal, Two-dimensional...

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[2015]

Enriquez, Quasi-reﬂection algebras and cyclotomic associators , Selecta Math

[Enr07] B. Enriquez, Quasi-reﬂection algebras and cyclotomic associators , Selecta Math. (N.S.) 13 (2007), no. 3, 391–463. [EO04] P. Etingof and V. Ostrik, Finite tensor categories , Mosc. Math. J. 4 (2004), no. 3, 627–654, 782–783. [FRS02] J. Fuchs, I. Runkel, and C. Schweigert, TFT construction of RCFT correlators. I. Partition functio ns, Nuclear Phys....

work page 2007

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Andruskiewitsch and J

[AM07] N. Andruskiewitsch and J. M. Mombelli, On module categories over ﬁnite-dimensional Hopf algebras , J. Algebra 314 (2007), no. 1, 383–418. [BK01] B. Bakalov and A. Kirillov Jr., Lectures on tensor categories and modular functors , University Lecture Series, vol. 21, American Mathematical Society, Providenc e, RI,

work page 2007

[2] [4]

Kolb, Braided module categories via quantum symmetric pairs , Proc

NONDEGENERATE MODULE CATEGORIES 37 [Kol20] S. Kolb, Braided module categories via quantum symmetric pairs , Proc. Lond. Math. Soc. (3) 121 (2020), no. 1, 1–31. [L WY23] R. Laugwitz, C. Walton, and M. Yakimov, Reﬂective centers of module categories and quantum K-matri ces, arXiv preprint arXiv:2307.14764 (2023). [M¨ ug03] M. M¨ uger,From subfactors to cate...

arXiv 2020

[3] [36]

Shimizu, Recent developments of the categorical Verlinde formula , Proceedings of the Meeting for Study of Number Theory, Hopf Algebras and Related Topics, 20 19, pp

[Shi19b] K. Shimizu, Recent developments of the categorical Verlinde formula , Proceedings of the Meeting for Study of Number Theory, Hopf Algebras and Related Topics, 20 19, pp. 197–222. [Shi20] K. Shimizu, Further results on the structure of (co)ends in ﬁnite tensor categories, Appl. Categ. Structures 28 (2020), no. 2, 237–286. [Shi23] K. Shimizu, Relat...

work page 2020

[4] [2001]

Balagovi´ c and S

[BK19] M. Balagovi´ c and S. Kolb, Universal K-matrix for quantum symmetric pairs , J. Reine Angew. Math. 747 (2019), 299–353. [BL V11] A. Brugui` eres, S. Lack, and A. Virelizier, Hopf monads on monoidal categories , Adv. Math. 227 (2011), no. 2, 745–800. [BM21a] N. Bortolussi and M. Mombelli, The character algebra for module categories over Hopf algeb r...

work page 2019

[5] [2010]

Wen, Colloquium: zoo of quantum-topological phases of matter , Rev

[Wen17] X.-G. Wen, Colloquium: zoo of quantum-topological phases of matter , Rev. Modern Phys. 89 (2017), no. 4, 041004,

work page 2017

[6] [2012]

[Rad94] D. E. Radford, On Kauﬀman ’s knot invariants arising from ﬁnite-dimensiona l Hopf algebras , Advances in Hopf algebras (Chicago, IL, 1992), 1994, pp. 205–266. [RSTS88] N. Yu. Reshetikhin and M. A. Semenov-Tian-Shansky , Quantum R-matrices and factorization problems , J. Geom. Phys. 5 (1988), no. 4, 533–550 (1989). [Seg89] G. Segal, Two-dimensional...

work page 1988

[7] [2015]

Enriquez, Quasi-reﬂection algebras and cyclotomic associators , Selecta Math

[Enr07] B. Enriquez, Quasi-reﬂection algebras and cyclotomic associators , Selecta Math. (N.S.) 13 (2007), no. 3, 391–463. [EO04] P. Etingof and V. Ostrik, Finite tensor categories , Mosc. Math. J. 4 (2004), no. 3, 627–654, 782–783. [FRS02] J. Fuchs, I. Runkel, and C. Schweigert, TFT construction of RCFT correlators. I. Partition functio ns, Nuclear Phys....

work page 2007