Pith. sign in

REVIEW 3 major objections 5 minor 37 references

Biproduct Quasi-Hopf Algebras of Rank 2

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper classifies every 2-dimensional braided Hopf algebra over a quasi-Hopf algebra, proving it is either the trivial group algebra or one explicit two-parameter family.

desk verdict Solid quasi-Hopf analogue of Radford's rank-2 classification, with an external 3-cocycle completeness assumption in the group case that needs checking. read the letter →

arxiv 2508.00597 v1 pith:OOLZN2UH submitted 2025-08-01 math.QA math.RA

Daniel Bulacu , Matteo Misurati This is my paper
classification math.QAmath.RA MSC 16T0518M15
keywords quasi-HopfalgebraYetter-DrinfeldmodulebraidedHopfbiproduct3-cocycletwistequivalenceNicholsfiniteabeliangroup
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 classifies the 2-dimensional Hopf algebras living in the braided category of left Yetter-Drinfeld modules over an arbitrary quasi-Hopf algebra H, over a field of characteristic different from 2. It proves that each such braided Hopf algebra is isomorphic either to the trivial group Hopf algebra k[C2] or to a braided Hopf algebra B_{σ,v} determined by an algebra map σ:H→k and an element v∈H with σ(v)=-1 and ε(v)=1. Consequently, every biproduct quasi-Hopf algebra of rank 2 is either the tensor product k[C2]⊗H or a twisted extension H(θ)_{σ,v} built from the same data. The classification is then applied to function algebras on groups with 3-cocycles, to finite abelian groups, to the double dihedral group, and to quasi-Hopf algebras with radical of codimension 2, producing new basic quasi-Hopf algebras of even dimension and new tensor categories.

What carries the argument

The central object is the two-parameter family B_{σ,v}: a 2-dimensional braided Hopf algebra in the Yetter-Drinfeld category over H whose H-action and H-coaction are encoded by an algebra map σ:H→k and an element v∈H with σ(v)=-1, ε(v)=1, plus the structural relations Δ(v)=σ(y1x3X2)x1X1vy2⊗x2vX3y3 and σ(h2)h1v=σ(h1)vh2. In the group-function case H=k^G_ω these constraints collapse to the requirement that the 2-cocycle ♭gω(x,y)=ω(g,x,y)ω(x,y,g)/ω(x,g,y) is a coboundary ∂ρ with ρ(g)=-1; the paper then reduces the existence of such pairs to a linear system XA=0 for finite abelian groups and to rational-point questions on conic sections for products of two cyclic groups.

What would settle it

Take a finite abelian group G and verify the completeness of the list {ω_a}_{a∈A} supplied by [25] by comparing the number of representative cocycles with |$H^{3}$(G,k*)|; a mismatch would show the parametrization in Proposition 4.8 is incomplete, since some pair (g,ρ) could then satisfy ♭gω=∂ρ outside the listed representatives.

Watch

Extended reading notes

Core claim

Over a field k of characteristic different from 2, let H be a quasi-Hopf algebra with bijective antipode. The main theorem (Theorem 2.6) states that every 2-dimensional braided Hopf algebra B in the category of left Yetter-Drinfeld modules over H is isomorphic either to the group Hopf algebra k[C2] with trivial H-action and coaction, or to a braided Hopf algebra B_{σ,v} on basis {1,n} with h·n=σ(h)n, coaction n↦v⊗n, multiplication $n^{2}$=0, and comultiplication Δ(n)=n⊗1+1⊗n, where σ:H→k is an algebra map and v∈H satisfies σ(v)=-1, ε(v)=1 together with the Yetter-Drinfeld relations Δ(v)=σ(y1x3X2)x1X1vy2⊗x2vX3y3 and σ(h2)h1v=σ(h1)vh2 for all h∈H. Consequently every rank-2 biproduct quasi-Hopf algebra is either k[C2]⊗H or the algebra H(θ)_{σ,v} generated by θ and H with $θ^{2}$=0 and hθ=σ(h1)θh2. In the group-function case H=k^G_ω, the non-trivial B_{σ,v} are parametrized by central elements g∈G and maps ρ:G→k* with ρ(e)=1, ρ(g)=-1 and ♭gω=∂ρ; for finite abelian G this reduces to solving a linear system of congruences, and explicit families are produced for cyclic groups, the Klein four group, products of two cyclic groups, and the double dihedral group.

Load-bearing premise

The classification of the abelian cases rests on the assumption that the list of representative normalized 3-cocycles on a finite abelian group given in [25] is complete; if that list misses any cohomology class, the parametrization by pairs (g,ρ) in Proposition 4.3 could omit genuine examples.

Editorial extensions

If this is right

  • Every quasi-Hopf algebra that is a free right module of rank 2 over a quasi-Hopf subalgebra H is either the tensor product k[C2]⊗H or one of the algebras H(θ)_{σ,v}; no other form exists (Proposition 2.7).
  • A rank-2 biproduct is semisimple exactly in the trivial k[C2]⊗H case; the H(θ)_{σ,v} algebras are nonsemisimple (Corollary 2.8).
  • In dimension 4 over an algebraically closed field of characteristic 0, every nonsemisimple quasi-Hopf algebra is twist equivalent to Sweedler's H4, yet there are infinitely many pairwise non-isomorphic such algebras, indexed by a scalar parameter a∈k* (Proposition 3.4 and Remark 3.5).
  • For finite abelian G, non-trivial rank-2 braided Hopf algebras in k^G_ω can exist only when N=LCM(m_j^2, m_s m_t) is even, and their enumeration reduces to solving a homogeneous linear system XA=0; for products of two cyclic groups the problem becomes finding rational points on a conic (Propositions 4.8 and Section 6).
  • Among quasi-Hopf algebras with radical of codimension 2, non-trivial rank-2 biproducts occur exactly for those twist equivalent to a Nichols Hopf algebra H_{2n}, where H_{2n}(θ)_{σ,g}≅H_{2n+1}; for H(2), H±(8) and H(32) only the trivial biproduct appears, while H_q(8)(θ) gives H4⊗H_q(8) (Propositions 5.2 and 5.3).

Reading between the lines

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

  • The completeness of the representative 3-cocycle list in [25] is load-bearing for the abelian classification; a natural next step is to verify that list against the known order of H^3(G,k*) for small finite abelian groups, which would upgrade Propositions 4.4 and 4.8 from conditional to unconditional.
  • The conic-section reduction of Section 6 suggests a counting problem: the number of admissible pairs (g,ρ) for G=C_{m1}×C_{m2} is governed by rational points on Pell-type curves, so a density statement for how often non-trivial rank-2 braided Hopf algebras exist as m1,m2 grow appears within reach.
  • The same strategy should extend to other non-abelian groups with explicit 3-cocycle representatives and non-trivial center; the double dihedral example shows the method works whenever such a cocycle presentation is available.
  • The infinite family of 4-dimensional nonsemisimple quasi-Hopf algebras twist equivalent to H4 indicates that twist equivalence classes can be much larger than isomorphism classes, and leaves open the question of how the H(θ)_{σ,v} families organize under twist equivalence for larger groups.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper studies 2-dimensional Hopf algebras in the braided category H_H YD of left Yetter-Drinfeld modules over a quasi-Hopf algebra H. Section 2 reduces the Yetter-Drinfeld Hopf algebra axioms to explicit data: an algebra map σ : H → k, elements u, v ∈ H satisfying (2.8)–(2.9), and then proves in Theorem 2.6 that every such object is either the trivial k[C2] or one of the algebras B_{σ,v} with σ(v) = −1. Proposition 2.7 describes the resulting biproduct quasi-Hopf algebras. The rest of the paper applies this framework: Section 3 treats quasi-Hopf algebras of dimension 4, Section 4 works out the group case kG_ω with a detailed parametrization for finite abelian groups and an example for the double dihedral group, and Section 5 constructs examples from quasi-Hopf algebras with radical of codimension 2.

Significance. The reduction in Section 2 is a genuine contribution: it converts a categorical classification problem into explicit equations in the quasi-Hopf algebra, and it cleanly recovers Radford's Hopf algebra case while producing concrete quasi-Hopf analogues. The group-theoretic applications are explicit and go beyond existing results, giving families of braided Hopf algebras and biproduct quasi-Hopf algebras such as Hq(16) and Hq(8)(θ). Credit is due for the systematic treatment of the double dihedral group and for the detailed arithmetic conditions in Section 4. The paper is also honest about relying on external classification input for 3-cocycles on finite abelian groups; however, that reliance, together with several literal 'one-to-one' statements, needs tightening before the classification claims are fully established.

major comments (3)
  1. [Section 4, eqs. (4.3)–(4.4)] The representative 3-cocycle list in (4.3) uses the inclusive bounds 0 ≤ c_{ij} ≤ (m_i, m_j) and 0 ≤ c_{rst} ≤ (m_r, m_s, m_t). In the formula (4.4) these parameters appear as exponents of roots of unity, so the endpoint values are not independent under the usual normalization; as written, the set A contains extra parameter values. This issue is load-bearing because Proposition 4.8 and Example 4.12 use A as though it were a true set of representatives, so the asserted 'one-to-one' classification inherits the duplication. The authors should either replace the upper bounds by strict inequalities, in line with the cited [26, Prop. 3.8], or explicitly identify the equivalence relation on A that makes (4.4) a genuine list of representatives.
  2. [Section 4, Proposition 4.8] The correspondence asserted in Proposition 4.8 is not literally one-to-one. The map y ↦ f = Y_y U′ is linear in y with integer coefficients, and the pair (g, ρ) depends only on f modulo N and on λ modulo the orders m_i. Since y ranges over all of Z^n, each equivalence class of the data is represented infinitely often, and the same braided Hopf algebra is counted many times. Example 4.5 even writes the general solution as f_i = μ_i M′ with μ_i ∈ Z, and only in Example 4.10 are finite ranges imposed ad hoc. To obtain a classification statement, the authors need to state the periodicity or quotient on y (and λ), or restrict f to a fundamental domain for its action on (g, ρ). As it stands, Corollaries 4.9–4.11 and the table in Example 4.12 enumerate parameter tuples, not isomorphism classes.
  3. [Section 4, Propositions 4.3–4.8] The completeness of the abelian-group classification is imported from [25] and [26, Prop. 3.8] and is not reproved in this paper. If any cohomology class is missing from the list (4.3)–(4.4), or if the congruence system (4.8) is solved only for a subset of the representative classes, then the parametrization by pairs (g, ρ) in Proposition 4.3 is not exhaustive. Since [25] is an arXiv preprint, the authors should state explicitly which published theorem establishes the completeness of (4.3)–(4.4) in the exact normalization used here, or they should prove the needed completeness for the groups to which Proposition 4.8 is applied.
minor comments (5)
  1. [Definition 2.5] The phrase 'the second formulas in (2.8) and (2.9)' is ambiguous; please state explicitly that the required conditions are Δ(v) = σ(y_1 x_3 X_2) x_1 X_1 v y_2 ⊗ x_2 v X_3 y_3 and σ(h_2) h_1 v = σ(h_1) v h_2 for all h ∈ H.
  2. [Lemma 2.2 and Theorem 2.6, Case 1] The phrases 'the remaining details are left to the reader' and 'a simple inspection' cover nontrivial verification of the coalgebra and antipode axioms; please expand these passages or point to the precise equations that complete the verification.
  3. [Example 4.12] The parameter a is written as (c1, c2, c3), while the earlier notation in (4.3) for n = 2 is (c1, c2, c12); rename to avoid confusion.
  4. [Paragraph after (4.8)] The sentence 'A is an n + n by n matrix' uses n both for the number of cyclic factors and for the binomial coefficient C(n,2), which makes the dimension statement hard to read; please use separate symbols.
  5. [References] References [30] and [33] cite the same paper by Radford; please unify the citation to avoid duplication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rank-2 classification is derived from the Yetter–Drinfeld axioms, and the group-case cocycle input is external to the paper.

full rationale

The central derivation chain is self-contained. Lemma 2.1 derives the module-algebra data (σ, ψ, ω) from the H-module algebra axioms, Lemma 2.2 derives the coalgebra data and forces ψ = 0, Lemma 2.3 derives the Yetter–Drinfeld data (u, v) from the coaction axioms, and Proposition 2.4 derives (2.16) from the bialgebra compatibility condition (1.20). Theorem 2.6 then splits into the two cases b′ = 1 and b′ ≠ 1, obtaining respectively B_{σ,v} and the trivial k[C2] directly from these derived relations; the defining conditions of B_{σ,v} in Definition 2.5 are exactly the relations that the proof derives, not assumptions smuggled in as conclusions. The group-case application, Proposition 4.3, is a direct translation of Theorem 2.6 and Lemma 4.2 into pairs (g, ρ), with the coboundary condition ♭_gω = ∂ρ computed from the quasi-Hopf structure. The later parametrizations in Propositions 4.4 and 4.8 rely on the explicit representative 3-cocycles imported from [25] and [26, Proposition 3.8]; this is an external, load-bearing input, not a self-citation or a fitted quantity, so any possible incompleteness in that external list would be a correctness risk rather than a circularity. The paper's self-citations ([9], [10]) are used for technique, for a minor lemma in a side count of twist classes, and for the standard biproduct recognition theorem; none of these makes the main classification reduce to the authors' own prior outputs. No prediction is produced by fitting a parameter to the data it is said to predict, and no uniqueness claim is imported from the same authors as a substitute for proof. Overall, the derivation has no significant circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 3 invented entities

The paper does not fit any numeric constants; its classification parameters (σ, v), (g, ρ), (f, λ) are outputs rather than free inputs. The main external inputs are domain restrictions on the field, the bijective antipode assumption, and cited classification/3-cocycle results. Three new algebraic objects are introduced as constructed examples, each with theorems verifying their defining properties.

assumptions (7)
  • domain assumption The ground field k has characteristic different from 2.
    Stated at the start of Section 2 and used throughout; in Lemma 2.2 it is used to conclude ψ = 0 from ψσ = σψ = 0, and in Section 3 it is needed for H^3(C2, k*) ≅ C2.
  • domain assumption H is a quasi-Hopf algebra with bijective antipode.
    Needed for the Yetter-Drinfeld category to be braided and for the pR/qR computations in Section 1.2; stated at the start of Section 2.
  • domain assumption k is algebraically closed of characteristic 0 for Section 3 and parts of Section 4.
    Required to use the classification of 4-dimensional Hopf algebras and to have primitive roots of unity of arbitrary order m for the explicit 3-cocycles in Section 4.1.
  • standard math The classification of 4-dimensional Hopf algebras over k (Sweedler H4 and group algebras).
    Invoked in Lemma 3.1 and Proposition 3.2 to conclude that gr(A) is isomorphic to H4.
  • standard math The representative normalized 3-cocycles for finite abelian groups given in [25] form a complete set.
    Used in Proposition 4.4 and Proposition 4.8; the paper does not reprove their completeness.
  • standard math The classification of quasi-Hopf algebras with radical of codimension 2 from [17].
    Used in Section 5, especially Proposition 5.3, to characterize when H is twist equivalent to a Nichols Hopf algebra.
  • standard math Theorem 2.5 of [6] on the existence and uniqueness of left integrals in finite-dimensional quasi-Hopf algebras.
    Used in Corollary 2.8 to characterize semisimplicity of biproducts.
invented entities (3)
  • B_{σ,v} independent evidence
    purpose: The 2-dimensional braided Hopf algebra family that, together with k[C2], exhausts all such objects in H_H YD.
    Introduced in Definition 2.5; Theorem 2.6 proves that every non-trivial 2-dimensional braided Hopf algebra is isomorphic to one of these, giving an internal verification.
  • H(θ)_{σ,v} independent evidence
    purpose: The biproduct quasi-Hopf algebra produced from B_{σ,v} and H.
    Constructed in Proposition 2.7 with explicit comultiplication and antipode formulas, verifying the quasi-Hopf axioms.
  • Hq(16) and the 32-dimensional algebra Hq(8)(θ)(σ,v) independent evidence
    purpose: New even-dimensional quasi-Hopf algebras built from Hq(8) and rank-2 biproducts.
    Constructed in Proposition 5.2, with the 32-dimensional algebra shown to be isomorphic to H4 ⊗ Hq(8).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Biproduct Quasi-Hopf Algebras of Rank 2." pith.science (2026). https://pith.science/paper/OOLZN2UH

@misc{pith2026250800597,
  author       = {Pith},
  title        = {Pith review of: Biproduct Quasi-Hopf Algebras of Rank 2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OOLZN2UH}},
  note         = {Machine review of arXiv:2508.00597}
}
abstract

Inspired by the work of Radford, for $H$ an arbitrary quasi-Hopf algebra we describe all the Hopf algebras of dimension $2$ within the braided category of left Yetter-Drinfeld modules over $H$ and determine the biproduct quasi-Hopf algebras defined by them. Classes of such biproduct quasi-Hopf algebras are obtained by taking $H$ as the Hopf algebra of functions on a group $G$, endowed with the quasi-Hopf algebra structure provided by a non-trivial $3$-cocycle on $G$ (especially when $G$ is a finite cyclic group or the double dihedral group), or as being a quasi-Hopf algebra with radical of codimension two. In this way we uncover new classes of basic quasi-Hopf algebras of even dimension, as well as new classes of tensor categories.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

37 extracted references · 35 canonical work pages

  1. [25]

    Huang, G

    HL. Huang, G. Liu, Y. Yang and Y. Ye, On the Classification of Finite Quasi-Quantum Groups over abelian Groups, arXiv:2403.04455, 2024. doi:10.48550/arXiv.2403.04455

    arXiv doi:10.48550/arxiv.2403.04455 2024

  2. [1]

    Abe, ”Hopf algebras”, Cambridge: Cambridge University Press 1977

    E. Abe, ”Hopf algebras”, Cambridge: Cambridge University Press 1977

    work page 1977

  3. [2]

    New directions in Hopf algebras

    N. Andruskiewitsch, Pointed Hopf algebras , “New directions in Hopf algebras”, MSRI series Cambridge Univ. Press (2002), 1–68

    work page 2002

  4. [3]

    Angiono, Basic quasi-Hopf algebras over cyclic groups , Adv

    I. Angiono, Basic quasi-Hopf algebras over cyclic groups , Adv. Math. 225 (2010), 3545-–3575

    work page 2010

  5. [4]

    Angiono, A.G

    I. Angiono, A.G. Iglesias, Pointed Hopf algebras: a guided tour to the liftings , Revista Colombiana de Matem´ aticas53 (2019), 1–44. 38 D. BULACU AND M. MISURATI

    work page 2019

  6. [5]

    Bulacu, A structure theorem for quasi-Hopf bimodule coalgebras , Theory and Applications of Categories (TAC) 32(1) (2017), 1–30

    D. Bulacu, A structure theorem for quasi-Hopf bimodule coalgebras , Theory and Applications of Categories (TAC) 32(1) (2017), 1–30

    work page 2017

  7. [6]

    Bulacu, S

    D. Bulacu, S. Caenepeel, Integrals for (dual) quasi-Hopf algebras. Applications. J. Algebra 266 (2003), 552–583

    work page 2003

  8. [7]

    Bulacu, S

    D. Bulacu, S. Caenepeel, F. Panaite, F. Van Oystaeyen, ”Quasi-Hopf algebras: A monoidal approach”, Ency- clopedia Math. Appl.171, Cambridge University Press2019

    work page

Show all 37 references
  1. [8]

    Bulacu, S

    D. Bulacu, S. Caenepeel, B. Torrecillas, The braided monoidal structures on the category of vector spaces graded by the Klein group , Proc. Edinburgh Math. Soc. 54 (2011), 613–641

    2011

  2. [9]

    Bulacu, M

    D. Bulacu, M. Misurati, Quasi-Hopf algebras of dimension 6 , J. Pure Appl. Algebra 229 (2025), 107991

    2025

  3. [10]

    Bulacu, E

    D. Bulacu, E. Nauwelaerts, Radford’s biproduct for quasi-Hopf algebras and bosonization, J. Pure Appl. Algebra 174 (2002), 1–42

    2002

  4. [11]

    Bulacu, E

    D. Bulacu, E. Nauwelaerts, Quasitriangular and ribbon quasi-Hopf algebras. Comm. Algebra 31 (2003), 657–672

    2003

  5. [12]

    Bulacu, F

    D. Bulacu, F. Panaite and F. Van Oystaeyen, Quasi-Hopf algebra actions and smash products , Comm. Alg. 28 (2000), 631–651

    2000

  6. [13]

    Cohen, ”Number Theory Volume I: Tools and Diophantine Equations”, Graduate Texts in Mathematics239, New York: Springer-Verlag, 2007

    H. Cohen, ”Number Theory Volume I: Tools and Diophantine Equations”, Graduate Texts in Mathematics239, New York: Springer-Verlag, 2007

    2007

  7. [14]

    D˘ asc˘ alescu, C

    S. D˘ asc˘ alescu, C. N˘ ast˘ asescu, S ¸. Raianu, ”Hopf Algebras. An Introduction”,CRC Press, Boca Raton, 2000

    2000

  8. [15]

    Drinfeld, Quantum groups

    V.G. Drinfeld, Quantum groups. In ”Proc. Int. Cong. Math.” (Berkeley, 1986), 798-820. Amer. Math. Soc., Providence, RI, 1987

    1986

  9. [16]

    V. G. Drinfeld, Quasi-Hopf algebras. Leningrad Math. J. 1 (1990), 1419–1457

    1990

  10. [17]

    Etingof, S

    P. Etingof, S. Gelaki, Finite dimensional quasi-Hopf algebras with radical of codimension 2 , Math. Res. Lett. 11 (2004), 685–696

    2004

  11. [18]

    Etingof, S

    P. Etingof, S. Gelaki, On radically graded finite dimensional quasi-Hopf algebras , Mosc. Math. J. 5 (2005), 371-–378

    2005

  12. [19]

    Etingof, S

    P. Etingof, S. Gelaki, Liftings of graded quasi-Hopf algebras with radical of prime codimension , J. Pure Appl. Algebra 205 (2006), 310–322

    2006

  13. [20]

    Etingof, S

    P. Etingof, S. Gelaki, D. Nikshych, V. Ostrikr, ”Tensor categories”, Mathematical Surveys and Monographs 205, American Mathematical Society 2016

    2016

  14. [21]

    Green, ∅

    E.L. Green, ∅. Solberg, Basic Hopf algebras and quantum groups , Math. Z. 229, 45-–76, 1998

    1998

  15. [22]

    Hamada, On a free resolution of a dihedral group , Tohoku Mathematical Journal 15(3) (1963), 212–219

    S. Hamada, On a free resolution of a dihedral group , Tohoku Mathematical Journal 15(3) (1963), 212–219

    1963

  16. [23]

    Handel, On products in the cohomology of the dihedral groups , Tohoku Mathematical Journal 45(1) (1993), 13–42

    D. Handel, On products in the cohomology of the dihedral groups , Tohoku Mathematical Journal 45(1) (1993), 13–42

    1993

  17. [24]

    Hausser and F

    F. Hausser and F. Nill, Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11, 553–629, 1999

    1999

  18. [26]

    Huang, G

    HL. Huang, G. Liu, Y. Yang, Y. Ye, Finite quasi-quntum groups of diagonal type . J. Reine. Angew. Math. 759 (2020), 201-243

    2020

  19. [27]

    Majid, Quantum Double for quasi-Hopf algebras , Letters in Mathematical Physics 45(1) (1998), 1–9

    S. Majid, Quantum Double for quasi-Hopf algebras , Letters in Mathematical Physics 45(1) (1998), 1–9

    1998

  20. [28]

    Majid, ”Foundations of quantum group theory”, Cambridge University Press, 1995

    S. Majid, ”Foundations of quantum group theory”, Cambridge University Press, 1995

    1995

  21. [29]

    Matthews, The Diophantine equation x2 − Dy2 = N , D >0, Expositiones Mathematicae 18 (2000), 323–331

    K. Matthews, The Diophantine equation x2 − Dy2 = N , D >0, Expositiones Mathematicae 18 (2000), 323–331

    2000

  22. [30]

    D. E. Radford, The structure of Hopf algebras with a projection , J. Algebra 92(1985), 322-347

    1985

  23. [31]

    Panaite, A Maschke-type theorem for quasi-Hopf Algebras , in ”Rings, Hopf Algebras and Brauer Groups”, S

    F. Panaite, A Maschke-type theorem for quasi-Hopf Algebras , in ”Rings, Hopf Algebras and Brauer Groups”, S. Caenepeel and A. Verschoren (eds.), Lecture Notes in Pure and Appl. Math., Marcel Dekker, 1998

    1998

  24. [32]

    Panaite, F

    F. Panaite, F. Van Oystaeyen, A structure theorem for quasi-Hopf comodule algebras , Proc. Amer. Math. Soc. 135, 1669–1677, 2007

    2007

  25. [33]

    Radford, The structure of Hopf algebras with a projection , J

    D.E. Radford, The structure of Hopf algebras with a projection , J. Algebra 92, 322–347, 1985

    1985

  26. [34]

    D. E. Radford, Biproducts and Kashina’s Examples , Comm. Alg. 44 (2015), 174–204

    2015

  27. [35]

    D. E. Radford, Yetter-Drinfel’d categories associated to an arbitrary bialgebra, J. Pure Appl. Algebra 87 (1993), 259–279

    1993

  28. [36]

    Propitius, Topological Interactions in Broken Gauge Theories , PhD Thesis, University of Amsterdam; arXiv:hep-th/9511195v1 27 Nov 1995

    W. Propitius, Topological Interactions in Broken Gauge Theories , PhD Thesis, University of Amsterdam; arXiv:hep-th/9511195v1 27 Nov 1995

    1995 arXiv

  29. [37]

    D. N. Yetter, Quantum groups and representations of monoidal categories , Math. Proc. Cambridge Philos. Soc. 108 (1990), 261-–290. F aculty of Mathematics and Informatics, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania Email address: daniel.bulacu@f...

    1990

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.