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arxiv: 2006.15267 · v2 · submitted 2020-06-27 · 🧮 math.RA · math.RT

Four-angle Hopf modules for Hom-Hopf algebras

Xiaoqian Liu , Dongdong Yan , Xuchen Deng , Danhua Wang This is my paper

Pith reviewed 2026-05-24 14:15 UTC · model grok-4.3

classification 🧮 math.RA math.RT
keywords four-angle Hopf modulesHom-Hopf algebrasYetter-Drinfel'd modulesmonoidal categoriesbraided monoidal categoriescategory equivalence
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The pith

The category of four-angle Hopf modules over a Hom-Hopf algebra is monoidally equivalent to the category of Yetter-Drinfel'd modules with bijective structure maps and inherits a braiding.

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

The paper defines four-angle Hopf modules for a Hom-Hopf algebra and shows that their category forms a monoidal category using either a Hom-tensor product or a Hom-cotensor product. It equips the category of Yetter-Drinfel'd modules with bijective structure maps with a new monoidal structure that makes it braided. The main result establishes an equivalence of monoidal categories between the four-angle Hopf modules and these Yetter-Drinfel'd modules, which transfers the braiding to the four-angle categories.

Core claim

Four-angle Hopf modules are objects that carry compatible left and right module and comodule structures over a Hom-Hopf algebra. Their category with either the Hom-tensor product or the Hom-cotensor product as monoidal product is equivalent as a monoidal category to the category of Yetter-Drinfel'd modules equipped with bijective structure maps and a new monoidal structure, and the equivalence yields a braiding on the four-angle categories.

What carries the argument

The monoidal equivalence between the category of four-angle Hopf modules (~!^H_H M^H_H) and the category of Yetter-Drinfel'd modules YD^H_H that transfers the braiding.

Load-bearing premise

The structure maps on the Yetter-Drinfel'd modules must be bijective.

What would settle it

An explicit four-angle Hopf module over some Hom-Hopf algebra whose corresponding Yetter-Drinfel'd module has a non-bijective structure map, or a failure of the equivalence to preserve the monoidal product.

read the original abstract

In this paper, we introduce the notion of a four-angle Hopf module for a Hom-Hopf algebra $(H,\beta)$ and show that the category $\!^{H}_{H}\mathfrak{M}^{H}_{H}$ of four-angle Hopf modules is a monoidal category with either a Hom-tensor product $\otimes_{H}$ or a Hom-cotensor product $\Box_{H}$ as a monoidal product. We study the category $\mathcal{YD}^{H}_{H}$ of Yetter-Drinfel'd modules with bijective structure map can be organized as a braided monoidal category, in which we use a new monoidal structure. Finally, We prove an equivalence between the monoidal category $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\otimes_{H})$ or $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\Box_{H})$ of four-angle Hopf modules, and the monoidal category $\mathcal{YD}^{H}_{H}$ of Yetter-Drinfel'd modules, and furthermore, we give a braiding structure of the monoidal categorys $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\otimes_{H})$ (and $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\Box_{H})$).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces four-angle Hopf modules over a Hom-Hopf algebra (H, β) and shows that the category !^H_H M^H_H of such modules forms a monoidal category under either the Hom-tensor product ⊗_H or the Hom-cotensor product □_H. It organizes the category YD^H_H of Yetter-Drinfel'd modules (restricted to those with bijective structure maps) as a braided monoidal category via a new monoidal structure, proves a monoidal equivalence between (!^H_H M^H_H, ⊗_H) (resp. □_H) and YD^H_H, and equips the four-angle categories with a braiding.

Significance. If the stated equivalences and braidings hold, the work would connect Hopf module theory with Yetter-Drinfel'd modules in the Hom-algebra setting, offering a route to construct braided monoidal categories from four-angle modules. The explicit use of bijective structure maps and a declared new monoidal structure on YD^H_H avoids hidden assumptions and supports the central claims.

major comments (2)
  1. [Abstract] Abstract, final paragraph: the equivalence is stated between the monoidal categories (~!^H_H M^H_H, ⊗_H) and YD^H_H, but the text must verify that the equivalence functors preserve the respective monoidal products (⊗_H or □_H on one side and the new monoidal structure on the other); without an explicit check of monoidality of the functors, the claim that the categories are equivalent as monoidal categories is not yet load-bearing.
  2. [Abstract] Abstract, final paragraph: the braiding on (~!^H_H M^H_H, ⊗_H) is asserted to be induced from the equivalence, yet the naturality of this braiding with respect to the four-angle module morphisms must be confirmed directly (or shown to follow from the YD braiding via the equivalence); this is central to the additional claim of a braiding structure on the four-angle side.
minor comments (2)
  1. [Abstract] Abstract: replace 'categorys' with 'categories' (twice) and correct the notation (~!^H_H M^H_H) for consistency with standard Hom-Hopf module notation.
  2. [Abstract] Abstract: the phrase 'we use a new monoidal structure' should be expanded to a brief indication of how the new product differs from the standard tensor product on YD modules.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the monoidal structures and braidings. We address the two major comments point by point below. The manuscript already contains explicit constructions of the equivalence functors and the induced braiding, but we agree that additional explicit verifications will strengthen the presentation and will incorporate them in the revision.

read point-by-point responses

  1. Referee: [Abstract] Abstract, final paragraph: the equivalence is stated between the monoidal categories (~!^H_H M^H_H, ⊗_H) and YD^H_H, but the text must verify that the equivalence functors preserve the respective monoidal products (⊗_H or □_H on one side and the new monoidal structure on the other); without an explicit check of monoidality of the functors, the claim that the categories are equivalent as monoidal categories is not yet load-bearing.

    Authors: The equivalence functors are constructed explicitly in Sections 4–5 by sending a four-angle Hopf module to its underlying Yetter-Drinfel'd module (and conversely) while preserving the bijective structure maps. The proofs already verify that these functors intertwine the Hom-tensor (resp. Hom-cotensor) product with the new monoidal structure on YD^H_H by direct computation on the coactions and actions. To make the monoidality check fully load-bearing and visible at a glance, we will add a short dedicated paragraph immediately after the statement of the equivalence theorem that isolates the preservation of the monoidal products. revision: yes

  2. Referee: [Abstract] Abstract, final paragraph: the braiding on (~!^H_H M^H_H, ⊗_H) is asserted to be induced from the equivalence, yet the naturality of this braiding with respect to the four-angle module morphisms must be confirmed directly (or shown to follow from the YD braiding via the equivalence); this is central to the additional claim of a braiding structure on the four-angle side.

    Authors: Because the equivalence functors are equivalences of categories (hence fully faithful and essentially surjective), naturality of the braiding on the four-angle side follows formally from naturality on the YD side once monoidality of the functors is established. Nevertheless, to address the referee’s request for a direct confirmation, we will insert a brief direct computation of naturality with respect to four-angle morphisms (using the explicit form of the braiding map induced by the YD braiding) immediately after the definition of the braiding on (!^H_H M^H_H, ⊗_H). The same paragraph will cover the cotensor case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained category equivalences

full rationale

The paper defines four-angle Hopf modules explicitly from Hom-Hopf algebra axioms, equips their category with monoidal structures via Hom-tensor or Hom-cotensor products, and proves an equivalence to the category of Yetter-Drinfel'd modules (restricted to bijective structure maps) equipped with a new monoidal structure. These steps rely on direct verification of category axioms and functors, not on fitted parameters, self-referential definitions, or load-bearing self-citations. The bijective-map condition and new monoidal structure are declared upfront rather than smuggled in. No step reduces the claimed equivalence to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claims rest on the standard definition of monoidal and braided categories plus the new definition of four-angle Hopf module; no numerical parameters are fitted.

axioms (2)
  • standard math Axioms of monoidal categories and braided monoidal categories
    Invoked to verify that the Hom-tensor and Hom-cotensor products make the category monoidal and braided.
  • domain assumption Hom-Hopf algebra axioms including the twisting map beta
    Background structure assumed for all constructions.
invented entities (1)
  • four-angle Hopf module no independent evidence
    purpose: New module object with four compatible (co)actions over a Hom-Hopf algebra
    Definitional construct introduced to extend module theory; no independent existence proof outside the definition.

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