Globalization and the biactegory of partial modules

Eliezer Batista; Joost Vercruysse; William Hautekiet

REVIEW 3 major objections 4 minor 24 references

Globalization and the biactegory of partial modules

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The category of partial modules is a bimodule category over global modules, and for pointed Hopf algebras with finite grouplikes the dilation functor is a Hom-object.

desk verdict Worth a serious referee: new categorical results on partial modules and dilations, with one terse lemma that deserves a closer look. read the letter →

arxiv 2506.18451 v1 pith:4CDYMPU2 submitted 2025-06-23 math.RA math.CTmath.QAmath.RT

Eliezer Batista , William Hautekiet , Joost Vercruysse This is my paper · ORCID

classification math.RAmath.CTmath.QAmath.RT MSC 16T0518D2018D25

keywords partialmodulesrepresentationsdilationglobalizationbiactegorymodulecategoryenrichmentHopfalgebroid

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

Partial representations of a Hopf algebra satisfy a weakened version of the module axioms, and the paper establishes that the category they form is a two-sided module category (a biactegory) over the category of global modules, via the ordinary tensor product. Since the actions have right adjoints, partial modules are enriched over global modules, with Hom-objects built from partially linear maps. The paper then ties this categorical structure to the classical globalization problem: for any partial module there is a natural injective map from its standard dilation into the Hom-object $\{A_{\mathrm{par}},M\}$, and for pointed Hopf algebras with finitely many grouplikes over characteristic zero this map is an isomorphism. The consequence is that dilation becomes an exact functor with adjoints, and a new Hopf algebroid $H_{\mathrm{glob}}$ is constructed whose modules, under a properness condition, recover partial modules.

What carries the argument

The biactegory structure is carried by the diagonal tensor product actions, while the central object is the base algebra $A_{\mathrm{par}}\subseteq H_{\mathrm{par}}$, generated by $\varepsilon_h=[h_{(1)}][S(h_{(2)})]$, together with the Hom-object $\{M,N\}=\mathrm{Hom}_{H\mathrm{PMod}}(M\otimes H,N)$. The Hom-object $\{A_{\mathrm{par}},M\}$ computes a minimal dilation of $M$, with projection $T=\theta\circ\kappa$. The proof that $\Xi:\mathrm{D}\Rightarrow\{A_{\mathrm{par}},-\}$ is an isomorphism for pointed $H$ with finite grouplikes runs through Lemma 4.10, which uses the coradical filtration of a pointed Hopf algebra to show that partially $H$-linear maps are determined by their restriction to the coradical $H_0=kG$. This restriction lemma is what upgrades the injective natural transformation into an isomorphism.

What would settle it

Take the four-dimensional noncommutative noncocommutative Hopf algebra $H_4$ over a field of characteristic zero and the trivial partial module $k$; compute $\{A_{\mathrm{par}},k\}=\mathrm{Hom}_{H\mathrm{PMod}}(A_{\mathrm{par}}\otimes H_4,k)$ and compare it with the standard dilation $\mathrm{D}(k)$ under $\Xi_k$. A dimension mismatch, or a partially $H_4$-linear map $H_4\to k$ that vanishes on $kG$ without vanishing everywhere, would falsify Theorem 4.11; Lemma 4.10 predicts no such map exists.

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Extended reading notes

Core claim

The paper's central discovery is that partial modules are not an isolated category: for any Hopf algebra $H$, the category $H\mathrm{PMod}$ is a biactegory over $H\mathrm{Mod}$ for the ordinary tensor product (Theorem 2.2), and the resulting actions have right adjoints, making $H\mathrm{PMod}$ enriched over $H\mathrm{Mod}$ with Hom-objects $[M,N]$ and $\{M,N\}$ (Theorem 3.4). The main structural result is Theorem 4.11: when $H$ is a pointed Hopf algebra with finitely many grouplikes over a field of characteristic zero, the natural transformation $\Xi:\mathrm{D}\Rightarrow\{A_{\mathrm{par}},-\}$ is a natural isomorphism. In concrete terms, every partial module $M$ has its standard dilation $\mathrm{D}(M)$ isomorphic to $\mathrm{Hom}_{H\mathrm{PMod}}(A_{\mathrm{par}}\otimes H,M)$, the Hom-object from the partial Hopf algebra's base algebra. The paper then constructs the Hopf algebroid $H_{\mathrm{glob}}=\{A_{\mathrm{par}},A_{\mathrm{par}}\}\#H$ and, assuming the globalization of $A_{\mathrm{par}}$ is proper, proves $H_{\mathrm{glob}}$ is Morita equivalent to $H_{\mathrm{par}}$ and that $\{A_{\mathrm{par}},-\}$ is an equivalence onto modules over $\{A_{\mathrm{par}},A_{\mathrm{par}}\}$ in $H\mathrm{Mod}$.

Load-bearing premise

The load-bearing premise is Lemma 4.10: a partially $H$-linear map $g:H\to M$ is completely determined by its restriction to the coradical $H_0=kG$; if two such maps agreed on all grouplike elements but differed higher in the filtration, the dilation-to-Hom-object isomorphism would fail.

Editorial extensions

If this is right

For every pointed Hopf algebra with finitely many grouplikes over characteristic zero, the standard dilation functor is naturally isomorphic to $\{A_{\mathrm{par}},-\}$, so dilation is exact and has both a left and a right adjoint.
For any Hopf algebra, the natural transformation $\Xi:\mathrm{D}\Rightarrow\{A_{\mathrm{par}},-\}$ has injective components, so each standard dilation embeds canonically into the corresponding Hom-object, with surjectivity under the integral condition of Proposition 4.8.
For finite-dimensional pointed Hopf algebras, dilations of partial modules can be viewed as modules over the single Hopf algebroid $H_{\mathrm{glob}}=\{A_{\mathrm{par}},A_{\mathrm{par}}\}\#H$.
When the globalization of $A_{\mathrm{par}}$ is proper, $H_{\mathrm{glob}}$ is Morita equivalent to $H_{\mathrm{par}}$ and the enrichment functor $\{A_{\mathrm{par}},-\}$ is an equivalence onto $\{A_{\mathrm{par}},A_{\mathrm{par}}\}$-modules in $H\mathrm{Mod}$.
For a finite group $G$, the chain of isomorphisms $k_{\mathrm{glob}}G\cong B(G)\#kG\cong (kG)_{\mathrm{glob}}$ identifies the new construction with the previously studied groupoid algebra of globalized partial actions.

Reading between the lines

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

The pointed hypothesis in Theorem 4.11 enters only through the coradical restriction lemma and the normalized integral on $kG$; an immediate next test is whether pointed Hopf algebras with infinite grouplike groups, or non-semisimple pointed examples, still satisfy the isomorphism, possibly after replacing the average over $G$ by another summation.
Proposition 4.8 converts the isomorphism question into checking whether some $b\in A_{\mathrm{par}}$ satisfies $t\cdot b=1_{A_{\mathrm{par}}}$; checking this elementwise for non-semisimple pointed Hopf algebras would give a cheap, explicit certificate for globalizability.
The groupoid-algebra description for finite groups suggests that $H_{\mathrm{glob}}$ may have a combinatorial basis for other finite-dimensional pointed Hopf algebras, indexed by orbits of the grouplike action on suitable subsets; if that holds, dilations become computable linear algebra.

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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

Worth a serious referee. The paper actually delivers something new: it turns the category of partial H-modules into a biactegory over HMod, works out the resulting enrichment, and shows that for pointed Hopf algebras with finite grouplikes the standard dilation functor is naturally isomorphic to the Hom-object {Apar,-}. The Hopf algebroid Hglob, Morita equivalent to Hpar in the finite-dimensional pointed case, is a nice addition and generalizes the group case. The authors build on their own prior work [3,5,6] for the standard dilation and the Hopf algebroid structure of Hpar, but those carry proofs, so the reliance is legitimate rather than circular.

The main thing I'd want a referee to check is Lemma 4.10, because Theorem 4.11 rests on it. The lemma says a partially H-linear map out of H is determined by its restriction to the coradical. The proof is terse and the displayed coradical filtration looks nonstandard—H_n is written as Δ^{-1}(H⊗H0+H_{n-1}⊗H) rather than the usual H⊗H_{n-1}+H0⊗H. I chased the Taft algebra example in the stress-test note: the claimed counterexample doesn't hold up, since x^2 lands in the printed H_2. So this is either a harmless reformulation or a typo that the Radford structural decomposition repairs. The argument is likely correct but deserves explicit cleaning up before publication.

A few "direct checks" are left to the reader in the biactegory and Hglob sections, but they look routine. The Morita equivalence proof depends on properness of the globalization in a way that's stated as a hypothesis in Theorem 5.10; for the pointed case Theorem 4.11 supplies it. That dependency is handled honestly.

Bottom line: this is a careful algebra paper with real new structure, not a repackaging. It should go to a serious referee, with a request to pin down the proof of Lemma 4.10 and expand the most compressed 'direct checks.' I would cite it if I work in this area; I'd also bring it to a reading group on categorical partial actions.

Referee Report

3 major / 4 minor

Summary. The paper develops a categorical framework for partial modules over a Hopf algebra H. Its main claims are: (1) the category HPMod of partial H-modules is a biactegory over the monoidal category HMod of global H-modules (Theorem 2.2); (2) HPMod is enriched over HMod and over HMod^rev via the Hom-objects [M,N] and {M,N} (Theorem 3.4); (3) for pointed Hopf algebras with finite group of grouplikes, the standard dilation functor D is naturally isomorphic to the Hom-object functor {Apar,−} (Theorem 4.11); and (4) for finite-dimensional H, the algebra Hglob = {Apar,Apar}#H is a Hopf algebroid and is Morita equivalent to Hpar, with the group case recovering known constructions such as kglobG (Theorem 5.6, Theorem 5.10, Proposition 5.11).

Significance. The paper gives a conceptual bridge between globalization of partial representations and enrichment of partial modules over global modules. If the main theorem holds, it provides a new structural explanation for the standard dilation and connects the partial Hopf algebra Hpar to a Hopf algebroid Hglob. The paper is strong on explicit categorical constructions and includes a useful analysis of the finite group case, where the results recover and unify known groupoid-algebra descriptions. The main theorems are supported by lengthy explicit proofs, and the group-case computations are concrete and checkable. However, the proof of the central pointed-case isomorphism in Section 4.2 contains gaps that need repair before the main claim is fully established.

major comments (3)

[§4.2, Lemma 4.10] The coradical filtration is printed as H_n = Δ^{-1}(H⊗H0 + H_{n-1}⊗H). This is not the standard coradical filtration used in the cited source; the standard form is H_n = Δ^{-1}(H⊗H_{n-1} + H0⊗H) (with the convention of Radford). With the printed definition, the union of the H_n need not exhaust H, so the induction showing that g|H0 = 0 implies g = 0 is not established as written. Since Lemma 4.10 is the step that extends equality on H0 to equality on all of H in Theorem 4.11, the filtration must be corrected and the citation to [21, Eq. 4.5, Prop. 4.3.1] checked against that corrected definition.
[§4.2, Proposition 4.8 and Theorem 4.11] Theorem 4.11 defines t = (1/|G|)∑_{g∈G} g and calls it a left integral in the coradical H0 = kG. Proposition 4.8, however, is stated for a left integral t of H itself. These are different conditions: for the Sweedler Hopf algebra H4, G = {1,g} and t = (1/2)(1+g) is not a left integral of H4 (with the usual relations, xt = (1/2)(x − gx) ≠ 0 while ε(x)t = 0). Moreover, the proof of Proposition 4.8 uses the displayed equality f(h(1)t(1)•b ⊗ h(2)t(2)S(t(3))) = f(h(1)t•b ⊗ h(2)) without stating which identity of t justifies it. The authors need to either replace t by a genuine left integral of H for which t•b = 1, or state and prove a variant of Proposition 4.8 whose hypotheses are satisfied by the normalized sum of grouplikes.
[§5.1, Lemma 5.3 and Proposition 5.11] Lemma 5.3 is load-bearing for the construction of the coaction on {Apar,Apar} and for Proposition 5.11, but its proof omits the verification that the inverse map Θ^{-1} lands in {M,N} and that Θ is H-linear, saying only 'one can verify' and 'direct check'. Since this is a central structural lemma, the omitted verification should either be supplied in full or replaced by a precise reference to [15, Lemma 7.9.4] with the relevant translation to the present partial-module setting.

minor comments (4)

[Abstract and §4.2] The abstract states the isomorphism result for finite-dimensional pointed Hopf algebras, while Theorem 4.11 states only that H is pointed with finite group of grouplikes. These statements should be aligned, especially because the proof uses a distinguished element t whose status as an integral depends on the setting.
[§4.2, Proposition 4.8] The parenthetical remark that invertibility of ε(t) implies that ε(t) is invertible in k and hence 'H is necessarily semisimple' is imprecise: without a finite-dimensionality assumption, invertibility of a scalar ε(t) does not imply semisimplicity of H, and this assertion is not used later.
[Introduction] There are several typos, e.g. 'there in a natural injective morphism' should read 'there is a natural injective morphism', and the phrase 'a left integral in the coradical H0 = kG' in Theorem 4.11 should specify that t is a left integral of kG, not of H.
[§4.2, Lemma 4.10 proof] In the induction step, the statement 'Since c,d,z_i ∈ H_{n-1}' should more precisely read 'c,d ∈ H0 ⊆ H_{n-1} and z_i ∈ H_{n-1}', since c and d are grouplike.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main isomorphism is derived from explicit integral and coradical arguments, not assumed by construction.

full rationale

The paper's central claims are proved from the definitions and from previously published external results, with no fitted parameter renamed as a prediction and no target result built into an input. The biactegory structure (Theorem 2.2) is verified from the H-bicomodule algebra structure on Hpar (Lemma 2.1); the enrichment (Theorem 3.4) is obtained from explicit adjunctions (Proposition 3.3). The comparison D ⇒ {Apar,−} is constructed in Proposition 4.3 as an injective natural transformation using the universal property of the standard dilation; the surjectivity in the pointed case (Theorem 4.11) is proved via the integral condition of Proposition 4.8, the explicit inverse of ε_t in the group case (Lemma 4.9), and the coradical-filtration lemma 4.10, which cites Radford's textbook for the generator decomposition. None of these steps defines {Apar,M} as D(M) or assumes the isomorphism. Prior work [3,5,6] supplies definitions, the existence of standard dilations, and the Hopf algebroid structure of Hpar; these are independent published inputs, and their use does not make Theorem 4.11 equivalent to an assumption. The possible misstatement of the coradical filtration in Lemma 4.10 is a correctness or checkability concern, not a circularity: the printed proof cites a structural decomposition rather than presupposing the conclusion it is used to prove.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper constructs new algebras Hglob and {Apar,Apar}, but these are explicitly defined from existing data (Hom-modules and smash products) and carry no independent postulation, so there are no invented entities in the sense of unexplained new objects.

assumptions (6)

domain assumption H is a Hopf algebra over a field k with invertible antipode
Standing assumption at the start of Section 1; all partial module theory and the biactegory theorem depend on it.
domain assumption Hpar is a Hopf algebroid over Apar and HPMod is monoidal with unit Apar
From [5]; it provides the monoidal structure and is used throughout, especially in Section 3 and in the definition of {Apar,-}.
domain assumption Every partial H-module has a proper minimal standard dilation and the category of dilations is equivalent to HPMod
From [6, Theorem 4.3]; this is the definition of D used in Proposition 4.3 and Theorem 4.11.
standard math For pointed Hopf algebras, the coradical filtration has the generator property stated in [21, Proposition 4.3.1]
Used in Lemma 4.10 to prove that partially linear maps are determined on the coradical.
domain assumption H is pointed with finite grouplikes over a field of characteristic zero (Theorem 4.11 and Corollary 4.12)
Explicit hypothesis; guarantees epsilon_t invertible in Apar(kG) by Lemma 4.9.
domain assumption H is finite-dimensional (Theorem 5.6 and Section 5)
Needed for the Yetter-Drinfeld structure on {Apar,Apar} and for Lemma 5.3.

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Pith. "Pith review of Globalization and the biactegory of partial modules." pith.science (2026). https://pith.science/paper/4CDYMPU2

@misc{pith2026250618451,
  author       = {Pith},
  title        = {Pith review of: Globalization and the biactegory of partial modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4CDYMPU2}},
  note         = {Machine review of arXiv:2506.18451}
}

abstract

We show that the category of partial modules over a Hopf algebra $H$ is a biactegory (a bimodule category) over the category of global $H$-modules. The corresponding enrichment of partial modules over global modules is described, and the close relation between the dilation of partial modules and Hom-objects arising from this enrichment is investigated. In particular, for finite-dimensional pointed Hopf algebras, the standard dilation of a partial module $M$ is isomorphic to the Hom-object from the monoidal unit to $M$.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Abadie: Enveloping actions and Takai duality for partial actions , J

F. Abadie: Enveloping actions and Takai duality for partial actions , J. Funct. Anal.197 (2003) 14-67. 2

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Abadie: Dilations of interaction groups that extend actions of Ore semigroups , J

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