{"id":"51fb1c3a-2281-426b-88f9-c78136a84307","arxiv_id":"2506.18451","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Partial H-modules form a biactegory over H-modules, and for pointed Hopf algebras with finitely many grouplikes the standard dilation functor is naturally isomorphic to the Hom-object {Apar, -}.","lead":"This paper proves that the category of partial modules over a Hopf algebra is acted on by global modules from both sides, and that the standard way of turning a partial module into a global one is actually an internal Hom construction. It connects partial representation theory with enriched category theory and Hopf algebroids, yielding a new algebra Hglob that is Morita equivalent to the partial Hopf algebra Hpar in many cases.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.10 is load-bearing; its proof misstates the coradical filtration, so the induction's exhaustiveness is not established as written.","rationale":"The paper's central claim is a natural isomorphism between the dilation functor and the Hom-object {Apar,−}. The proof reduces to showing that equality on H0=kG extends to all of H; that extension is exactly Lemma 4.10. I checked the surrounding arguments and found them largely sound: Proposition 4.8's integral computation is valid, and for h∈G it does not require t to be a left integral of H; Proposition 4.5's identification of {Apar,M} with partially H-linear maps is correct; and the induction in Lemma 4.10 works once the standard coradical filtration and the Radford generator decomposition are used. The genuine issue is that the filtration is misprinted: H_n should be Δ^{-1}(H⊗H_{n-1}+H0⊗H), not Δ^{-1}(H⊗H0+H_{n-1}⊗H). With the printed definition, H is not exhausted in general, so the proof as written is incomplete. This is a concrete, repairable gap rather than a refutation; it does not change the likely validity of the main theorem but it does mean the theorem's proof requires correction. Hence I recommend conditional acceptance: the authors should fix the filtration definition and explicitly confirm the cited generator decomposition, after which the argument appears coherent.","tokens_in":30289,"tokens_out":45054,"duration_ms":423169,"concrete_test":"Replace the definition of H_n in Lemma 4.10 by the standard H_n=Δ^{-1}(H⊗H_{n-1}+H0⊗H), and run the induction explicitly for the Taft algebra T_9 (q a primitive 3rd root): verify that H_1=span{g^i, g^i x}, H_2=span{g^i x^j, j≤2}, that x^2 satisfies Δ(x^2)=g^2⊗x^2+(gx+xg)⊗x+x^2⊗1 with mixed terms in H_1⊗H_1, and that the displayed equation in Lemma 4.10 with h=x^2 c^{-1} gives g(x^2)=0 from g|H0=0. If the induction fails on this example, Lemma 4.10 and Theorem 4.11 are unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 4.11's isomorphism D≅{Apar,−} depends entirely on Lemma 4.10, which asserts that a partially H-linear map g:H→M is determined by its restriction to H0=kG. The proof inducts over the coradical filtration, but the filtration is misstated: it defines H_n=Δ^{-1}(H⊗H0+H_{n-1}⊗H), whereas the standard coradical filtration is H_n=Δ^{-1}(H⊗H_{n-1}+H0⊗H). With the printed definition, ∪H_n need not exhaust H; for example, in the Taft algebra at a primitive cube root, x^2 is in H but is not contained in any printed H_n. Thus, taken literally, the induction would not prove g=0. The intended argument is recovered by citing Radford [21, Eq. 4.5, Prop. 4.3.1] for the generator decomposition Δ(x)=c⊗x+x⊗d+Σy_i⊗z_i with c,d grouplike and y_i,z_i∈H_{n-1}. That citation is doing the essential work: if the decomposition fails, or if the filtration is taken literally, Lemma 4.10 breaks and the equality of ΨΞ_M(y) and Ψ(f) on H0 in Theorem 4.11 does not extend to H. The argument is plausible and likely repairable, but the proof as printed is not fully checkable without correcting the filtration and verifying the cited structural fact.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":30547,"tokens_out":20410,"duration_ms":199418,"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":[{"comment":"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.","section":"§4.2, Lemma 4.10"},{"comment":"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.","section":"§4.2, Proposition 4.8 and Theorem 4.11"},{"comment":"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.","section":"§5.1, Lemma 5.3 and Proposition 5.11"}],"minor_comments":[{"comment":"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.","section":"Abstract and §4.2"},{"comment":"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.","section":"§4.2, Proposition 4.8"},{"comment":"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.","section":"Introduction"},{"comment":"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.","section":"§4.2, Lemma 4.10 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is a substantial and interesting contribution, and the categorical framework in Sections 2–3 is well executed. The main issue is concentrated in Section 4.2: the proof of the pointed-case isomorphism is not fully checkable as printed because of the misstated coradical filtration and the mismatch between the element t used in Theorem 4.11 and the hypothesis of Proposition 4.8. Both issues appear repairable, so I recommend major revision rather than rejection. The referee report should ask for a corrected proof of Lemma 4.10 and a clear statement of the exact identity of t used in Proposition 4.8, together with verification that this identity holds in the pointed case."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nA 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.\n\nBottom 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.","headline":"Worth a serious referee: new categorical results on partial modules and dilations, with one terse lemma that deserves a closer look.","tokens_in":31116,"tokens_out":10649,"would_cite":true,"duration_ms":87532,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16T05","18D20","18D25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["partial modules","partial representations","dilation","globalization","biactegory","module category","enrichment","Hopf algebroid"],"falsifier":"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.","tokens_in":30065,"feed_emoji":"🔗","tokens_out":12730,"duration_ms":112681,"temperature":0.7,"pith_summary":"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.","feed_headline":"For pointed Hopf algebras, dilation of partial modules is a Hom-object","feed_subtitle":"Standard dilation equals the Hom-object from Apar, tying globalization to enrichment, exactness, Morita equivalence.","key_machinery":"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.","core_discovery":"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 $H\\mathrm{Mod}$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces partial representations of Hopf algebras, the universal algebra $H_{\\mathrm{par}}$, and the base algebra $A_{\\mathrm{par}}$ used throughout.","marker":"[5]"},{"why":"Constructs the standard dilation functor $\\mathrm{D}$ and the projection/c-condition formalism that the paper compares with Hom-objects.","marker":"[6]"},{"why":"Supplies the globalization theorem for partial Hopf actions and the Morita-equivalence theorem between partial and global smash products used in Section 5.","marker":"[3]"},{"why":"Provides the coradical filtration and generation result for pointed Hopf algebras that powers Lemma 4.10.","marker":"[21]"},{"why":"Supplies the tensor-category setting for module categories, enrichment, and Hom-objects.","marker":"[15]"},{"why":"Establishes partial group representations and the groupoid algebra $k_{\\mathrm{par}}G$, the base case generalised to $(kG)_{\\mathrm{glob}}$.","marker":"[14]"},{"why":"Gives the theorem used to show that $H_{\\mathrm{glob}}$ is a Hopf algebroid from a braided commutative algebra in Yetter-Drinfeld modules.","marker":"[8]"},{"why":"Provides the actegory machinery that turns right adjoints to the action functors into an enrichment of partial modules over global modules.","marker":"[17]"}],"fun_headline_variants":["Dilation of partial modules equals Hom-object in pointed Hopf case","Partial modules: a biactegory with dilation as Hom-object","Globalization of partial modules: dilation is Hom-object","Biactegory for partial modules over Hopf algebras","Morita equivalence from partial module dilation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Dilation of partial modules equals Hom-object in pointed Hopf case","Partial modules: a biactegory with dilation as Hom-object","Globalization of partial modules: dilation is Hom-object","Biactegory for partial modules over Hopf algebras","Morita equivalence from partial module dilation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000196,"raw_usage":{"total_tokens":1362,"prompt_tokens":951,"completion_tokens":411,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":329}},"tokens_in":567,"tokens_out":411,"duration_ms":4118,"temperature":1.0,"reasoning_tokens":329,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:50:05.382743+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces partial representations of Hopf algebras, the universal algebra $H_{\\mathrm{par}}$, and the base algebra $A_{\\mathrm{par}}$ used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs the standard dilation functor $\\mathrm{D}$ and the projection/c-condition formalism that the paper compares with Hom-objects."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the globalization theorem for partial Hopf actions and the Morita-equivalence theorem between partial and global smash products used in Section 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the coradical filtration and generation result for pointed Hopf algebras that powers Lemma 4.10."},{"cited_title":"Etingof, S","cited_arxiv_id":null,"evidence_quote":"Supplies the tensor-category setting for module categories, enrichment, and Hom-objects."},{"cited_title":"Dokuchaev, R","cited_arxiv_id":null,"evidence_quote":"Establishes partial group representations and the groupoid algebra $k_{\\mathrm{par}}G$, the base case generalised to $(kG)_{\\mathrm{glob}}$."},{"cited_title":"Brzezi´ nski, G","cited_arxiv_id":null,"evidence_quote":"Gives the theorem used to show that $H_{\\mathrm{glob}}$ is a Hopf algebroid from a braided commutative algebra in Yetter-Drinfeld modules."},{"cited_title":"Janelidze, G.M","cited_arxiv_id":null,"evidence_quote":"Provides the actegory machinery that turns right adjoints to the action functors into an enrichment of partial modules over global modules."}],"review_version":2}