{"id":"99c8786e-bb68-4eb6-9c83-d9614791f57b","arxiv_id":"1908.07620","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"The core of a group-like element in a cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over the group ring of Z2×Z2 need not be completely trivial.","lead":"The authors construct two explicit eight-dimensional algebra families over the group Z2×Z2 and show that a substructure called the core is an ordinary group algebra, yet the group acts and coacts on it non-trivially. The result answers a question left open from the prime-order case and sharpens the structure theory of Yetter-Drinfel'd Hopf algebras.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader identified the long verification of the coproduct and the reliance on [13] as the weakest assumption. My stress-test agrees that these are the only places where an error could plausibly hide, but I found no actual inconsistency. The direct action computation g2.ω2 = ω3 is simple and independent of [13], so the core is clearly not completely trivial once the core is correctly identified. The identification of the core as Span(ω1,...,ω4) is supported both by the cited dualized theory and by the explicit observation that all products η_iη_j lie in that span, matching Proposition 4.1. Because the examples are concrete and the key verifications are laid out in detail, the moderate-confidence ACCEPT verdict remains appropriate.","tokens_in":27585,"tokens_out":18094,"duration_ms":181931,"concrete_test":"Recompute the braided tensor product relations for Proposition 2.3 (and 3.3) with a computer algebra system: construct the 8-dimensional algebra A with the given generators and relations, define the K[Z2×Z2]-action and the coaction from the bicharacter, define Δ on x and y by the stated formulas, and verify that Δ is an algebra homomorphism A→A⊗A with the braided product by checking all 64 basis products. Then verify the group-like basis ω_i, η_i satisfies Δ(ω_i)=ω_i⊗ω_i and Δ(η_i)=η_i⊗η_i, and confirm the multiplication tables used in Section 4. If any of these relations fail, the counterexample collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No specific flaw found. The central claim rests on the explicit construction in Sections 2 and 3, with the most delicate point being the verification in Proposition 2.3/3.3 that the proposed coproduct is an algebra homomorphism into the braided tensor product. I re-examined the argument structure: the braided product formula is applied consistently, the required relations for x' and y' are checked in detail, and the subsequent H-linearity and coassociativity arguments are sound. The non-complete triviality of the core is supported by direct equations g2.ω2 = ω3 and the displayed formulas for δ(ω2), which are elementary and do not depend on the external core theory. The only substantive external dependency is the dualized core theory of [13]; it is invoked in a standard way, with the relevant properties of the core (subalgebra, antipode stability, orbit span) all explicitly matching the direct computations in the examples. I do not find a load-bearing gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that the core of a group-like element in a finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over the group ring of a finite abelian group need not be completely trivial. To prove this, it constructs two explicit eight-dimensional examples over K[Z2×Z2], one commutative and one noncommutative, depending on a fourth root of unity ζ. In each case the algebra is equipped with an action and a coaction of K[Z2×Z2] via automorphisms and a bicharacter, and a braided Hopf coproduct is defined by explicit formulas. Sections 2 and 3 verify the defining relations, H-linearity and H-colinearity, coassociativity via a basis of group-like elements, and the antipode. Section 4 reviews a dualized version of the core theory from [13] and identifies the core of the group-like element η1 with Span(ω1,...,ω4), which is isomorphic as an ordinary Hopf algebra to K[Z2×Z2]. Direct computations show that g2.ω2 = ω3 and that δ(ω2) = 1/2(g1+g3)⊗ω2 + 1/2(g1-g3)⊗ω3, so the core is not completely trivial. The paper closes with the conjecture that cores are nevertheless always trivial over the index group.","tokens_in":27677,"tokens_out":38980,"duration_ms":541790,"significance":"If the constructions are correct, the paper settles a natural question left open by the prime-order theory in [11,12]: complete triviality of cores fails in general for cocommutative cosemisimple Yetter-Drinfel'd Hopf algebras over finite abelian group rings. The counterexample is robust: it is verified by explicit action and coaction formulas, multiplication tables, and antipode computations, not by an abstract existence argument, and it covers all choices of the parameter ζ, including non-primitive fourth roots of unity. The paper is also commendably explicit about the delicate braided tensor-product verifications in Propositions 2.3 and 3.3, which are the most likely places for an error to hide. The conjecture at the end is clearly labeled as such and is consistent with the examples, whose cores are trivial but not completely trivial.","major_comments":[],"minor_comments":[{"comment":"The sentence 'This fact can be shown as in Lemma 2.1, or in fact be deduced from the re' is incomplete; it should end with 'from the relations' or a similar phrase.","section":"3.1"},{"comment":"There are typographical slips ('comp letely' in the abstract and 'It comes at no surprise' in Paragraph 2.1); these should be corrected to 'completely' and 'It comes as no surprise'.","section":"Abstract and 2.1"},{"comment":"The proof of Lemma 1.2 is too compressed: the first displayed equality appears to apply the antipode to a product and then multiply by the unsymmetrized factor, and the order of multiplication is not explained. Please expand the computation or replace it with a precise citation to the original argument.","section":"1.2"},{"comment":"The assertion that the stated hypotheses imply that A has a basis consisting of group-like elements is made without proof or reference; a precise citation to [13] or to an earlier paper would remove a gap for readers who want to apply the core theory beyond the explicit examples.","section":"4.1"},{"comment":"In the proof of Proposition 3.5, after verifying η_i S_A(η_i) = ω1 for all i, the conclusion that S_A(η_i)η_i = ω1 uses the finite-dimensionality of A, since a right inverse in a finite-dimensional algebra is two-sided; this fact should be stated explicitly.","section":"3.5"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is sound in its central claim. The dependence of Section 4 on [13], written by the second author, is transparent, and the counterexample part is verified directly in Sections 2 and 3. The remaining issues are presentation and clarification items only."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know this paper gives the first explicit counterexamples showing that cores in Yetter-Drinfel'd Hopf algebras over finite abelian groups of composite order need not be completely trivial, even though they are trivial as ordinary Hopf algebras. The examples are eight-dimensional, over K[Z2×Z2], one commutative and one noncommutative, uniform in a fourth-root-of-unity parameter ζ.\n\nThe construction is the real content, and it is solid. The paper defines the algebras by generators and relations, verifies the braided tensor-product compatibility for the coproduct in detail, exhibits group-like bases, computes antipodes, and checks semisimplicity. I spot-checked the coaction formula for δ(x), the product tables, and the non-complete triviality equations (g2.ω2 = ω3 and the mixed coaction on ω2). Everything holds up. The core is shown to be isomorphic to K[Z2×Z2], hence trivial in the braided sense, but the group action and coaction restrict nontrivially to it. That is exactly the claimed phenomenon, and it contrasts with the prime-order theory in [11,12].\n\nThe soft spots are minor. Section 4 imports the dualized core theory from [13], written by the second author, rather than reproving it. That is a real dependency, but it is applied in a standard way, and the properties used for the examples are verified directly from the multiplication tables. The base-field conventions shift between sections (primitive eighth root in Section 2, primitive fourth root in Section 3, algebraically closed characteristic zero in Section 4); the implicit base change to an algebraically closed field is standard but never stated. Also, the authors acknowledge the examples are crossed products over Z2 and defer a systematic treatment to a forthcoming paper. That is honest, though it means the paper is not the full structure theory.\n\nThe conjecture at the end—that cores are always trivial, though not completely trivial—is clearly labeled and supported only by these examples plus unpublished work. Fine.\n\nNet: the paper does what it claims. The construction is explicit, checkable, and new. It deserves a serious referee; some referee time on the long verifications would be useful, but I do not expect a fatal flaw. I would cite this if I worked in the area, and would probably bring it to reading group.\n\nRecommendation: send it out for peer review.","headline":"A concrete, checkable counterexample showing that cores in finite abelian Yetter-Drinfel'd Hopf algebras can be trivial as Hopf algebras yet not completely trivial; the construction is sound and deserves peer review.","tokens_in":28248,"tokens_out":1861,"would_cite":true,"duration_ms":477434,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs explicit eight-dimensional Yetter-Drinfel'd Hopf algebras over the group ring of $\\mathbb{Z}_2\\times\\mathbb{Z}_2$ whose group-like element has a core that is trivial as an ordinary Hopf algebra but not completely…","keywords":["Yetter-Drinfel'd Hopf algebras","core of a group-like element","cocommutative cosemisimple","finite abelian group","elementary abelian group Z2 x Z2","braided tensor product","group-like elements","triviality"],"falsifier":"A direct symbolic recomputation of the defining relations for $x'$ and $y'$ in $A\\hat{\\otimes}A$ in either example would settle the central claim: if any of $x'^4=1$, the relevant $y'^2$ relation, or the commutation relation $x'y'=y'x'^3$ in the noncommutative case fails, the construction collapses. If all relations pass, one can then compute $\\delta(\\omega_2)$ directly and check whether it equals $\\frac{1}{2}(g_1+g_3)\\otimes\\omega_2 + \\frac{1}{2}(g_1-g_3)\\otimes\\omega_3$; equality with $1\\otimes\\omega_2$ would show the core is completely trivial after all.","tokens_in":27334,"feed_emoji":"","tokens_out":9509,"duration_ms":89251,"temperature":0.7,"pith_summary":"The paper constructs explicit eight-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebras over the group ring of $\\mathbb{Z}_2\\times\\mathbb{Z}_2$, in which a group-like element has a core that is trivial as an ordinary Hopf algebra but not completely trivial: the group both acts and coacts nontrivially on the core. This settles in the negative the natural extension of the prime-order case, where the core is always completely trivial. If the construction is correct, the core of a group-like element cannot in general be reduced to an ordinary Hopf algebra with no extra Yetter-Drinfel'd structure. The authors also conjecture that the core is nevertheless always trivial as a Yetter-Drinfel'd Hopf algebra over the index group.","feed_headline":"Cores of group-like elements need not be completely trivial","feed_subtitle":"Eight-dimensional examples: the core can be an ordinary Hopf algebra while the group action and coaction stay nontrivial.","key_machinery":"The load-bearing construction is a pair of explicit algebras $A$ with generators $x,y$ and relations that are commutative in Section 2 and skew in Section 3, made into Yetter-Drinfel'd Hopf algebras over $K[G]$ by letting $g_2,g_3$ act by two commuting order-two automorphisms $\\varphi,\\varphi'$ and defining the coaction from a nondegenerate symmetric bicharacter $\\theta$ on $G$, i.e. a bi-multiplicative pairing of group elements into the base field. The coproduct is prescribed on generators, and the whole argument depends on verifying that the proposed images $x'$ and $y'$ satisfy the same defining relations in the braided tensor product $A\\hat{\\otimes}A$. Coassociativity is obtained by exhibiting a basis of group-like elements, and the core of $\\eta_1$ is then identified with the subalgebra spanned by $\\omega_1,\\dots,\\omega_4$. The nontriviality of the Yetter-Drinfel'd structure on this core is read off directly from the formulas for the $G$-action and the $\\theta$-coaction.","core_discovery":"Working over a field containing suitable roots of unity, the authors exhibit two eight-dimensional Yetter-Drinfel'd Hopf algebras $A$ over $H=K[G]$ with $G=\\mathbb{Z}_2\\times\\mathbb{Z}_2$, one commutative and one noncommutative. Each has a basis of group-like elements, split into $\\omega_1,\\dots,\\omega_4$ and $\\eta_1,\\dots,\\eta_4$. For the group-like element $\\eta_1$, the core is $\\operatorname{Span}(\\omega_1,\\dots,\\omega_4)$, which is isomorphic as a Hopf algebra to $K[G]$ and is therefore trivial in the braided sense. Yet the action and coaction are nontrivial: $g_2.\\omega_2 = \\omega_3$, and $\\delta(\\omega_2) = \\frac{1}{2}(g_1+g_3)\\otimes\\omega_2 + \\frac{1}{2}(g_1-g_3)\\otimes\\omega_3$, in both examples. This is the first demonstration that complete triviality fails beyond the prime-order case.","pith_inferences":["Beyond the paper: a natural stress test is to repeat the construction with larger elementary abelian groups and higher-rank bicharacters; if nontrivial cores appear there, the phenomenon is not an artefact of dimension eight.","Beyond the paper: the parameter $\\zeta$ runs through fourth roots of unity, so the two examples are really small families; comparing the commutative and noncommutative members suggests that the core's nontriviality depends on the action and coaction data, not on the commutativity of $A$.","Beyond the paper: if the concluding conjecture holds, classifying finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebras over abelian groups would naturally split into classifying trivial cores over index groups and classifying the extensions built on them, with the present examples serving as first instances of nontrivial extensions over a trivial core."],"forward_implications":["The prime-order pattern does not extend: for finite abelian groups with more than one element, complete triviality of cores is not automatic.","The core can be an ordinary Hopf algebra while the ambient Yetter-Drinfel'd Hopf algebra is genuinely braided, so the failure of complete triviality is exactly a failure of the $G$-action and coaction to be trivial on that subalgebra.","Any classification of finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebras over abelian group rings must record not only the core's algebra structure but also the induced action and coaction over the index group.","The authors conjecture that the core is nevertheless always trivial as a Yetter-Drinfel'd Hopf algebra over the index group, so the only non-complete-triviality would occur through the index-group action rather than through the core's internal Hopf structure."],"supporting_citations":[{"why":"Supplies the theory of cores and index groups in the commutative case, which the paper dualizes to define and compute cores for cocommutative algebras.","marker":"[13]"},{"why":"Establishes the prime-order classification in which cores are always completely trivial, the result the present examples contradict.","marker":"[12]"},{"why":"Earlier exposition of the prime-order classification and of the definitions of trivial and completely trivial Yetter-Drinfel'd Hopf algebras.","marker":"[11]"},{"why":"Provides the dual Yetter-Drinfel'd Hopf algebra structure on $A^*$ used in the dualization argument in Proposition 4.1.","marker":"[10]"},{"why":"Gives the criterion that a Yetter-Drinfel'd Hopf algebra that is an ordinary Hopf algebra is trivial, used when calling the core trivial.","marker":"[8]"}],"fun_headline_variants":["Core trivial as Hopf algebra, yet action and coaction nontrivial","First counterexample: core trivial, Yetter-Drinfeld structure not","Cores needn't be completely trivial: new explicit examples","Trivial core, nontrivial action: first examples over Z2×Z2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire counterexample rests on the long computation in Proposition 2.3 and its noncommutative analogue showing that the proposed coproduct is a well-defined algebra homomorphism into the braided tensor product; if that verification has a hidden mistake, the constructed object is not a Yetter-Drinfel'd Hopf algebra and the claimed counterexample collapses.","fun_headline_variants_meta":{"raw":{"variants":["Core trivial as Hopf algebra, yet action and coaction nontrivial","First counterexample: core trivial, Yetter-Drinfeld structure not","Cores needn't be completely trivial: new explicit examples","Trivial core, nontrivial action: first examples over Z2×Z2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00075,"raw_usage":{"total_tokens":3266,"prompt_tokens":800,"completion_tokens":2466,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":416,"completion_tokens_details":{"reasoning_tokens":2385}},"tokens_in":416,"tokens_out":2466,"duration_ms":18110,"temperature":1.0,"reasoning_tokens":2385,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:01:43.967569+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct symbolic recomputation of the defining relations for $x'$ and $y'$ in $A\\hat{\\otimes}A$ in either example would settle the central claim: if any of $x'^4=1$, the relevant $y'^2$ relation, or the commutation relation $x'y'=y'x'^3$ in the noncommutative case fails, the construction collapses. If all relations pass, one can then compute $\\delta(\\omega_2)$ directly and check whether it equals $\\frac{1}{2}(g_1+g_3)\\otimes\\omega_2 + \\frac{1}{2}(g_1-g_3)\\otimes\\omega_3$; equality with $1\\otimes\\omega_2$ would show the core is completely trivial after all.","supporting_citations":[{"cited_title":"Sommerh¨ auser: Triviality theorems for Yetter-Drinfel’d Ho pf algebras, J","cited_arxiv_id":null,"evidence_quote":"Supplies the theory of cores and index groups in the commutative case, which the paper dualizes to define and compute cores for cocommutative algebras."},{"cited_title":"Sommerh¨ auser: Yetter-Drinfel’d Hopf algebras over grou ps of prime order, Lect","cited_arxiv_id":null,"evidence_quote":"Establishes the prime-order classification in which cores are always completely trivial, the result the present examples contradict."},{"cited_title":"Sommerh¨ auser: Yetter-Drinfel’d Hopf algebras over grou ps of prime order, Dissertation, M¨ unchen, 1999","cited_arxiv_id":null,"evidence_quote":"Earlier exposition of the prime-order classification and of the definitions of trivial and completely trivial Yetter-Drinfel'd Hopf algebras."},{"cited_title":"Sommerh¨ auser: Deformed enveloping algebras, New York J","cited_arxiv_id":null,"evidence_quote":"Provides the dual Yetter-Drinfel'd Hopf algebra structure on $A^*$ used in the dualization argument in Proposition 4.1."},{"cited_title":"Schauenburg: On the braiding on a Hopf algebra in a braided cat egory, New York J","cited_arxiv_id":null,"evidence_quote":"Gives the criterion that a Yetter-Drinfel'd Hopf algebra that is an ordinary Hopf algebra is trivial, used when calling the core trivial."}],"review_version":1}