On Cores in Yetter-Drinfel'd Hopf Algebras

Yevgenia Kashina; Yorck Sommerhaeuser

On Cores in Yetter-Drinfel'd Hopf Algebras

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 1908.07620 v1 pith:25KQSC4P submitted 2019-08-20 math.RA math.QAmath.RT

Yevgenia Kashina , Yorck Sommerhaeuser This is my paper · ORCID

classification math.RAmath.QAmath.RT MSC 16T05

keywords Yetter-Drinfel'dHopfalgebrascoreofagroup-likeelementcocommutativecosemisimplefiniteabeliangroupelementaryZ2xbraidedtensorproductelementstriviality

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

Dear colleague,

You 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.

The 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].

The 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.

The 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.

Net: 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.

Recommendation: send it out for peer review.

Referee Report

0 major / 5 minor

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.

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.

minor comments (5)

[3.1] 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.
[Abstract and 2.1] 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'.
[1.2] 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.
[4.1] 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.
[3.5] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the counterexample is established by direct action and coaction computations, and the imported core theory from [13] is background, not an input that forces the conclusion.

full rationale

The derivation's central claim is an explicit counterexample. Sections 2 and 3 define A by generators and relations, define the automorphisms phi and phi', and construct the coaction from a nondegenerate bicharacter; they then verify directly that the proposed coproduct is an algebra homomorphism into A hat-tensor A, that the listed omega_i and eta_j form a basis of group-like elements, and that the displayed antipode formulas satisfy the antipode equations. The decisive nontriviality of the core is likewise direct: Lemma 2.4 and Lemma 3.4 give g2.omega2 = omega3, and the displayed equations delta(omega2) = 1/2(g1+g3) tensor omega2 + 1/2(g1-g3) tensor omega3 and delta(omega3) = 1/2(g1-g3) tensor omega2 + 1/2(g1+g3) tensor omega3 are derived from Proposition 2.2 and its Section 3 analogue without appeal to [13]. Section 4 reviews the definition and structural properties of the core from [13], which is a publication of the second author, and the identification of Span(omega1,...,omega4) as the core of eta1 uses that theory. This is a self-citation, but it is not circular here: the cited results are general theorems about cores, not restatements of the counterexample, and the example-specific consequences they yield (multiplication in the span, antipode stability, orbit equalities) are also displayed directly in the paper's multiplication tables and antipode formulas. The paper fits no parameter, renames no known empirical pattern, and invokes no uniqueness theorem to exclude alternatives. The final conjecture and the reference to unpublished work are explicitly conjectural and carry no weight in the derivation. Hence the derivation chain is self-contained at the point where it matters: the counterexample is verified by explicit equations.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The only free parameter is the fourth root of unity ζ, which parametrizes the family of examples. The main external input is the core theory of [13], used to define and identify cores; the constructed algebras themselves are not invented entities in the sense of unexplained postulates, since they are fully defined and verified.

free parameters (1)

ζ
Arbitrary fourth root of unity, not necessarily primitive; the constructions in Sections 2 and 3 are uniform in ζ and the conclusion holds for all choices. It is hand-chosen as a parameter of the family, not fitted to data.

assumptions (4)

domain assumption The base field K contains a primitive eighth root of unity in Section 2.
Used to define ι=ξ^2 and to write down the eight characters in Proposition 2.1 (Paragraph 2.1).
domain assumption The base field K contains a primitive fourth root of unity in Section 3.
Needed for the parameter ι in the definition of the elements ω_i and for the character table in Proposition 3.6 (Paragraph 3.1).
domain assumption The base field K is algebraically closed of characteristic zero in Section 4.
Assumed at the start of Section 4 for the dualized core theory from [13] to apply (Paragraph 4.1).
domain assumption The core theory and dualization results from [13] (Lem. 3.2, Thm. 3.3, Prop. 3.7, Thm. 3.8, Def. 3.8) are correct and apply in the cocommutative setting after dualization.
The paper imports these results rather than reproving them; they identify the core as Span(ω1,...,ωm). The explicit formulas showing non-complete triviality do not depend on this assumption.

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Cite this review

Pith. "Pith review of On Cores in Yetter-Drinfel'd Hopf Algebras." pith.science (2026). https://pith.science/paper/25KQSC4P

@misc{pith2026190807620,
  author       = {Pith},
  title        = {Pith review of: On Cores in Yetter-Drinfel'd Hopf Algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/25KQSC4P}},
  note         = {Machine review of arXiv:1908.07620}
}

read the original abstract

By constructing explicit examples, we show that the core of a group-like element in a cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over the group ring of a finite abelian group is not always completely trivial.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [13]

Sommerh¨ auser: Triviality theorems for Yetter-Drinfel’d Ho pf algebras, J

Y. Sommerh¨ auser: Triviality theorems for Yetter-Drinfel’d Ho pf algebras, J. Algebra 454 (2016), 475–519 37

work page 2016

[2] [1]

Andruskiewitsch/S

N. Andruskiewitsch/S. Natale: Braided Hopf algebras arising fro m matched pairs of groups, J. Pure Appl. Algebra 182 (2003), 119–149

work page 2003

[3] [2]

B. Farb/R. K. Dennis: Noncommutative algebra, Grad. Texts Ma th., Vol. 144, Springer, Berlin, 1993

work page 1993

[4] [3]

Jacobson: Lectures in abstract algebra III: Theory of ﬁeld s and Galois theory, Grad

N. Jacobson: Lectures in abstract algebra III: Theory of ﬁeld s and Galois theory, Grad. Texts Math., Vol. 32, Springer, Berlin, 1976

work page 1976

[5] [4]

Kashina/Y

Y. Kashina/Y. Sommerh¨ auser: Extension theory for Yetter- Drinfel’d Hopf algebras, in preparation

work page

[6] [5]

Kassel: Quantum groups, Grad

C. Kassel: Quantum groups, Grad. Texts Math., Vol. 155, Spring er, Berlin, 1995

work page 1995

[7] [6]

F. W. Long: The Brauer group of dimodule algebras, J. Algebra 30 (1974), 559–601

work page 1974

[8] [7]

Montgomery: Hopf algebras and their actions on rings, 2nd re vised printing, Reg

S. Montgomery: Hopf algebras and their actions on rings, 2nd re vised printing, Reg. Conf. Ser. Math., Vol. 82, Am. Math. Soc., Providenc e, 1997

work page 1997