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Juan Cuadra

Publications and source records attributed to Juan Cuadra.

15 recordsLinked to original sources
arXiv2025

Existence of integral Hopf orders in twists of group algebras

We find a group-theoretical condition under which a twist of a group algebra, in Movshev's way, admits an integral Hopf order. Let $K$ be a (large enough) number field with ring of integers $R$. Let $G$ be a finite group and $M$ an abelian subgroup of $G$ of central type. Consider the twist $J$ for $K\hspace{-0.8pt}G$ afforded by a non-degenerate $2$-cocycle on the character group $\widehat{M}$. We show that if there is a Lagrangian decomposition $\widehat{M} \simeq L \times \widehat{L}$ such that $L$ is contained in a normal abelian subgroup $N$ of $G$, then the twisted group algebra $(K\hspace{-0.8pt}G)_J$ admits a Hopf order $X$ over $R$. The Hopf order $X$ is constructed as the $R$-submodule generated by the primitive idempotents of $K\hspace{-1.1pt}N$ and the elements of $G$. It is indeed a Hopf order of $K\hspace{-0.8pt}G$ such that $J^{\pm 1} \in X \otimes_R X$. Furthermore, we give some criteria for this Hopf order to be unique. We illustrate this construction with several families of examples. As an application, we provide a further example of simple and semisimple complex Hopf algebra that does not admit integral Hopf orders.

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arXiv2022

Non-existence of integral Hopf orders for twists of several simple groups of Lie type

Let $p$ be a prime number and $q=p^m$, with $m \geq 1$ if $p \neq 2,3$ and $m>1$ otherwise. Let $Ω$ be any non-trivial twist for the complex group algebra of $\mathbf{PSL}_2(q)$ arising from a $2$-cocycle on an abelian subgroup of $\mathbf{PSL}_2(q)$. We show that the twisted Hopf algebra $(\mathbb{C} \mathbf{PSL}_2(q))_Ω$ does not admit a Hopf order over any number ring. The same conclusion is proved for the Suzuki groups, and for $\mathbf{SL}_3(p)$ when the twist stems from an abelian $p$-subgroup. This supplies new families of complex semisimple (and simple) Hopf algebras that do not admit a Hopf order over any number ring. The strategy of the proof is formulated in a general framework that includes the finite simple groups of Lie type. As an application, we combine our results with two theorems of Thompson and Barry and Ward on minimal simple groups to establish that for any finite non-abelian simple group $G$ there is a twist $Ω$ for $\mathbb{C} G$, arising from a $2$-cocycle on an abelian subgroup of $G$, such that $(\mathbb{C} G)_Ω$ does not admit a Hopf order over any number ring. This partially answers in the negative a question posed by Meir and the second author.

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arXiv2018

Non-existence of Hopf orders for a twist of the alternating and symmetric groups

We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfel'd twists of group algebras for the following groups: $A_n$, the alternating group on $n$ elements, with $n \geq 5$; and $S_{2m}$, the symmetric group on $2m$ elements, with $m \geq 4$ even. The twist for $A_n$ arises from a $2$-cocycle on the Klein four-group contained in $A_4$. The twist for $S_{2m}$ arises from a $2$-cocycle on a subgroup generated by certain transpositions which is isomorphic to $\mathbb{Z}_2^m$. This provides more examples of complex semisimple Hopf algebras that can not be defined over number rings. As in the previous family known, these Hopf algebras are simple.

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arXiv2018

Orders of Nikshych's Hopf algebra

Let $p$ be an odd prime number and $K$ a number field having a primitive $p$-th root of unity $ζ.$ We prove that Nikshych's non-group theoretical Hopf algebra $H_p$, which is defined over $\mathbb{Q}(ζ)$, admits a Hopf order over the ring of integers $\mathcal{O}_K$ if and only if there is an ideal $I$ of $\mathcal{O}_K$ such that $I^{2(p-1)} = (p)$. This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over $\mathcal{O}_K$ exists, it is unique and we describe it explicitly.

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arXiv2016

Finite dimensional Hopf actions on Weyl algebras

We prove that any action of a finite dimensional Hopf algebra H on a Weyl algebra A over an algebraically closed field of characteristic zero factors through a group action. In other words, Weyl algebras do not admit genuine finite quantum symmetries. This improves a previous result by the authors, where the statement was established for semisimple H. The proof relies on a refinement of the method previously used: namely, considering reductions of the action of H on A modulo prime powers rather than primes. We also show that the result holds, more generally, for algebras of differential operators. This gives an affirmative answer to a question posed by the last two authors.

math.RA↗
arXiv2016

Finite dimensional Hopf actions on central division algebras

Let $\mathbb{k}$ be an algebraically closed field of characteristic zero. Let $D$ be a division algebra of degree $d$ over its center $Z(D)$. Assume that $\mathbb{k}\subset Z(D)$. We show that a finite group $G$ faithfully grades $D$ if and only if $G$ contains a normal abelian subgroup of index dividing $d$. We also prove that if a finite dimensional Hopf algebra coacts on $D$ defining a Hopf-Galois extension, then its PI degree is at most $d^2$. Finally, we construct Hopf-Galois actions on division algebras of twisted group algebras attached to bijective cocycles.

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arXiv2015

Semisimple Hopf actions on Weyl algebras

We study actions of semisimple Hopf algebras H on Weyl algebras A over a field of characteristic zero. We show that the action of H on A must factor through a group algebra; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.

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arXiv2014

On the existence of orders in semisimple Hopf algebras

We show that there is a family of complex semisimple Hopf algebras that do not admit a Hopf order over any number ring. They are Drinfel'd twists of certain group algebras. The twist contains a scalar fraction which makes impossible the definability of such Hopf algebras over number rings. We also prove that a complex semisimple Hopf algebra satisfies Kaplansky's sixth conjecture if and only if it admits a weak order, in the sense of Rumynin and Lorenz, over the integers.

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arXiv2013

On two finiteness conditions for Hopf algebras with nonzero integral

A Hopf algebra is co-Frobenius when it has a nonzero integral. It is proved that the composition length of the indecomposable injective comodules over a co-Frobenius Hopf algebra is bounded. As a consequence, the coradical filtration of a co-Frobenius Hopf algebra is finite; this confirms a conjecture by Sorin Dăscălescu and the first author. The proof is of categorical nature and the same result is obtained for Frobenius tensor categories of subexponential growth. A family of co-Frobenius Hopf algebras that are not of finite type over their Hopf socles is constructed, answering so in the negative another question by the same authors.

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arXiv2012

On the structure of (co-Frobenius) Hopf algebras

We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra is co-Frobenius if and only if its Hopf coradical is so and the diagram is finite dimensional. We also prove that the standard filtration of such Hopf algebras is finite. Finally, we show that extensions of co-Frobenius (resp. cosemisimple) Hopf algebras are co-Frobenius (resp. cosemisimple).

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arXiv2009

A sequence to compute the Brauer group of certain quasi-triangular Hopf algebras

A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra $B$ in a braided monoidal category $\C$, and under certain assumptions on the braiding (fulfilled if $\C$ is symmetric), we construct a sequence for the Brauer group $\BM(\C;B)$ of $B$-module algebras, generalizing Beattie's one. It allows one to prove that $\BM(\C;B) \cong \Br(\C) \times \Gal(\C;B),$ where $\Br(\C)$ is the Brauer group of $\C$ and $\Gal(\C;B)$ the group of $B$-Galois objects. We also show that $\BM(\C;B)$ contains a subgroup isomorphic to $\Br(\C) \times \Hc(\C;B,I),$ where $\Hc(\C;B,I)$ is the second Sweedler cohomology group of $B$ with values in the unit object $I$ of $\C$. These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct $B \times H$, where $H$ is a usual Hopf algebra over a field $K$, the Hopf subalgebra generated by the quasi-triangular structure $\R$ is contained in $H$ and $B$ is a Hopf algebra in the category ${}_H\M$ of left $H$-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that $\BM(K,H,\R) \times \Hc({}_H\M;B,K)$ is a subgroup of the Brauer group $\BM(K,B \times H,\R),$ confirming the suspicion that a certain cohomology group of $B \times H$ (second lazy cohomology group was conjectured) embeds into $\BM(K,B \times H,\R).$ New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.

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arXiv2009

On the subgroup structure of the full Brauer group of Sweedler Hopf algebra

We introduce a family of three parameters 2-dimensional algebras representing elements in the Brauer group BQ(k,H_4) of Sweedler Hopf algebra H_4 over a field k. They allow us to describe the mutual intersection of the subgroups arising from a quasitriangular or coquasitriangular structure. We also introduce a new subgroup of BQ(k,H_4) whose elements are represented by algebras for which the two natural Z_2-gradings coincide. We construct an exact sequence relating this subgroup to the Brauer group of Nichols 8-dimensional Hopf algebra E(2) with respect to the quasitriangular structure attached to the 2x2-matrix N with 1 in the (1,2)-entry and zero elsewhere.

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arXiv2007

On the Hopf-Schur group of a field

Let k be any field. We consider the Hopf-Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a Hopf algebra over k, revealing so that the Hopf-Schur group can be much larger than the Schur group of k.

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