Sonia Natale

National University of Córdoba | UNC · Center of Research and Studies in Mathematics, Córdoba (CIEM)

A finite-dimensional Hopf algebra is called quasi-split if it is Morita equivalent to a split abelian extension of Hopf algebras. Combining results of Schauenburg and Negron, it is shown that every quasi-split finite-dimensional Hopf algebra satisfies the finite generation cohomology conjecture of Etingof and Ostrik. This is applied to a family of...

We present a rich source of Hopf algebras starting from a cofinite central extension of a Noetherian Hopf algebra and a subgroup of the algebraic group of characters of the central Hopf subalgebra. The construction is transparent from a Tannakian perspective. We determine when the new Hopf algebras are co-Frobenius, or cosemisimple, or Noetherian,...

We present an overview of the notions of exact sequences of Hopf algebras and tensor categories and their connections. We also present some examples illustrating their main features; these include simple fusion categories and a natural question regarding composition series of finite tensor categories.

We study exact sequences of finite tensor categories of the form Rep G → 𝒞 → 𝒟, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual actions by permutations ⊳ : Γ × G → G and ⊲ : Γ × G → Γ that make ( G , Γ) into matched pair of groups endowed with a natural crossed action on 𝒟 such that 𝒞 is equiva...

We study a class of strictly weakly integral fusion categories $\mathfrak{I}_{N, \zeta}$, where $N \geq 1$ is a natural number and $\zeta$ is a $2^N$th root of unity, that we call $N$-Ising fusion categories. An $N$-Ising fusion category has Frobenius-Perron dimension $2^{N+1}$ and is a graded extension of a pointed fusion category of rank 2 by the...

Let C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {C}$\end{document} be a modular category of Frobenius-Perron dimension dqⁿ, where q > 2 is a prime number a...

We study exact sequences of finite tensor categories of the form $\Rep G \to \C \to \D$, where $G$ is a finite group. We show that, under suitable assumptions, there exists a group $\Gamma$ and mutual actions by permutations $\rhd: \Gamma \times G \to G$ and $\lhd: \Gamma \times G \to \Gamma$ that make $(G, \Gamma)$ into matched pair of groups endo...

We show that the core of a weakly group-theoretical braided fusion category $\C$ is equivalent as a braided fusion category to a tensor product $\B \boxtimes \D$, where $\D$ is a pointed weakly anisotropic braided fusion category, and $\B \cong \vect$ or $\B$ is an Ising braided category. In particular, if $\C$ is integral, then its core is a point...

We show that the core of a weakly group-theoretical braided fusion category $\C$ is equivalent as a braided fusion category to a tensor product $\B \boxtimes \D$, where $\D$ is a pointed weakly anisotropic braided fusion category, and $\B \cong \vect$ or $\B$ is an Ising braided category. In particular, if $\C$ is integral, then its core is a point...

Let $p$ and $q$ be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $pq^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocy...

Let $p$ and $q$ be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $pq^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocy...

We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category $\C$ to be equivalent as $\C$-module categories.

We classify integral modular categories of dimension pq 4 and p 2 q 2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q 2 . In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum...

We classify integral modular categories of dimension pq^4 and p^2q^2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q^2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum gr...

We classify integral modular categories of dimension pq 4 and p 2 q 2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q 2 . In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum...

We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable non-degenerate braided fusion categories. We also give some sufficient conditions for a...

We determine the fusion rules of the equivariantization of a fusion category \documentclass[12pt]{minimal}\begin{document}${\mathcal {C}}$\end{document}C under the action of a finite group G in terms of the fusion rules of \documentclass[12pt]{minimal}\begin{document}${\mathcal {C}}$\end{document}C and group-theoretical data associated to the group...

We show that a weakly integral braided fusion category C such that every simple object of C has Frobenius-Perron dimension at most 2 is solvable. In addition, we prove that such a fusion category is group-theoretical in the extreme case where the universal grading group of C is trivial.

We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization under a finite group or finite group scheme action. We introduce the notion of central exact sequence of tensor categories and use it in order to pres...

We establish some relations between the orders of simple objects in a fusion category and the structure of its universal grading group. We consider fusion categories which have a faithful simple object and show that its universal grading group must be cyclic. As for the converse, we prove that a braided nilpotent fusion category with cyclic univers...

We give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups. We also give examples of finite quantum groups which are not quantum permutation groups: one such example occurs as a split abelian extension associated to the exact factoriza...

We prove some results on the structure of certain classes of integral fusion categories and semisimple Hopf algebras under restrictions on the set of its irreducible degrees.

We show that bicrossed product Hopf algebras arising from exact factorizations in almost simple finite groups, so in particular, in simple and symmetric groups, admit no quasitriangular structure. Comment: 11 pages, amslatex

We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of tensor categories generalize strictly exact sequences of Hopf algebras as defined by Schneider, and in particular, exact sequences of (finite) groups. We classify exact sequences of tensor categories (such that is finite) in te...

We determine the isomorphism classes of semisimple Hopf algebras of dimension 60 which are simple as Hopf algebras.

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z_2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a streng...

We give a general description of the structure of a discrete double groupoid (with an extra, quite natural, filling condition) in terms of groupoid factorizations and groupoid 2-cocycles with coefficients in abelian group bundles. Our description goes as follows: to any double groupoid, we associate an abelian group bundle and a second double group...

We determine the structure of Hopf algebra extensions of a group algebra by the cyclic group of order 2. We study the corepresentation theory of such Hopf algebras, which provide a generalization, at the Hopf algebra level, of the so called Tambara-Yamagami fusion categories. As a byproduct, we show that every semisimple Hopf algebra of dimension $...

Let G be a finite group and let π : G → G′ be a surjective group homomorphism. Consider the cocycle deformation L = H σ of the Hopf algebra H = k G of k-valued linear functions on G, with respect to some convolution invertible 2-cocycle σ. The (normal) Hopf subalgebra \(k^{G'} \subseteq k^{G}\) corresponds to a Hopf subalgebra \(L' \subseteq L\). O...

We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and p^2q^2, for prime numbers p, q with q dividing p-1. We also show that certain twisting deformation of the symmetric...

We study a natural construction of Hopf algebra quotients canonically associated to an R-matrix in a finite-dimensional Hopf algebra. We apply this construction to show that a quasitriangular Hopf algebra whose dimension is odd and square-free is necessarily cocommutative.

We describe the exponent of a group-theoretical fusion category $\mathcal C = \mathcal C(G, \omega, F, \alpha)$ associated to a finite group $G$ in terms of group cohomology. We show that the exponent of $\C$ divides both $e(\omega) \exp G$ and $(\exp G)^2$, where $e(\omega)$ is the cohomological order of the 3-cocycle $\omega$. In particular $\exp...

We show that fusion categories $\Rep(\ku^{\sigma}_{\tau} \Tc)$ of representations of the weak Hopf algebra coming from a vacant double groupoid $\Tc$ and a pair $(\sigma, \tau)$ of compatible 2-cocyles are group-theoretical.

The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. This extends previous work that appeared in math.QA/0308228. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal group-like element and...

Let p and q be distinct prime numbers. We prove a result on the existence of nontrivial group-like elements in a certain class of semisimple Hopf algebras of dimension pq r . We conclude the classification of semisimple Hopf algebras A of dimension pq 2 over an algebraically closed field k of characteristic zero, such that both A and A * are of Fr...

We give an explicit description, up to gauge equivalence, of group-theoretical quasi-Hopf algebras. We use this description to compute the Frobenius-Schur indicators for group-theoretical fusion categories.

We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the Dijkgraaf-Pasquier-Roche quasi-Hopf algebra D-omega(Sigma), for some finite group Sigma and some omega is an element of Z(3) (Sigma, k(x)). We show that semisimple Hopf algebras obtained as bicrossed products from an exact factoriza...

We give the construction of a class of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data. This generalizes a now well-known construction for Hopf algebras, first studied by G. I. Kac in the sixties. Our approach is based on the notion of double groupoids, as introduced by Ehresmann.

Let be a field. Let also (F,G) be a matched pair of groups. We give necessary and sufficient conditions on a pair (σ,τ) of 2-cocycles in order that the crossed product algebra and the crossed coproduct coalgebra combine into a braided Hopf algebra. We also discuss diagonal realizations of such braided Hopf algebras in the category of Yetter–Drinfel...

We prove that every semisimple Hopf algebra of dimension less than 60 over an algebraically closed field k of characteristic zero is either upper or lower semisolvable up to a cocycle twist.

We announce recent progress on the question about the semisolv- ability of semisimple Hopf algebras of dimension < 60.

We conclude the classification of Hopf algebras of dimension 12 over an algebraically closed field of characteristic zero.

Let k be a field. Let also (F, G) be a matched pair of groups. We give necessary and sufficient conditions on a pair (\sigma, \tau) of 2-cocycles in order that the crossed product algebra and the crossed coproduct coalgebra k^G{}^{\tau}#_{\sigma} kF combine into a braided Hopf algebra. We also discuss diagonal realizations of such braided Hopf alge...

We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the Dijkgraaf-Pasquier-Roche quasi-Hopf algebra D^{omega}(Sigma), for some finite group Sigma and some 3-cocycle omega on Sigma. We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finit...

Let p and q be odd prime numbers. It is shown that all quasitriangular Hopf algebras of dimension pq over an algebraically closed field k of characteristic zero are semisimple and therefore isomorphic to a group algebra.

http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=528121&tocid=100080702&page=187-201

We obtain further classification results for semisimple Hopf algebras of dimension pq 2 over an algebraically closed field k of characteristic zero. We complete the classification of semisimple Hopf algebras of dimension 28.

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