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Journals and Conferences
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 grouptheoretical fusion categories.
These notes contain the material presented in a series of five lectures at the University of Córdoba in September 1994. The intent of this brief course was to give a quick introduction to Hopf algebras and to prove as directly as possible (to me) some recent results on finitedimensional Hopf algebras conjectured by Kaplansky in 1975. In particular, in the… (More)
We survey some aspects of the theory of Hopf-Galois objects that may studied advantageously by using the language of cogroupoids. These are the notes for a series of lectures given at Cordobá University, may 2010. The lectures are part of the course “HopfGalois theory” by Sonia Natale.
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(Σ), for some finite group Σ and some ω ∈ Z(Σ, k×). We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Σ are group theoretical. We… (More)
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. Received by the editor September 2006. 1991 Mathematics Subject Classification. Primary 16W30; Secondary 17B37.
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 pq, for prime numbers p, q with q|p − 1. We also show that certain twisting deformation of the symmetric group is simple as a… (More)
We determine the structure of Hopf algebras that admit an extension 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… (More)
The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal grouplike element and quantum and Frobenius-Perron dimensions of simple objects are computed.
We describe the exponent of a group-theoretical fusion category C = C(G,ω, F, α) associated to a finite group G in terms of group cohomology. We show that the exponent of C divides both e(ω) expG and (expG), where e(ω) is the cohomological order of the 3-cocycle ω. In particular exp C divides (dim C).
We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. 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 strengthening of… (More)