Weakly group-theoretical and solvable fusion categories

Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes of fusion categories generated by a single non-invertible simple object: near groups, those fusion categories with one non-invertible simple object, and Haagerup-Izumi categories, those with one non-invertible simple object for every invertible object. Examples of both types arise as representations of finite or quantum groups or as Jones standard invariants of finite-depth Murray-von Neumann subfactors. We utilize the Evans-Gannon computation of the tube algebras to obtain formulae for the Frobenius-Schur indicators of objects in both of these families.

Given an action of a finite group G on a fusion category \({\mathcal{C}}\) we give a criterion for the category of G-equivariant objects in \({\mathcal{C}}\) to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non-group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in [7]. Namely, we show that certain \({\mathbb{Z}}\)/2\({\mathbb{Z}}\)-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami [26] are equivalent to representation categories of non-group-theoretical semisimple Hopf algebras. We describe these Hopf algebras as extensions and show that they are upper and lower semisolvable.

Hopf algebras have important connections to quantum theory, Lie algebras, knot and braid theory, operator algebras and other areas of physics and mathematics. They have been intensely studied in the past; in particular, the solution of a number of conjectures of Kaplansky from the 1970s has led to progress on the classification of semisimple Hopf algebras and on the structure of pointed Hopf algebras. Among the topics covered are results toward the classification of finite-dimensional Hopf algebras (semisimple and non-semisimple), as well as what is known about the extension theory of Hopf algebras. Some papers consider Hopf versions of classical topics, such as the Brauer group, while others are closer to work in quantum groups. The book also explores the connections and applications of Hopf algebras to other fields.

We use a version of Haboush's theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic $p$ to characteristic zero (arXiv/math:0203060, Section 9), showing that, moreover, any isomorphism between such structures can be reduced modulo $p$. This fills a gap in arXiv/math:0203060, Subsection 9.3. We also show that lifting of semisimple cosemisimple Hopf algebras is a fully faithful functor, and prove that lifting induces an isomorphism on Picard and Brauer-Picard groups. Finally, we show that a subcategory or quotient category of a separable multifusion category is separable (resolving an open question from arXiv/math:0203060, Subsection 9.4), and use this to show that certain classes of tensor functors between lifts of separable categories to characteristic zero can be reduced modulo $p$.

The paper considers a classification of semisimple Hopf algebras having exactly one irreducible non-one-dimensional representation under a certain condition on the number of group elements.

Weights for symmetric and general linear groups

Variants of some of the Brauer-Fowler theorems

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.

Finite Groups With Many Conjugate Elements

Let K be a commutative noetherian ring. It is proved that a representation of a finite group on a K-module of finite length or on a K-module of finite exponent has a finite basis for its identities. In particular, this implies an earlier result of Nguyen Hung Shon and the author stating that every representation of a finite group over a field is finitely based. The problem whether every representation of a finite group over a commutative noetherian ring is finitely based still remains open.

The notion of globally irreducible representations of finite groups was introduced by B.H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell–Weil lattices of elliptic curves. It has been observed by R. Gow and Gross that irreducible Weil representations of certain finite classical groups lead to globally irreducible representations. In this paper we classify all globally irreducible representations coming from Weil representations of finite classical groups.

We consider a subclass of the class of group-theoretical fusion categories: To every finite group G and subgroup H one can associate the category of G-graded vector spaces with a two-sided H-action compatible with the grading. We derive a formula that computes higher Frobenius-Schur indicators for the objects in such a category using the combinatorics and representation theory of the groups involved in their construction. We calculate some explicit examples for inclusions of symmetric groups.

A computational method for determining the automorphism group of a finite solvable group

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 action. As an application we obtain a formula for the fusion rules in an equivariantization of a pointed fusion category in terms of group-theoretical data. This entails a description of the fusion rules in any braided group-theoretical fusion category.