WEAKLY GROUP-THEORETICAL AND SOLVABLE FUSION CATEGORIES

By Pavel Etingof, D. Nikshych, V. Ostrik. 2008

269 citations recorded by the source corpus.

We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups – weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group-theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq2, where p,q,r are distinct primes.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cc55e9dadb93fc7ba09e75ce625cc5d357d6c676

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases (2019, background): “Conjecturally [18], all integral fusion categories A are Morita equivalent (i.”
Anyon condensation and tensor categories (2013, methodology): “ively clear that this bulk-to-boundary map LM is given by (51). As a consequence, we have Cloc A ≃Vect [DMNO]. Moreover, the functor L∨ induces a monoidal equivalence: L∨ : C∗ M ≃CA deﬁned by F→L∨(F) [ENO2]. Namely, any quasiparticle Fin C∗ M can be realized in CA thus also in Z(C) as L∨(F). This im”
On finite-dimensional Hopf algebras (2014, background): “To my knowledge, all examples of semisimple Hopf algebras arise from group algebras by 129 the preceding constructions; this was proved in [68, 70] in low dimensions and in [32] for di130 mensions pq, pqr, where p, q and r are primes.”
Centers of graded fusion categories (2009, unknown): “Recall from [Etingof et al. 2008] that a fusion category C is called weakly integral if its Frobenius–Perron dimension is an integer.”
Exact sequences of tensor categories (2010, background): “10]: a fusion category is simple in the sense of [16] if it has no proper fusion subcategories, that is, vectk and C are the only replete fusion subcategories of C.”
Fusion categories and homotopy theory (2009, unknown): “HYCH, AND VICTOR OSTRIK 5.2. A functor Ψ : EqBr(Z(C)) → BrPic(C). Let α be a braided tensor autoequivalence of Z(C). Below we recall a construction of an invertible C-bimodule category from αgiven in [ENO3]. Let F : Z(C) → C and I: C → Z(C) denote the forgetful functor and its adjoint. Given an alge”
The classification of subfactors of index at most 5 (2013, methodology): “gives a self-dual 2221 subfactor. In addition to Xu’s construction, this argument uses the characterization of fusion categories Morita equivalent to group extensions of a given fusion category, from [ENO11]. As shown in [CMS11, Theorem 1.0.1], the 2221 fusion category yields the smallest dimension ”
A Finiteness Property for Braided Fusion Categories (2009, background): “-hard (where the last dichotomy assumes P 6= NP ).”