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Weakly group-theoretical and solvable fusion categories

Pavel Etingof, Dmitri Nikshych, Victor Ostrik

TL;DR

The paper introduces weakly group-theoretical and solvable fusion categories, establishing that the former satisfy a strong Frobenius property and proving a Burnside-type result that dimensions with at most two prime divisors imply solvability. It develops a center-based Morita framework, analyzes module categories over equivariantized categories, and proves foundational theorems linking Morita equivalence, centers, and de-equivariantization. These results yield powerful consequences for classifying fusion categories and semisimple Hopf algebras of small or square-free dimension, including complete classifications for dimensions pq^2 and pqr and a detailed picture at dimension 60. The work significantly enhances the connection between group-theoretical data and fusion category structure, providing new tools for structural and classification problems in modular and nondegenerate fusion categories.

Abstract

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 pq^2, where p,q,r are distinct primes.

Weakly group-theoretical and solvable fusion categories

TL;DR

The paper introduces weakly group-theoretical and solvable fusion categories, establishing that the former satisfy a strong Frobenius property and proving a Burnside-type result that dimensions with at most two prime divisors imply solvability. It develops a center-based Morita framework, analyzes module categories over equivariantized categories, and proves foundational theorems linking Morita equivalence, centers, and de-equivariantization. These results yield powerful consequences for classifying fusion categories and semisimple Hopf algebras of small or square-free dimension, including complete classifications for dimensions pq^2 and pqr and a detailed picture at dimension 60. The work significantly enhances the connection between group-theoretical data and fusion category structure, providing new tools for structural and classification problems in modular and nondegenerate fusion categories.

Abstract

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 pq^2, where p,q,r are distinct primes.

Paper Structure

This paper contains 32 sections, 45 theorems, 24 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Graded categories
  4. Frobenius-Perron dimensions in a module category
  5. Weakly integral and integral categories
  6. Tannakian categories
  7. Morita equivalence
  8. Equivariantization and de-equivariantization (BM3 and DGNO2)
  9. The crossed product fusion category
  10. The Müger centralizer
  11. Müger's theorem (M4, DGNO2)
  12. Slightly degenerate categories
  13. Interpretation of extensions and equivariantizations in terms of the center (see BM1M3DGNO2GNN)
  14. The divisibility theorems
  15. Kac algebras (abelian extensions) N1
  16. ...and 17 more sections

Key Result

Theorem 1.3

Let $\mathcal{D}$ be a fusion category and let $G$ be a finite group. A fusion category $\mathcal{C}$ is Morita equivalent to a $G$-extension of $\mathcal{D}$ if and only if its Drinfeld center $\mathcal{Z}(\mathcal{C})$ contains a Tannakian subcategory $\mathcal{E}=\text{Rep}(G)$ such that the de-e

Theorems & Definitions (98)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Proposition 2.2
  • Remark 2.3
  • Theorem 2.4
  • ...and 88 more