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Classification of fusion categories of dimension pq

Pavel Etingof, Shlomo Gelaki, Viktor Ostrik

TL;DR

We classify fusion categories over $\mathbb{C}$ with Frobenius-Perron dimension $pq$ for distinct primes $p<q$, showing a dichotomy: if $p=2$ the category is Tambara-Yamagami of dimension $2q$, otherwise it is group-theoretical. The analysis combines pseudounitary/fusion-category theory, Drinfeld centers, and cohomology $H^3(G,\mathbb{C}^*)$ to enumerate all invertible-object cases, and uses quasi-Hopf algebra methods to treat integer-dimension cases, yielding a complete classification. As a by-product, we classify finite-dimensional semisimple quasi-Hopf algebras with irreps of dimensions $1$ and $n$ whose $1$-dimensional reps form a cyclic group, showing they are group-theoretical, and we describe families with exactly one non-invertible object of dimension $n$. The results connect fusion categories of small FP dimension to explicit group-theoretical constructions and Frobenius-group data, enabling concrete listings and corollaries for Hopf/quasi-Hopf representation theories.

Abstract

In this paper we provide a complete classification of fusion categories of Frobenius-Perron (FP) dimension pq, where p<q are distinct primes, thus giving a categorical generalization of math.QA/9801129. As a corollary we also obtain the classification of semisimple quasi-Hopf algebras of dimension pq. A concise formulation of our main result is: Let C be a fusion category over the complex numbers of FP dimension pq. Then either p=2 and C is a Tambara-Yamagami category of dimension 2q, or C is group-theoretical in the sense of math.QA/0203060 (which easily yields the full classification). As a by-product, we obtain the classification of finite dimensional semisimple quasi-Hopf (in particular, Hopf) algebras whose irreducible representations have dimensions 1 and n, such that the 1-dimensional representations form a cyclic group of order n. All such quasi-Hopf algebras turn out to be group-theoretical. We also classify fusion categories whose invertible objects form a cyclic group of order n>1 and which have only one non-invertible object of dimension n.

Classification of fusion categories of dimension pq

TL;DR

We classify fusion categories over with Frobenius-Perron dimension for distinct primes , showing a dichotomy: if the category is Tambara-Yamagami of dimension , otherwise it is group-theoretical. The analysis combines pseudounitary/fusion-category theory, Drinfeld centers, and cohomology to enumerate all invertible-object cases, and uses quasi-Hopf algebra methods to treat integer-dimension cases, yielding a complete classification. As a by-product, we classify finite-dimensional semisimple quasi-Hopf algebras with irreps of dimensions and whose -dimensional reps form a cyclic group, showing they are group-theoretical, and we describe families with exactly one non-invertible object of dimension . The results connect fusion categories of small FP dimension to explicit group-theoretical constructions and Frobenius-group data, enabling concrete listings and corollaries for Hopf/quasi-Hopf representation theories.

Abstract

In this paper we provide a complete classification of fusion categories of Frobenius-Perron (FP) dimension pq, where p<q are distinct primes, thus giving a categorical generalization of math.QA/9801129. As a corollary we also obtain the classification of semisimple quasi-Hopf algebras of dimension pq. A concise formulation of our main result is: Let C be a fusion category over the complex numbers of FP dimension pq. Then either p=2 and C is a Tambara-Yamagami category of dimension 2q, or C is group-theoretical in the sense of math.QA/0203060 (which easily yields the full classification). As a by-product, we obtain the classification of finite dimensional semisimple quasi-Hopf (in particular, Hopf) algebras whose irreducible representations have dimensions 1 and n, such that the 1-dimensional representations form a cyclic group of order n. All such quasi-Hopf algebras turn out to be group-theoretical. We also classify fusion categories whose invertible objects form a cyclic group of order n>1 and which have only one non-invertible object of dimension n.

Paper Structure

This paper contains 7 sections, 20 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Categories with non-integer dimensions
  4. Categories with 1-dimensional simple objects
  5. Categories with integer dimensions, not all equal to $1$
  6. Categories with integer dimensions
  7. Categories with simple objects of dimension $1$ and $n$

Key Result

Theorem 1.1

Let $\mathcal{C}$ be a fusion category over $\Bbb C$ of FP dimension $pq$, where $p<q$ are distinct primes. Then either $p=2$ and $\mathcal{C}$ is a Tambara-Yamagami category of dimension $2q$ (ty), or $\mathcal{C}$ is group-theoretical in the sense of eno.

Theorems & Definitions (44)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • proof
  • Theorem 5.1
  • Lemma 5.2
  • ...and 34 more