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Quantum subgroups of the Haagerup fusion categories

Pinhas Grossman, Noah Snyder

TL;DR

This work classifies Ocneanu quantum subgroups for the Haagerup fusion categories $\mathscr{H}_1$ and $\mathscr{H}_2$, revealing exactly three simple module categories for each and introducing a new Morita-equivalent fusion category $\mathscr{H}_3$ with the same fusion ring as $\mathscr{H}_2$. It determines all subfactors whose principal even part is one of $\mathscr{H}_1$, $\mathscr{H}_2$, or $\mathscr{H}_3$, and computes the full lattice of intermediate subfactors, the outer automorphism groups, and the Brauer-Picard groupoid, showing trivial outer automorphisms and a three-object Morita 3-groupoid. The paper also generalizes these results to Izumi subfactors associated with odd-order cyclic groups, notably treating the $\mathbb{Z}/5\mathbb{Z}$ case with a complete set of Morita equivalences, module categories, and principal graphs. Collectively, these results illuminate the quantum-subgroup structure and Morita theory of Haagerup and Izumi fusion categories and provide a framework for understanding related subfactor lattices.

Abstract

We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the `"quantum subgroups" in the sense of Ocneanu), we find all subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equivalent to each of them. This third fusion category has six simple objects and the same fusion rules as one of the even parts of the Haagerup subfactor, but has not previously appeared in the literature. We also find the full lattice of intermediate subfactors for every subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors.

Quantum subgroups of the Haagerup fusion categories

TL;DR

This work classifies Ocneanu quantum subgroups for the Haagerup fusion categories and , revealing exactly three simple module categories for each and introducing a new Morita-equivalent fusion category with the same fusion ring as . It determines all subfactors whose principal even part is one of , , or , and computes the full lattice of intermediate subfactors, the outer automorphism groups, and the Brauer-Picard groupoid, showing trivial outer automorphisms and a three-object Morita 3-groupoid. The paper also generalizes these results to Izumi subfactors associated with odd-order cyclic groups, notably treating the case with a complete set of Morita equivalences, module categories, and principal graphs. Collectively, these results illuminate the quantum-subgroup structure and Morita theory of Haagerup and Izumi fusion categories and provide a framework for understanding related subfactor lattices.

Abstract

We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the `"quantum subgroups" in the sense of Ocneanu), we find all subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equivalent to each of them. This third fusion category has six simple objects and the same fusion rules as one of the even parts of the Haagerup subfactor, but has not previously appeared in the literature. We also find the full lattice of intermediate subfactors for every subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors.

Paper Structure

This paper contains 12 sections, 46 theorems, 3 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Translating between fusion category language and subfactor language
  4. Fusion categories, module categories, and bimodule categories
  5. Algebra objects
  6. Subfactors
  7. The Haagerup subfactor
  8. Algebra objects, principal graphs and subfactors in the Haagerup categories
  9. The intermediate subfactor lattices
  10. Outer automorphisms and the Brauer-Picard Groupoid
  11. The Izumi subfactors
  12. The Izumi subfactor for $\mathbb{Z} / 5 \mathbb{Z}$

Key Result

Theorem 1.1

There are exactly three quantum subgroups of each of $\mathscr{H}_1$ and $\mathscr{H}_2$. The quantum subgroups of $\mathscr{H}_1$ have the following graphs for fusion with the object of dimension $\frac{1+\sqrt{13}}{2}$. $$$$$$ The quantum subgroups of $\mathscr{H}_2$ have the following graphs for

Figures (2)

  • Figure 1: Schematic representation of the graph computation
  • Figure 2: The intermediate subfactor lattice of the $\frac{33+9\sqrt{13}}{2}$ subfactor: $G = Gal(M/N)=\{1, \alpha, \alpha^2 \}$, $H=Gal(N/N_{-1})=\{1, \beta, \beta^2 \}$, $S=N \ltimes H$, $T= M^G$, $[P;N]=[Q:N]=[R:N]=\frac{5+\sqrt{13}}{2}$ .

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 89 more