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Some non-braided fusion categories of rank 3

Tobias J. Hagge, Seung-Moon Hong

Abstract

We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, concrete and useful variation of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary re-proof of the fact that the quadruple dual functor is naturally isomorphic to the identity.

Some non-braided fusion categories of rank 3

Abstract

We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, concrete and useful variation of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary re-proof of the fact that the quadruple dual functor is naturally isomorphic to the identity.

Paper Structure

This paper contains 21 sections, 4 theorems, 16 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main theorem and outline
  3. Preliminaries and notational conventions
  4. Skeletization
  5. Strictification
  6. Strictified skeletal fusion categories
  7. Remarks
  8. Proof of Theorem \ref{['thm_main']} part \ref{['main_tensor']}:possible tensor category structures
  9. Setting up the pentagon equations
  10. Normalizations
  11. Associativity matrices
  12. Pentagon equations with $1 \times 1$ matrices
  13. Pentagon equations with $2 \times 2$ or $6 \times 6$ matrices
  14. The pentagon equation with $16 \times 16$ matrices
  15. Solutions
  16. ...and 6 more sections

Key Result

Theorem 1

Consider the set of fusion rules with three simple object types, $x$, $y$ and ${\mathbf 1}$. Let ${\mathbf 1}$ be the trivial object, and let $x \otimes x \cong x \oplus x \oplus y \oplus {\mathbf 1}$, $x \otimes y \cong y \otimes x \cong x$ and $y \otimes y \cong {\mathbf 1}$. Then the following ho

Figures (7)

  • Figure 1: (a) Pentagon equality and (b) corresponding equality
  • Figure 2: Graphical notation of $v^1_{xx}$ and $v^{xx}_1$ and property
  • Figure 3: Definitions of $b_x$ and $d_x$, and elementary properties
  • Figure 4: Hexagon equalities
  • Figure 5: Isomorphisms $R^{z}_{x,y}$ and $\bar{R}^{z }_{x,y}$
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 1: Main Theorem
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4
  • proof