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On the center of near-group fusion category of type $\mathbb{Z}_3+6$

Zhiqiang Yu

Abstract

Let $\mathcal{A}$ be a near-group fusion category of type $\mathbb{Z}_3+6$. We show that there is a modular tensor equivalence $\mathcal{Z}(\mathcal{A})\cong\mathcal{C}(\mathbb{Z}_3,η)\boxtimes\mathcal{C}(\mathfrak{sl}_3,9)_{\mathbb{Z}_3}^0$. Moreover, we construct two non-trivial faithful extensions of $\mathcal{A}$ explicitly, whose Drinfeld centers can also be obtained from representation categories quantum groups at root of unity.

On the center of near-group fusion category of type $\mathbb{Z}_3+6$

Abstract

Let be a near-group fusion category of type . We show that there is a modular tensor equivalence . Moreover, we construct two non-trivial faithful extensions of explicitly, whose Drinfeld centers can also be obtained from representation categories quantum groups at root of unity.
Paper Structure (5 sections, 10 theorems, 48 equations)

This paper contains 5 sections, 10 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Center and extensions of near-group fusion categories
  4. Modular data of $\mathcal{C}(\mathfrak{sl}_3,9)_{\mathbb{Z}_3}^0$ and Drinfeld center
  5. Faithful extensions of $\mathcal{A}$

Key Result

Lemma 3.1

The isomorphism classes of simple objects of $\mathcal{C}(\mathfrak{sl}_3,9)_{\mathbb{Z}_3}^0$ are $\text{FPdim}(X_i)=\text{FPdim}(Y_j)=d$ for $1\leq i,j\leq 3$, $\text{FPdim}(Y_4)=2d+2$ and $\text{FPdim}(Y_5)=2d+1$, where $d:=3+2\sqrt{3}$.

Theorems & Definitions (20)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • Theorem 3.6
  • ...and 10 more