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

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 $\mathcal{Z}(\mathcal{A})\cong \mathcal{C}({\mathbb{Z}}_3,\eta )⊠\mathcal{C}({\mathfrak{sl}}_3,9{)}_{{\mathbb{Z}}_3}^0$. 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.

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