Title: On the Representation Categories of Weak Hopf Algebras Arising from Levin-Wen Models Show Chinese title

Title: 论 Levin-Wen 模型中弱 Hopf 代数的表示范畴

Abstract: In their study of Levin-Wen models [Commun. Math. Phys. 313 (2012) 351-373], Kitaev and Kong proposed a weak Hopf algebra associated with a unitary fusion category $\mathcal{C}$ and a unitary left $\mathcal{C}$-module $\mathcal{M}$, and sketched a proof that its representation category is monoidally equivalent to the unitary $\mathcal{C}$-module functor category $\mathrm{Fun}^{\mathrm{u}}_{\mathcal{C}}(\mathcal{M},\mathcal{M})^\mathrm{rev}$. We give an independent proof of this result without the unitarity conditions. In particular, viewing $\mathcal{C}$ as a left $\mathcal{C} \boxtimes \mathcal{C}^{\mathrm{rev}}$-module, we obtain a quasi-triangular weak Hopf algebra whose representation category is braided equivalent to the Drinfeld center $\mathcal{Z}(\mathcal{C})$. In the appendix, we also compare this quasi-triangular weak Hopf algebra with the tube algebra $\mathrm{Tube}_{\mathcal{C}}$ of $\mathcal{C}$ when $\mathcal{C}$ is pivotal. These two algebras are Morita equivalent by the well-known equivalence $\mathrm{Rep}(\mathrm{Tube}_{\mathcal{C}})\cong\mathcal{Z}(\mathcal{C})$. However, we show that in general there is no weak Hopf algebra structure on $\mathrm{Tube}_{\mathcal{C}}$ such that the above equivalence is monoidal. Show Chinese abstract

Abstract: 在研究Levin-Wen模型[Commun. Math. Phys. 313 (2012) 351-373]时，Kitaev和Kong提出了一个与酉融合范畴 $\mathcal{C}$ 和酉左 $\mathcal{C}$-模 $\mathcal{M}$相关的弱Hopf代数，并概述了其表示范畴在幺乘意义下等价于酉 $\mathcal{C}$-模函子范畴 $\mathrm{Fun}^{\mathrm{u}}_{\mathcal{C}}(\mathcal{M},\mathcal{M})^\mathrm{rev}$的证明。我们给出了在去掉酉性条件下的这一结果的独立证明。 特别地，将$\mathcal{C}$视为左$\mathcal{C} \boxtimes \mathcal{C}^{\mathrm{rev}}$- 模块时，我们得到一个拟三角弱Hopf代数，其表示范畴与Drinfeld中心$\mathcal{Z}(\mathcal{C})$在辫子意义下等价。 在附录中，我们也比较了当$\mathcal{C}$是 pivotal 时，这个拟三角弱Hopf代数与$\mathcal{C}$的管代数$\mathrm{Tube}_{\mathcal{C}}$之间的关系。 通过著名的等价关系$\mathrm{Rep}(\mathrm{Tube}_{\mathcal{C}})\cong\mathcal{Z}(\mathcal{C})$，这两个代数是Moroita等价的。 然而，我们证明一般情况下不存在弱Hopf代数结构于$\mathrm{Tube}_{\mathcal{C}}$上，使得上述等价为张量范畴的。