On the Representation Categories of Weak Hopf Algebras Arising from Levin-Wen Models

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 ${\mathrm{F}\mathrm{u}\mathrm{n}}_{\mathcal{C}}^{\mathrm{u}}(\mathcal{M},\mathcal{M}{)}^{\mathrm{r}\mathrm{e}\mathrm{v}}$. We give an independent proof of this result without the unitarity conditions. In particular, viewing $\mathcal{C}$ as a left $\mathcal{C}⊠{\mathcal{C}}^{\mathrm{r}\mathrm{e}\mathrm{v}}$-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 ${\mathrm{T}\mathrm{u}\mathrm{b}\mathrm{e}}_{\mathcal{C}}$ of $\mathcal{C}$ when $\mathcal{C}$ is pivotal. These two algebras are Morita equivalent by the well-known equivalence $\mathrm{R}\mathrm{e}\mathrm{p}({\mathrm{T}\mathrm{u}\mathrm{b}\mathrm{e}}_{\mathcal{C}})\cong \mathcal{Z}(\mathcal{C})$. However, we show that in general there is no weak Hopf algebra structure on ${\mathrm{T}\mathrm{u}\mathrm{b}\mathrm{e}}_{\mathcal{C}}$ such that the above equivalence is monoidal.