On the Representation Categories of Weak Hopf Algebras Arising from Levin-Wen Models
Ansi Bai, Zhi-Hao Zhang
TL;DR
This work provides an explicit, unitary-free reconstruction framework for weak Hopf algebras associated with a fusion category C and a finite C-module M, proving a monoidal equivalence Rep(A_{M}^{C})≅Fun_C(M,M). The authors develop the six-structure maps of the reconstructed weak Hopf algebra via End(F) for a faithful exact separable Frobenius functor F: to Vec_k, and show Rep(A_{M}^{C}) realizes the category of C-module endofunctors on M, with internal Homs playing a central role. A key result identifies a quasi-triangular structure on A_{C}^{C⊗C^{rev}} that corresponds to the braided Drinfeld center Z(C), yielding a braided equivalence Rep(A_{C}^{C⊗C^{rev}})≅Z(C). The paper also compares this reconstruction with Ocneanu’s tube algebra Tub e_C, showing Morita equivalence but highlighting that Tube_C need not carry a weak Hopf algebra structure compatible with the monoidal center equivalence in general. Collectively, the results provide a coalgebraic perspective on boundaries and defects in Levin-Wen models and offer explicit presentations useful for computations and generalizations in higher-categorical settings.
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.