Beijing Institute of Mathematical Sciences and Applications

Ansi Bai , Hank Chen , Liang Kong , Yilong Wang , Zhi-Hao Zhang , Hao Zheng

Ansi Bai

Tuesday, December 31, 2024 9:45 AM - 11:45 AM

A3-2-301

Zoom 518 868 7656 (BIMSA)

Recently, boundary tube algebras associated with a fusion category C and a left C-module M, a class of weak Hopf algebras originally introduced in Kitaev and Kong's work arXiv: 1104.5027, have drawn broad attention from physicists. In this talk, we approach these algebras from a different perspective. We first present the reconstruction theorem for finite-dimensional weak Hopf algebras, rooted in Szlachányi's and Hayashi's works, which establishes a bijection between the isomorphism classes of weak Hopf algebras and the equivalence classes of weak fiber functors. Then we construct a weak fiber functor Fun_C(M,M) -> Vec from the category of C-module functors M -> M to the category of vector spaces, and study the reconstructed weak Hopf algebra A(C,M). We find that A(C,M) gives rise to a non-C*-version of the boundary tube algebra. This leads to a rigorous proof that the representation category of A(C,M) is Fun_C(M,M), a non-C*-version of a corresponding result announced in Kitaev and Kong's work, of which the proof was only sketched there. A special case is Fun_{C\boxtimes C^\ rev}(C,C)=Z(C), where Z(C) denotes the Drinfeld center. In this case, we use the braiding on Z(C) to endow A(C\boxtimes C^\ rev,C) with a quasi-triangular structure. We explicitly investigate the scenario where C=Vec_G. All content of this talk will appear in a joint survey with Zhi-Hao Zhang, which will be available on arXiv shortly.