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

2025/03/09 by Bai, Ansi, Zhang, Zhi-Hao · 1 citation
#16T05 (Primary) 18M20 #81R50 (Secondary) #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Strongly Correlated Electrons (cond-mat.str-el)

paper · doi:10.48550/arxiv.2503.06731

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 C and a unitary left C-module M, and sketched a proof that its representation category is monoidally equivalent to the unitary C-module functor category Fun^u_C(M,M)^rev. We give an independent proof of this result without the unitarity conditions. In particular, viewing C as a left C \boxtimes C^rev-module, we obtain a quasi-triangular weak Hopf algebra whose representation category is braided equivalent to the Drinfeld center Z(C). In the appendix, we also compare this quasi-triangular weak Hopf algebra with the tube algebra Tube_C of C when C is pivotal. These two algebras are Morita equivalent by the well-known equivalence Rep(Tube_C)\congZ(C). However, we show that in general there is no weak Hopf algebra structure on Tube_C such that the above equivalence is monoidal.

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