Centers of Braided Tensor Categories

2021/08/20 by Zhimin Liu, Liu, Zhimin, Shenglin Zhu +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2108.09102

openalex publication_date 2021/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let C be a finite braided multitensor category. Let B be Majid's automorphism braided group of C, then B is a cocommutative Hopf algebra in C. We show that the center of C is isomorphic to the category of left B-comodules in C, and the decomposition of B into a direct sum of indecomposable C-subcoalgebras leads to a decomposition of B-\operatorname*Comod_C into a direct sum of indecomposable C-module subcategories. As an application, we present an explicit characterization of the structure of irreducible Yetter-Drinfeld modules over semisimple quasi-triangular weak Hopf algebras. Our results generalize those results on finite groups and on quasi-triangular Hopf algebras.

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