Science China Mathematics

, Volume 62, Issue 3, pp 417–446 | Cite as

On generalized symmetries and structure of modular categories

  • Shawn Xingshan Cui
  • Modjtaba Shokrian Zini
  • Zhenghan WangEmail author


Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category symmetries of modular categories, which include also categorical Hopf algebras as special cases. As an application, we propose an analogue of the classification of finite simple groups to modular categories, where we define simple modular categories as the prime ones without any nontrivial normal algebras.


modular category category symmetry Hopf monad tensor functor 


18D10 81R05 


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The first author was supported by the Simons Foundation. The third author was sup- ported by National Science Foundation of USA (Grants Nos. DMS-1411212 and FRG-1664351).


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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Shawn Xingshan Cui
    • 1
    • 2
  • Modjtaba Shokrian Zini
    • 3
  • Zhenghan Wang
    • 3
    • 4
    Email author
  1. 1.Stanford Institute for Theoretical PhysicsStanford UniversityStanfordUSA
  2. 2.Department of MathematicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  3. 3.Department of MathematicsUniversity of California at Santa BarbaraSanta BarbaraUSA
  4. 4.Microsoft Station QSanta BarbaraUSA

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