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Hopf algebras and congruence subgroups

About this Title

Yorck Sommerhäuser, University of South Alabama, Department of Mathematics and Statistics, 411 University Blvd. N, Mobile, Alabama 36688 and Yongchang Zhu, Hong Kong University of Science and Technology, Department of Mathematics, Clear Water Bay, Kowloon, Hong Kong

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 219, Number 1028
ISBNs: 978-0-8218-6913-0 (print); 978-0-8218-9108-7 (online)
Published electronically: February 7, 2012
Keywords:Modular group, congruence subgroup, Hopf algebra, Drinfel’d element, ribbon element, Frobenius-Schur indicator, Jacobi symbol, Hopf symbol.
MSC: Primary 16T05; Secondary 17B37

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Table of Contents


  • Introduction
  • Chapter 1. The modular group
  • Chapter 2. Quasitriangular Hopf algebras
  • Chapter 3. Factorizable Hopf algebras
  • Chapter 4. The action of the modular group
  • Chapter 5. The semisimple case
  • Chapter 6. The case of the Drinfel’d Double
  • Chapter 7. Induced modules
  • Chapter 8. Equivariant Frobenius-Schur indicators
  • Chapter 9. Two congruence subgroup theorems
  • Chapter 10. The action of the Galois group
  • Chapter 11. Galois groups and indicators
  • Chapter 12. Galois groups and congruence subgroups


We prove that the kernel of the action of the modular group on the center of a semisimple factorizable Hopf algebra is a congruence subgroup whenever this action is linear. If the action is only projective, we show that the projective kernel is a congruence subgroup. To do this, we introduce a class of generalized Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one.

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