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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)
DOI: https://doi.org/10.1090/S0065-9266-2012-00649-6
Published electronically: February 7, 2012
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Corrected version of record:PDF
Corrigenda:PDF
Keywords: Modular group,
congruence subgroup,
Hopf algebra,
Drinfel’d element,
ribbon element,
Frobenius-Schur indicator,
Jacobi symbol,
Hopf symbol.
MSC: Primary 16T05; Secondary 17B37
Table of Contents
Chapters
- Introduction
- 1. The Modular Group
- 2. Quasitriangular Hopf Algebras
- 3. Factorizable Hopf Algebras
- 4. The Action of the Modular Group
- 5. The Semisimple Case
- 6. The Case of the Drinfel’d Double
- 7. Induced Modules
- 8. Equivariant Frobenius-Schur Indicators
- 9. Two Congruence Subgroup Theorems
- 10. The Action of the Galois Group
- 11. Galois Groups and Indicators
- 12. Galois Groups and Congruence Subgroups
Abstract
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.- Marcelo Aguiar, A note on strongly separable algebras, Bol. Acad. Nac. Cienc. (Córdoba) 65 (2000), 51–60 (English, with English and Spanish summaries). Colloquium on Homology and Representation Theory (Spanish) (Vaquerías, 1998). MR 1840439
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