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Quantum Groups and Noncommutative Geometry

  • Textbook
  • © 2018

Overview

Authors:
  1. Yuri I. Manin

    1. Max Planck Institute for Mathematics, Bonn, Germany

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  • Additional chapter by Raedschelders and Van den Bergh surveys recent work that focuses on the representation theory of a number of bi- and Hopf algebras
  • New edition of Manin's celebrated 1988 Montreal lectures
  • Systematically develops an approach to quantum groups as symmetry objects in noncommutative geometry

Part of the book series: CRM Short Courses (CRMSC)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-viii
    Download chapter PDF

  2. Introduction

    • Yuri I. Manin
    Pages 1-3

  3. The Quantum Group \({{\mathrm{GL}}}_q(2)\)

    • Yuri I. Manin
    Pages 5-9

  4. Bialgebras and Hopf Algebras

    • Yuri I. Manin
    Pages 11-17

  5. Quadratic Algebras as Quantum Linear Spaces

    • Yuri I. Manin
    Pages 19-24

  6. Quantum Matrix Spaces. I. Categorical Viewpoint

    • Yuri I. Manin
    Pages 25-28

  7. Quantum Matrix Spaces. II. Coordinate Approach

    • Yuri I. Manin
    Pages 29-35

  8. Adding Missing Relations

    • Yuri I. Manin
    Pages 37-41

  9. From Semigroups to Groups

    • Yuri I. Manin
    Pages 43-46

  10. Frobenius Algebras and the Quantum Determinant

    • Yuri I. Manin
    Pages 47-51

  11. Koszul Complexes and the Growth Rate of Quadratic Algebras

    • Yuri I. Manin
    Pages 53-62

  12. Hopf \(*\)-Algebras and Compact Matrix Pseudogroups

    • Yuri I. Manin
    Pages 63-65

  13. Yang–Baxter Equations

    • Yuri I. Manin
    Pages 67-71

  14. Algebras in Tensor Categories and Yang–Baxter Functors

    • Yuri I. Manin
    Pages 73-78

  15. Some Open Problems

    • Yuri I. Manin
    Pages 79-81

  16. The Tannaka–Krein Formalism and (Re)Presentations of Universal Quantum Groups

    • Theo Raedschelders, Michel Van den Bergh
    Pages 83-115

  17. Correction to: The Tannaka–Krein Formalism and (Re)Presentations of Universal Quantum Groups

    • Theo Raedschelders, Michel Van den Bergh
    Pages E1-E1

  18. Back Matter

    Pages 117-125
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Keywords

About this book

This textbook presents the second edition of Manin's celebrated 1988 Montreal lectures, which influenced a new generation of researchers in algebra to take up the study of Hopf algebras and quantum groups. In this expanded write-up of those lectures, Manin systematically develops an approach to quantum groups as symmetry objects in noncommutative geometry in contrast to the more deformation-oriented approach due to Faddeev, Drinfeld, and others.  This new edition contains an extra chapter by Theo Raedschelders and Michel Van den Bergh, surveying recent work that focuses on the representation theory of a number of bi- and Hopf algebras that were first introduced in Manin's lectures, and have since gained a lot of attention. Emphasis is placed on the Tannaka–Krein formalism, which further strengthens Manin's approach to symmetry and moduli-objects in noncommutative geometry.

Authors and Affiliations

  • Max Planck Institute for Mathematics, Bonn, Germany

    Yuri I. Manin

About the author

​Yuri I. Manin is a Professor at the Max Planck Institute for Mathematics in Bonn. Personal distinctions include: Principal Researcher, Steklov Mathematical Institute, 1960-1993; since 1993 Principal Researcher in absentia. Professor (Algebra Chair), University of Moscow 1965-1992. Professor, M.I.T. 1992-1993. Scientific Member, MPI for Mathematics since 1993. Director, MPI for Mathematics 1995-2005, now Professor Emeritus. Board of Trustees Professor, Northwestern University (Evanston, USA) 2002-2011, now Professor Emeritus. Lenin Prize 1967. Brouwer Medal 1987. Frederic Esser Nemmers Prize 1994. Rolf Schock Prize in Mathematics 1999. King Faisal International Prize in Mathematics 2002. Georg Cantor Medal 2002. Order pour le Mérite for Science and Art, Germany, 2007. Great Cross of Merit with Star, Germany, 2008. János Bolyai International Mathematical Prize, Hungarian Academy of Sciences, 2010. Member of nine Academies of Sciences. Honorary degrees at Sorbonne, Oslo, Warwick.Honorary Member of the London Math. Society.

Bibliographic Information

  • Book Title: Quantum Groups and Noncommutative Geometry

  • Authors: Yuri I. Manin

  • Series Title: CRM Short Courses

  • DOI: https://doi.org/10.1007/978-3-319-97987-8

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2018

  • Hardcover ISBN: 978-3-319-97986-1Published: 22 October 2018

  • Softcover ISBN: 978-3-030-07432-6Published: 20 December 2018

  • eBook ISBN: 978-3-319-97987-8Published: 11 October 2018

  • Series ISSN: 2522-5200

  • Series E-ISSN: 2522-5219

  • Edition Number: 2

  • Number of Pages: VIII, 125

  • Number of Illustrations: 82 b/w illustrations, 1 illustrations in colour

  • Additional Information: Originally published by Centre de Recherches Mathématiques, Montreal, 1988

  • Topics: Associative Rings and Algebras, Group Theory and Generalizations, Category Theory, Homological Algebra

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