Quantum Groups and Noncommutative Geometry

1. Front Matter

   Pages i-viii

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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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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.

• quantum groups
• Hopf algebras
• Tanaka-Krein
• coalgebras
• bialgebras
• monoidal categories
• noncommutative geometry
• Yuri Manin textbook

• Max Planck Institute for Mathematics, Bonn, Germany

  Yuri I. Manin

​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.

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