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.
Author(s): Yuri I. Manin
Series: CRM Short Courses
Edition: Second Edition
Publisher: Springer
Year: 2018
Language: English
Pages: 120
Front Matter ....Pages i-viii
Introduction (Yuri I. Manin)....Pages 1-3
The Quantum Group \({{\mathrm{GL}}}_q(2)\) (Yuri I. Manin)....Pages 5-9
Bialgebras and Hopf Algebras (Yuri I. Manin)....Pages 11-17
Quadratic Algebras as Quantum Linear Spaces (Yuri I. Manin)....Pages 19-24
Quantum Matrix Spaces. I. Categorical Viewpoint (Yuri I. Manin)....Pages 25-28
Quantum Matrix Spaces. II. Coordinate Approach (Yuri I. Manin)....Pages 29-35
Adding Missing Relations (Yuri I. Manin)....Pages 37-41
From Semigroups to Groups (Yuri I. Manin)....Pages 43-46
Frobenius Algebras and the Quantum Determinant (Yuri I. Manin)....Pages 47-51
Koszul Complexes and the Growth Rate of Quadratic Algebras (Yuri I. Manin)....Pages 53-62
Hopf \(*\)-Algebras and Compact Matrix Pseudogroups (Yuri I. Manin)....Pages 63-65
Yang–Baxter Equations (Yuri I. Manin)....Pages 67-71
Algebras in Tensor Categories and Yang–Baxter Functors (Yuri I. Manin)....Pages 73-78
Some Open Problems (Yuri I. Manin)....Pages 79-81
The Tannaka–Krein Formalism and (Re)Presentations of Universal Quantum Groups (Theo Raedschelders, Michel Van den Bergh)....Pages 83-115
Correction to: The Tannaka–Krein Formalism and (Re)Presentations of Universal Quantum Groups (Theo Raedschelders, Michel Van den Bergh)....Pages E1-E1
Back Matter ....Pages 117-125
