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This book introduces the reader to quantum groups, focusing on the simplest ones, namely the closed subgroups of the free unitary group.
Although such quantum groups are quite easy to understand mathematically, interesting examples abound, including all classical Lie groups, their free versions, half-liberations, other intermediate liberations, anticommutation twists, the duals of finitely generated discrete groups, quantum permutation groups, quantum reflection groups, quantum symmetry groups of finite graphs, and more.
The book is written in textbook style, with its contents roughly covering a one-year graduate course. Besides exercises, the author has included many remarks, comments and pieces of advice with the lone reader in mind. The prerequisites are basic algebra, analysis and probability, and a certain familiarity with complex analysis and measure theory. Organized in four parts, the book begins with the foundations of the theory, due to Woronowicz, comprising axioms, Haar measure, Peter-Weyl theory, Tannakian duality and basic Brauer theorems. The core of the book, its second and third parts, focus on the main examples, first in the continuous case, and then in the discrete case. The fourth and last part is an introduction to selected research topics, such as toral subgroups, homogeneous spaces and matrix models.
Introduction to Quantum Groups offers a compelling introduction to quantum groups, from the simplest examples to research level topics.
List of contents
Part I. Quantum groups.- Chapter 1. Quantum spaces.- Chapter 2. Quantum groups.- Chapter 3. Representation theory.- Chapter 4. Tannakian duality.- Part II. Quantum rotations.- Chapter 5. Free rotations.- Chapter 6. Unitary groups.- Chapter 7. Easiness, twisting.- Chapter 8. Probabilistic aspects.- Part III. Quantum permutations.- Chapter 9. Quantum permutations.- Chapter 10. Quantum reflections.- Chapter 11. Classification results.- Chapter 12. The standard cube.- Part IV. Advanced topics.- Chapter 13. Toral subgroups.- Chapter 14. Amenability, growth.- Chapter 15. Homogeneous spaces.- Chapter 16. Modelling questions.- Bibliography.- Index.
About the author
Teo Banica is Professor of Mathematics at the University of Cergy-Pontoise. As one of the leading experts in quantum groups, he has done extensive research on the subject since the mid 90s, with about 100 papers written on the subject with numerous collaborators, and with many research activities organized throughout the 90s and 00s. Professor Banica now enjoys living in the countryside, preparing his classes, doing some research, and spending most of his time in writing mathematics and physics books.
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This book would serve as an excellent resource for those who want to develop a thorough understanding of the basic theory of quantum groups. There are exercises which encourage the reader to try calculations and work out some details for themselves. The author includes clear references to the literature where further details can be found, and also gives the interested reader information about the historical development of the subject. (Andrew McKee, Mathematical Reviews, January, 2025)
This book is for anyone with an interest or use for compact quantum groups, but the breadth of the exposition probably means that there is something interesting in there for everyone. (J. P. McCarthy, Irish Mathematical Society Bulletin, Issue 93, 2024)
