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.
Author(s): Teo Banica
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 425
City: Cham, Switzerland
Tags: Quantum Groups, Operator Algebras, Hopf Algebras, Tannakian Duality, Quantum Rotations, Quantum PermutationsQ
Preface
Contents
Part I Quantum Groups
Chapter 1 Quantum Spaces
1.1 Operator Algebras
1.2 Gelfand’s Theorem
1.3 Algebraic Manifolds
1.4 Axiomatization Fix
1.5 Exercises
Chapter 2 Quantum Groups
2.1 Hopf Algebras
2.2 Axioms, Theory
2.3 Product Operations
2.4 Free Constructions
2.5 Exercises
Chapter 3 Representation Theory
3.1 Representations
3.2 Peter–Weyl Theory
3.3 The Haar Measure
3.4 More Peter–Weyl
3.5 Exercises
Chapter 4 Tannakian Duality
4.1 Tensor Categories
4.2 Abstract Algebra
4.3 The Correspondence
4.4 Brauer Theorems
4.5 Exercises
Part II Quantum Rotations
Chapter 5 Free Rotations
5.1 Gram Determinants
5.2 TheWigner Law
5.3 Clebsch–Gordan Rules
5.4 Symplectic Groups
5.5 Exercises
Chapter 6 Unitary Groups
6.1 Gaussian Laws
6.2 Circular Variables
6.3 Fusion Rules
6.4 Further Results
6.5 Exercises
Chapter 7 Easiness, Twisting
7.1 Partitions, Easiness
7.2 Basic Operations
7.3 Ad-Hoc Twisting
7.4 Schur–Weyl Twisting
7.5 Exercises
Chapter 8 Probabilistic Aspects
8.1 Free Probability
8.2 Laws of Characters
8.3 Truncated Characters
8.4 Gram Determinants
8.5 Exercises
Part III Quantum Permutations
Chapter 9 Quantum Permutations
9.1 Magic Matrices
9.2 Representations
9.3 Twisted Extension
9.4 Poisson Laws
9.5 Exercises
Chapter 10 Quantum Reflections
10.1 Finite Graphs
10.2 Reflection Groups
10.3 Complex Reflections
10.4 Bessel Laws
10.5 Exercises
Chapter 11 Classification Results
11.1 Uniform Groups
11.2 Twistability
11.3 Orientability
11.4 Ground Zero
11.5 Exercises
Chapter 12 The Standard Cube
12.1 Face Results
12.2 Edge Results
12.3 Beyond Easiness
12.4 Maximality Questions
12.5 Exercises
Part IV Advanced Topics
Chapter 13 Toral Subgroups
13.1 Diagonal Tori
13.2 The Skeleton
13.3 Generation Questions
13.4 Fourier Liberation
13.5 Exercises
Chapter 14 Amenability, Growth
14.1 Functional Analysis
14.2 Amenability
14.3 Growth
14.4 Toral Conjectures
14.5 Exercises
Chapter 15 Homogeneous Spaces
15.1 Quotient Spaces
15.2 Partial Isometries
15.3 Free Isometries
15.4 Integration Theory
15.5 Exercises
Chapter 16 Modeling Questions
16.1 Matrix Models
16.2 Stationarity
16.3 Weyl Matrices
16.4 Fourier Models
16.5 Exercises
References
Index
