Quantum Groups

Quantum Groups

Symmetries of topological spaces are classically described by groups. However, when transitioning to noncommutative spaces, this classical concept of a group reaches its limits. To make symmetries mathematically tractable in this broader framework, the theory of quantum groups is required.

This lecture offers a systematic introduction to compact quantum groups following S.L. Woronowicz. We begin by developing the mathematical foundations and translating classical symmetry concepts into the dual language of C^*-algebras and Hopf algebras. To subsequently bring the abstract theory to life, we study compact matrix quantum groups and a lot of other examples.

A central component of the course is the development of the associated representation theory. We investigate how representations behave in this noncommutative setting and how they can be decomposed into their elementary building blocks. This paves the way for one of the most important results of the theory: Woronowicz's Tannaka-Krein duality. This deep reconstruction theorem elegantly demonstrates how a compact quantum group can be entirely recovered from the monoidal category of its finite-dimensional representations.

Depending on the remaining time, an outlook on advanced topics, such as Discrete Quantum Groups, Drinfeld-Jimbo quantum groups or easy quantum groups, will conclude the course.

Lecture and exercise sessions

Lecturer: Luca Junk and Jonas Metzinger

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Lecture time: Mondays 12-14 and Wednesdays 12-14 in HS IV (E2.4)

Exercise sessions: We will look for a date in the first lecture.

Contents

• Revision of C*-algebra Theory
• Basics on Compact Quantum Groups
• Compact Matrix Quantum Groups
• Representation Theory
• Woronowicz' Tannaka-Krein Duality
• Optional: Discrete Quantum Groups, Drinfeld-Jimbo Quantum Groups, Easy/Banica-Speicher Quantum Groups, ...

Language

The course will be taught in English unless all participants speak German.

Prerequisites

Basic knowledge about C*-algebras and universal C*-algebras as in the ISEM 24 lecture notes will be useful. However we will repeat basic definitions and tools.
The course might also be suitable for physics students with a good mathematical background.

Exam and admission requirements

There will be an oral exam. You can obtain 9 CP. To be admitted, you must attend the lectures and exercise sessions regularly

 

Literature

(Compact) Quantum Groups:

• Sergey Neshveyev, Lars Tuset. Compact Quantum Groups and Their Representation Categories, 2013.
• Moritz Weber. Introduction to compact (matrix) quantum groups and Banica-Speicher (easy) quantum groups, 2017.
• Thomas Timmermann. An Invitation to Quantum Groups and Duality, 2008.
• Teo Banica. Introduction to Quantum Groups, 2022.
• Ann Maes, Alfons van Daele. Notes on Compact Quantum Groups, 1998.

Basics on C*-algebras and universal C*-algebras:

• ISEM 24 lecture notes
• Bruce Blackadar, Operator algebras. Theory of C*-algebras and von Neumann algebras, 2006.
• Kenneth Davidson, C*-algebras by example, 1996.
• Gerard Murphy, C*-algebras and operator theory, 1990.