Quantum Groups

Course Description

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.

Registration

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