Complex semisimple quantum groups and representation theory

TL;DR

This paper introduces complex semisimple quantum groups, focusing on classifying irreducible Harish-Chandra modules and exploring their connections to quantized universal enveloping algebras and locally compact quantum groups.

Contribution

It provides a comprehensive classification of irreducible Harish-Chandra modules for complex semisimple quantum groups, extending classical representation theory to the quantum setting.

Findings

Classification of irreducible Harish-Chandra modules achieved

Connections established between quantum groups and locally compact quantum groups

Extensive background on quantized universal enveloping algebras provided

Abstract

These notes contain an introduction to the theory of complex semisimple quantum groups. Our main aim is to discuss the classification of irreducible Harish-Chandra modules for these quantum groups, following Joseph and Letzter. Along the way we cover extensive background material on quantized universal enveloping algebras and explain connections to the analytical theory in the setting of locally compact quantum groups.