Discrete quantum groups and their duals

TL;DR

This paper reviews the theory of discrete quantum groups, presenting an updated, self-contained perspective that connects them with algebraic and locally compact quantum groups, emphasizing duality and recent developments.

Contribution

It offers a modern, comprehensive review of discrete quantum groups as special cases of algebraic quantum groups, integrating recent advances and broader duality frameworks.

Findings

Provides a unified view of discrete quantum groups within algebraic quantum groups.

Connects duality concepts across compact, discrete, and locally compact quantum groups.

Enhances understanding of quantum group dualities with a self-contained approach.

Abstract

Discrete quantum groups were introduced as duals of compact quantum groups by Podle\'s and Woronowicz in 1990. Shortly after, they were defined and studied intrinsically by Effros and Ruan, and by this author. In 1998, with the introduction of the multiplier Hopf algebras with integrals (also called algebraic quantum groups), the duality between discrete and compact quantum groups became part of the more general duality in the self-dual category of these algebraic quantum groups. Again a few years later the duality was extended to all locally compact quantum groups. In these notes, we give a new and a somewhat updated approach of the theory of discrete quantum groups. In particular, we view them as special cases of algebraic quantum groups. The duality between the compact quantum groups and the discrete quantum groups is seen in this larger context. This has a number of advantages as we will explain. On the one hand, we provide quite a bit of information about how all of this fits into the more general theory of algebraic quantum groups and its duality. Occasionally, we even go one step further and look at the most general case of locally compact quantum groups. Also sometimes, we compare with known results in pure Hopf algebra theory. On the other hand however, we have tried to make these notes highly self-contained. The aim of these notes in the first place is not to give new results but rather to review known results in a more modern perspective, taking into account recent developments. We believe this may be helpful for people who want to work with compact and discrete quantum groups now.