# From Hopf algebras to topological quantum groups. A short history, various aspects and some problems

## TL;DR

This paper reviews the development of quantum groups from Hopf algebras to topological quantum groups, discussing historical context, key concepts, difficulties, and open problems in the field.

## Contribution

It provides a comprehensive overview of the evolution of quantum groups, clarifying choices and difficulties faced in transitioning from algebraic to topological quantum groups.

## Key findings

- Clarification of the development stages of quantum groups
- Identification of key difficulties and choices in the theory
- Presentation of open problems in locally compact quantum groups

## Abstract

Quantum groups have been studied within several areas of mathematics and mathematical physics. This has led to different approaches, each of them with their own techniques and conventions. Starting with Hopf algebras, where there is a general consensus, moving in the direction of topological quantum groups, where there is no such consensus, it is easy to get lost. Not only many difficulties have to be overcome, but also several choices must be made. The way this is done is often confusing. Some choices even turn out to be rather annoying. As an introductory lecture at the conference on Topological quantum groups and Hopf algebras in 2016, we have explained these choices, difficulties and annoyances, encountered on the road from Hopf algebras to topological quantum groups. In these notes, we discuss more aspects of the development of locally compact quantum groups. We not only explain some of these difficulties in greater detail, but we also give background information about the different steps, combined with some historical comments. We start with finite quantum groups and continue with discrete quantum groups, compact quantum groups and algebraic quantum groups. Multiplicative unitaries are an important side track before we finally arrive at locally compact quantum groups. Along the way, we also formulate some interesting remaining problems in the theory.