Locally compact quantum groups. Radford's $S^4$ formula

TL;DR

This paper explores Radford's $S^4$ formula within the context of locally compact quantum groups, illustrating the progression from Hopf algebras to the broader framework of quantum groups using analytical and algebraic techniques.

Contribution

It demonstrates how Radford's formula applies across various quantum group theories, highlighting its role in connecting algebraic and analytical approaches.

Findings

Radford's $S^4$ formula extends to multiplier Hopf algebras.

Analytical forms of Radford's formula are established for locally compact quantum groups.

The paper illustrates the transition from Hopf algebras to locally compact quantum groups.

Abstract

Let $A$ be a finite-dimensional Hopf algebra. The left and the right integrals on $A$ are related by means of a distinguished group-like element $\delta$ of $A$. Similarly, there is this element $\hat\delta$ in the dual Hopf algebra $\hat A$. Radford showed that $$S^4(a)=\delta^{-1}(\hat\delta\triangleright a \triangleleft \hat\delta^{-1})\delta$$ for all $a$ in $A$ where $S$ is the antipode of $A$ and where $\triangleright$ and $\triangleleft$ are used to denote the standard left and right actions of $\hat A$ on $A$. The formula still holds for multiplier Hopf algebras with integrals (algebraic quantum groups). In the theory of locally compact quantum groups, an analytical form of Radford's formula can be proven (in terms of bounded operators on a Hilbert space). In this talk, we do not have the intention to discuss Radford's formula as such, but rather to use it, together with related formulas, for illustrating various aspects of the road that takes us from the theory of Hopf algebras (including compact quantum groups) to multiplier Hopf algebras (including discrete quantum groups) and further to the more general theory of locally compact quantum groups.