  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.
