# Hopf Algebras in Quantum Groups

In the theory of quantum groups Hopf algebras arise via the Fourier transform:

A third point of view is that Hopf algebras are the next simplest category after Abelian groups admitting Fourier transform.

At least for nice functions, a Fourier series is just a Laurent series on a circle (which means just substituting $z = re^{i\theta}$ into a Laurent series), so I can see Fourier analysis on Abelian groups as generalizing this simple example.

How do I see (in an easy way) that Hopf Algebras are the natural generalization of Fourier theory to algebras, in a way that motivates what a Hopf algebra is, so that I have some feel for quantum groups?

• The keyword to look for here is Tannaka-Krein duality (e.g. on nLab), a natural extension of Pontryagin duality. Perhaps someone else here is capable of giving some simple explanation of it, but I don't think I can do better than what's already on nLab or Wikipedia. – Logan M Oct 17 '14 at 3:37