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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?

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    $\begingroup$ 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. $\endgroup$ – Logan M Oct 17 '14 at 3:37
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From the point of view of physics, Fourier methods were originally a part of "harmonic analysis"; that is, the attempt to solve wave equations by splitting the wave into its harmonics. Thus the Fourier transform on the real line converts the degrees of freedom of a function on the line into a sum over various frequencies. This is a method of solution that takes advantage of the symmetry of the real line. We look for a transformation that breaks the symmetry but in the simplest possible ways. When it is not vibrating an infinite rope has translational symmetry; one wishes to analyze what happens when that symmetry is disturbed by a vibration. The sine waves are the simplest possible (single energy and so repetitive) vibrations on an object with the symmetry of a line.

when the underlying symmetry is that of a sphere, harmonic analysis uses "spherical harmonics" to do the same thing. These are the simplest way of describing vibrations that exist on a sphere (and so disturb the symmetry of the sphere). The spherical harmonics are a generalization of the Fourier transform. Again, one transforms the degrees of freedom of a function defined on the sphere in such a way that the new degrees of freedom are natural for the vibrations on a sphere.

Relating the spherical harmonics to the irreducible representations of SO(3) symmetry we see that we can generalize the Fourier transforms by giving up the necessity of explicitly describing the underlying geometry. That is, the irreps of SO(3) encode the vibrations on a sphere without explicitly specifying the dependency on position (x,y,z) or angle (theta,phi) as used in the spherical harmonics.

The same applies to the point group and space group symmetries; these are descriptions of how a crystal can vibrate in a way that is compatible with its crystal symmetry. Since the point group symmetries are finite, it's easy to count their degrees of freedom and to see that they do not change the number. The same preservation of the degrees of freedom are explicit in the analysis of the vibrations of a finite number of atoms on a circle.


Why should we generalize these symmetry applications to algebras?

The usual explanation is that Hopf algebras are useful for understanding Feynman diagrams (and renormalization).

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