# Hopf Algebras in Quantum Groups

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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?

This post imported from StackExchange Physics at 2014-10-11 09:47 (UTC), posted by SE-user bolbteppa

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I don't understand it well, and I don't think it can serve as a motivation for Hopf algebras (except for very abstract minded people).

But I guess the generalization Majid has in mind is his work on the subject; see Section 6 of his arXiv paper Algebraic {q}-Integration and Fourier Theory on Quantum and Braided Spaces and his paper Braided Groups and Quantum Fourier Transform. On p.525 of the later paper he argues in which sense his quantum Fourier transform generalized the ordinary Fourier transform.

You can find more papers in this direction by looking up the citations in scholar.google.com.

answered Aug 22, 2015 by (14,009 points)
edited Aug 22, 2015

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