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Hopf Algebras in Quantum Groups

  1. Oct 11, 2014 #1
    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 [itex]z = re^{i\theta}[/itex] 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?
  2. jcsd
  3. Oct 16, 2014 #2
    Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
  4. Oct 16, 2014 #3
    If we look at a Hopf algebra the way Baez explains it in the following quote it looks almost like a group except we're using algebras instead of sets in the domains of our functions and a co-multiplication instead of a multiplication. I wonder if there is a way to see the algebriac-Fourier-transform as coming out of the co-multiplication analogous to how the group-Fourier-transform arises from multiplication?

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