From Hopf Algebras to quantum groups I start with self study about quantum groups. Until now I covered Hopf $*$-algebras and there representations (for example by the book of Klymik and Schmüdgen). Now I want to understand the step from Hopf algebras to quantum groups (and especially to compact quantum groups).
The most book tread the theory of Hopf algebras and go directly to the example of $U_q(\mathfrak{sl}(2))$. But there is no definition of a quantum group, hence I don't know what a quantum group should be. Roughly speaking one says that a quantum group is a deformation of a Hopf algebra. But why isn't there a precise definition? Can you refer to literature where I can found more on this topic?! Why do one need the $q$-special functions?
If I look to the definition of a compact quantum group, I see the definition in the $C^*$-algebraic sense and the von Neumann algebraic sense (which are equivalent). But I don't see the theory of Hopf algebras in the definition. Why can it not be done like this (roughly speaking): A compact quantum group is a one-parameter deformation of a Hopf algebra $A$ with a topology on $A$ such that $A$ is compact. Why do we need weights and KMS-weights to study such compact quantum groups?
I am searching for literature which possibly give answers on this question (of approximate an answer). If there are any other answers and comments by you I would be very glad.
Thank you very much.
 A: I can mainly say something about compact quantum groups from the C*-algebraic point of view...
This C*-algebraic definition of compact quantum groups seems to be quite established by now. For an introduction from this starting point, including the basic results on existence of a Haar measure and representations, you can have a look at the paper by S.L. Woronowicz "Compact quantum groups": http://www.fuw.edu.pl/~slworono/PDF-y/CQG3.pdf, and the survey by A. Maes and A. Van Daele (which I started with when learning about the subject) "Notes on compact quantum groups": http://arxiv.org/pdf/math/9803122v1.pdf.
What you will find related to Hopf algebras in these articles is basicly the fact that with every compact quantum group, you can associate a canonical dense Hopf *-algebra (generated by the matrix elements of all finite-dimensional unitary representations of your compact quantum group).
For another point of view, you can check the paper by M.S. Dijkhuizen and T.H. Koornwinder "CQG Algebras: a Direct Algebraic Approach to Compact Quantum Groups": http://arxiv.org/pdf/hep-th/9406042.pdf. In this paper the authors start from a Hopf *-algebra with the special property that it is spanned by the coefficients of its finite-dimensional unitary (irreducible) corepresentations. (They call this a CQG algebra.) Such a Hopf *-algebra can then canonically be associated with a compact quantum group in duality with the above approach.
I hope this will help you a bit. It is of course perfectly possible that there are other (better or worse) things to find in the literature about the link with Hopf algebras which I am not aware of.
