# 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.