I'm a master's student waiting to start my phd in quantum groups and their represenation theory in march 2015. I love representation theory $\textit{per se}$, and looking for references on this work I stumbled upon a 5 volume monstrous bookset by karpilosky "group representations" (easy too google.) It's a beautifully written piece of work about group representations in general, but 90% of the focus is devoted to finite groups. It is also a very standard knowledge the power of finite groups and their representations.
I really wanted to have a reason to study this work (and others like it), but I feel studying finite groups just for fun would be a real waste of time (even though I want it. Maybe someone could counter-argue here but this is not my question).
So what I would really like to know is a relation between:
on one side: quantum groups, duality, braided (monoidal) categories, groupoids, operator algebras,
on the other side: finite groups.
More precisely speaking: How can each of the two sides "defined" above can help the other? Are there more subtle relations? Can they ellucidate each other's represetantion theory?
I'm aware that there are finite lie bialgebras which are quantum groups, but I want to know how explicitly their relations work.
I'm a PhD student in operator algebraic quantum groups, which is a generalisation of topological groups effectively. The other side of quantum groups are the purely algebraic versions, that is to say Hopf algebras. So quantum groups begin with the study of functions on a group (as it's a generalisation), in the Hopf algebra case we have polynomials, for compact quantum groups continuous functions, but let's look at polynomials for now. Let's just denote polynomials on $G$ by $\mathcal{P}(G)$ for convenience.
In a group we have a map $m : G \times G \to G$ but when we are looking at functions on the group we effectively lose this information. What we do then we is we put a comultiplication, a map $\Delta : \mathcal{P}(G) \to \mathcal{P}(G) \otimes \mathcal{P}(G)$ ($\otimes$ being the tensor product). It can be shown in the case of finite groups that $\mathcal{P}(G) \otimes \mathcal{P}(G) \cong \mathcal{P}(G \times G)$, i.e. functions on $G \times G$. So we can define $\Delta$ such that $$ \Delta(f)(x,y) = f(xy) $$ for $f \in \mathcal{P}(G)$, $x,y \in G$. So in this sense we recapture the group product with this new operation.
The problem is however if we generalise to infinite groups the relation $\mathcal{P}(G) \otimes \mathcal{P}(G) \cong \mathcal{P}(G \times G)$ does not necessarily hold any more so we cannot use this any more. This is where operator algebraic methods come in because we have tensor products in C$^*$-algebra setting that overcome this issue, I could go into further details as this is the area I'm interested in but if your interest is in finite groups it's probably not necessary.
Also with a Hopf algebra, we have a counit and an antipode which play the role of a group unit and inverse respectively. This can be found in any decent text on quantum groups. Maybe try Kassel for an undergraduate style introduction and Klymyk and Schmudgen or Majid's books for more details (search amazon, they'll all have quantum groups in the title). They will also give more details into the Lie algebraic deformations that are genuine quantum groups (i.e. don't have an underlying group structure).
I realise that's not answered all your questions but hopefully it's helped answer some bits.