For compact quantum groups there exists a rich representation theory.
What I still not well understand, how this helps in various calculations. My problem is somehow, that these representations are not representations of the whole Hopf-algebra structure, but are at least co-representations.
ED: removed { only algebra representations. (If this is correct.) }
Or one even only goes back to "matrix coefficients", and only a part of the quantum group is somehow "represented". (?)
Why then do the representations still help to understand the whole Hopf-algebra strucure, or at least help in various calculations?
ED:
I am particularly interested in $K$-theory computations.
Maybe the representations are only useful to make concrete calculations, which would be impossible without them.
Not sure if this is the kind of thing you are looking for...
Diaconis and Shahshahani have an upper bound for the distance between two probability measures on a group $G$. The upper bound is given as a sum over non-trivial irreducible representations. This allows one to calculate convergence rates for random walks on finite groups.
It is also possible to write down an upper bound for the distance between two states on the algebra of functions of a quantum group in the same fashion, except this time the sum is over non-trivial irreducible co-representations.