# Usefulness of representations of quantum groups

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.

• What's the difference between calculations and "calculations"? If you want a meaningful answer, you should be quite a bit more specific. – Professor Vector Dec 22 '17 at 8:37
• I want to emphasize that these calculations are not so simple but complexer things. – hänsel Dec 22 '17 at 8:50

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.
• Thanks for comment. It sounds interesting and useful, but is not the overall answer yet. I would be particularly interested in $K$-theory computations of quantum groups ($C^*$-algebras. Also, representations are co-representations; I wrote wrongly "algebra rpresentations" in my original question. – hänsel Jan 9 '18 at 17:44