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The most famous of all the quantum groups is $SU_q(2)$ - the Quantum special unitary group. The irreducible comodules of this quantum group are very well understood - they are labelled by integers (or half-integers) just as in the classical case. Now I am wondering myself what the finite-dimensional modules of $SU_q(2)$ looks like?? The irreducible $*$-representations have been described here. But what happens for finite-dimensional (not necessarily irreducible) modules. I know that in general the modules will not be decomposed into irreducible modules, but maybe there is still a classification?

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    $\begingroup$ When, many years ago, I took a seminar class on this (based on, e.g. arxiv.org/abs/q-alg/9603025 ), I remember that many of the modules that arose naturally were indeed completely reducible, in a form very similar to the $q=0$ versions. $\endgroup$
    – Buzz
    Aug 15, 2022 at 1:33

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