Let $U\_q(\frak{sl}\_2)$ denote the quantum universal enveloping algebra of $\frak{sl}\_2$, and consider the adjoint action
$$
\mathrm{ad}\_X: U\_q({\frak sl}\_2) \to U\_q({\frak sl}\_2), ~~ Y \mapsto S(X\_{(1)})YX\_{(2)},
$$
where we have used sumless Sweedler notation. This gives $U\_q({$\frak sl}\_2)$ the structure of a $U\_q({\frak sl}\_2)$-module. What is the structure of this module? Does it decompose into a direct sum of finite-dim irreps, or are there infinite-dim reps in there? If decomposes into a direct sum of finite-dim irreps, then does every irrep appear, and are there multiplicities. How does it compare to the classical situation?



The adjoint representation of $U_q({\frak sl}_2)$ on itself
Unlike what happens in the classical case, it is not locally finite dimensional, basically because it has invertible elements. However its ad-locally finite part $U'$ is very large and it decomposes as a direct sum
$$U'=\bigoplus\_{V} V^\* \otimes V$$
where the sum is over the irreducible finite dimensional modules (note this is a purely 'quantum' phenomena).


A standard reference is Joseph-Letzter "[Local finiteness of the adjoint action for quantized enveloping algebras](https://www.sciencedirect.com/science/article/pii/002186939290157H)".


