It was shown by Fock and Goncharov that moduli spaces of local systems on decorated surfaces provide examples of cluster varieties and thus admit canonical quantization. I will describe a joint work with Gus Schrader where we embed the quantum group U_q(sl_n) into the quantized moduli space of SL(n)-local systems on a punctured disk with two marked points. This embedding endows the quantum group with a system of quantum cluster coordinates. It also allows us to realize the adjoint action of the R-matrix as a half Dehn twist of a twice punctured disk and factor it into a sequence of cluster mutations. At the end I hope to discuss another joint work in progress, where we use the above construction to answer a long-standing question regarding the principal series representations of quantum groups.