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We construct an injective algebra homomorphism of the quantum group$U_q(\mathfrak{sl}_{n+1})$ into a quantum cluster algebra $\mathbf{L}_n$associated to the moduli space of framed $PGL_{n+1}$-local systems on a markedpunctured disk. We obtain a description of the coproduct of$U_q(\mathfrak{sl}_{n+1})$ in terms of the corresponding quantum clusteralgebra associated to the marked twice punctured disk, and express the actionof the $R$-matrix in terms of a mapping class group element corresponding tothe half-Dehn twist rotating one puncture about the other. As a consequence, werealize the algebra automorphism of $U_q(\mathfrak{sl}_{n+1})^{\otimes 2}$given by conjugation by the $R$-matrix as an explicit sequence of clustermutations, and derive a refined factorization of the $R$-matrix into quantumdilogarithms of cluster monomials.

A cluster realization of $U_q(\mathfrak{sl_n})$ from quantum character varieties