  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 marked
punctured disk. We obtain a description of the coproduct of
$U_q(\mathfrak{sl}_{n+1})$ in terms of the corresponding quantum cluster
algebra associated to the marked twice punctured disk, and express the action
of the $R$-matrix in terms of a mapping class group element corresponding to
the half-Dehn twist rotating one puncture about the other. As a consequence, we
realize the algebra automorphism of $U_q(\mathfrak{sl}_{n+1})^{\otimes 2}$
given by conjugation by the $R$-matrix as an explicit sequence of cluster
mutations, and derive a refined factorization of the $R$-matrix into quantum
dilogarithms of cluster monomials.
