A cluster realization of U q ( sl n ) from quantum character varieties

We construct an injective algebra homomorphism of the quantum group U q ( sl n + 1 ) into a quantum cluster algebra 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 ( 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 ( sl n + 1 ) ⊗ 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.