A cluster realization of $$U_q(\mathfrak {sl}_{\mathfrak {n}})$$ U q ( sl n ) from quantum character varieties

Abstract

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.