Cluster Realization of $U_q(\mathfrak{g})$ and Factorization of the Universal $R$-Matrix

Abstract

For each simple Lie algebra $\mathfrak{g}$, we construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into certain quantum torus algebra $D_\mathfrak{g}$ via the positive representations of split real quantum group. The quivers corresponding to $D_\mathfrak{g}$ is obtained from amalgamation of two basic quivers, where each of them is mutation equivalent to the cluster structure of the moduli space of framed $G$-local system on a disk with 3 marked points when $G$ is of classical type. We derive a factorization of the universal $R$-matrix into quantum dilogarithms of cluster variables, and show that conjugation by the $R$-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk.