Cluster realization of $$\mathcal {U}_q(\mathfrak {g})$$ U q ( g ) and factorizations of the universal R-matrix

Abstract

For each simple Lie algebra $$\mathfrak {g}$$ , we construct an algebra embedding of the quantum group $$\mathcal {U}_q(\mathfrak {g})$$ into certain quantum torus algebra $$\mathcal {D}_\mathfrak {g}$$ via the positive representations of split real quantum group. The quivers corresponding to $$\mathcal {D}_\mathfrak {g}$$ is obtained from an amalgamation of two basic quivers, each of which is mutation equivalent to one describing the cluster structure of the moduli space of framed G-local system on a disk with 3 marked points on its boundary when G is of classical type. We derive a factorization of the universal R-matrix into quantum dilogarithms of cluster monomials, 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.