Braiding operator via quantum cluster algebra

We construct a braiding operator in terms of the quantum dilogarithm function based on the quantum cluster algebra. We show that it is a q-deformation of the R-operator for which hyperbolic octahedron is assigned. Also shown is that, by taking q to be a root of unity, our braiding operator reduces to the Kashaev R^K-matrix up to a simple gauge-transformation. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Cluster algebras in mathematical physics'.