The holonomy braiding for $\mathcal{U}_\xi(\mathfrak{sl}_2)$ in terms of geometric quantum dilogarithms

Abstract

We derive an explicit formula for the holonomy $R$-matrix of quantum $\mathfrak{sl}_2$ at a root of unity. We show it factorizes into a product of four quantum dilogarithms and satisfies a holonomy Yang-Baxter equation. This factorization extends previously known results and we collect many existing results needed for our computation.