Geometrical Meaning of R-matrix Action for Quantum Groups at Roots of 1

The present work splits in two parts: first, we perform a straightforward generalization of results from [Re], proving autoquasitriangularity of quantum groups $U_q(\frak{g})$ and their unrestricted specializations at roots of 1, in particular the function algebra $F[H]$ of the Poisson group $H$ dual of $G$; second, as a main contribution, we prove the convergence of the (specialized) $R$--matrix action to a birational automorphism of a 2$\ell$--fold ramified covering of the specialization of $U_q(\frak{g})$ at a primitive $\ell$--th root of 1, and of a 2-fold ramified covering of $H$, thus giving a geometric content to the notion of triangularity (or autoquasitriangularity) for quantum groups.