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.