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 [Reshetikhin, N. "Quasitriangularity of quantum groups at roots of 1", Commun. Math. Phys. 170 (1995), 79-99], proving that quantum groups U_q^M(g) and their unrestricted specializations at roots of 1, in particular the function algebra F[H] of the Poisson group H dual to G, are braided. 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 {Spec(U_\varepsilon^M(g))}^{\times 2} when \varepsilon is 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 braiding) for quantum groups at roots of 1