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

Abstract:

The present work splits in two parts: first, we perform a straightforward generalization of results from [Re], proving that quantum groups \(\uqMg\) and their unrestricted specializations at roots of 1, in particular the function algebra F[H] of the Poisson group H dual of 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 \({Spec \left( U_\varepsilon^M \! (\gerg) \right)}^{\times 2}\) 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 braiding for quantum groups at roots of 1.