The R-matrix action of untwisted affine quantum groups at roots of 1

Abstract

Let gˆ be an untwisted affine Kac–Moody algebra. The quantum group Uq(gˆ) is known to be a quasitriangular Hopf algebra (to be precise, a braided Hopf algebra). Here we prove that its unrestricted specializations at odd roots of 1 are braided too: in particular, specializing q at 1 we have that the function algebra F[Hˆ] of the Poisson proalgebraic group Ĥ dual of Ĝ (a Kac–Moody group with Lie algebra gˆ) is braided. This in turn implies also that the action of the universal R-matrix on the tensor products of pairs of Verma modules can be specialized at odd roots of 1.