Quantization of Poisson groups

Let G(tau) be a connected simply connected semisimple algebraic group, endo wed with generalized Sklyanin-Drinfel'd structure of Poisson group; let H-t au be its dual Poisson group. By means of quantum double construction and d ualization via formal Hopf algebras, we construct new quantum groups U-q,ph i(M)(h) - dual of U-q,phi(M)'(g) - which yield infinitesimal quantization o f H-tau and G(tau); We study their specializations at roots of 1 (in partic ular, their classical limits), thus discovering new quantum Frobenius morph isms. The whole description dualize for H-tau what was known for G(tau), co mpleting the quantization of the pair (G(tau), H-tau).