Quantization of Poisson groups

Let G^\tau be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfel'd structure of Poisson group; let H^\tau be its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups U_{q,\phi}^M(h) - dual to the multiparameter quantum group U_{q,\phi}^{M'}(g) built upon g^\tau, with g = Lie(G) - which yield infinitesimal quantization of H^\tau and G^\tau ; we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for H^\tau what was known for G^\tau , completing the quantization of the pair (G^\tau,H^\tau) .