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 $U_{q,\varphi}^M(\frak{h})$ --- dual of $U_{q,\varphi}^{M'}(\frak{g})$ --- 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 $(G^\tau,H^\tau)$.