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) $.