Quantization of Poisson Groups

Abstract

Let $ G^τ$ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfel'd structure of Poisson group, let $ H^τ$ be its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups $ U_{q,φ}^M(\frak{h}) $ --- dual of $ U_{q,φ}^{M'}(\frak{g}) $ --- which yield infinitesimal quantization of $ H^τ$ and $ G^τ\, $, 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^τ$ what was known for $ G^τ$, completing the quantization of the pair $ (G^τ,H^τ) $.