Quantization of Poisson groups -- II

Abstract

Let $ G^τ$ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfeld structure of Poisson group; let $ H^τ$ be its dual Poisson group. By means of Drinfeld's 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) $.