Dual Affine Quantum Groups

Let $\hat{\frak g}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\frak h}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group $U_q(\hat{\frak h})$, dual of $U_q(\hat{\frak g})$. Studying its restricted and unrestricted integer forms and their specializations at roots of 1 (in particular, their classical limits), we prove that $U_q(\hat{\frak h})$ yields quantizations of $\hat{\frak h}$ and $\hat{G}^\infty$ (the formal group attached to $\hat{\frak g}$), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.