Dual affine quantum groups

Let g be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let h be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group U_q(h), dual of U_q(g). Studying its specializations at roots of 1 (in particular, its classical limits), we prove that it yields quantizations of h and G^\infty (the formal group attached to 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.