Dual affine quantum groups

Let (g) over cap be an untwisted affine Kac-Moody algebra, with its Sklyani n-Drinfel'd structure of Lie bialgebra, and let (h) over cap be the dual Li e bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group U-q ((h) over cap), dual of U- q((g) over cap). Studying its specializations at roots of I (in particular, its semi-classical limits), we prove that it yields quantizations of (h) o ver cap and (G) over cap(infinity) (the formal Poisson group attached to (g ) over cap), and we construct new quantum Frobenius morphisms. The whole pi cture extends to the untwisted affine case the results known for quantum gr oups of finite type. Mathematics Subject Classification (1991): 17B37, 81R5 0.