Quantizations of the Witt algebra and of simple Lie algebras in characteristic p

We quantize explicitly the Witt algebra in characteristic 0 equipped with its Lie bialgebra structures discovered by Taft. Then, we study the reduction modulo p of our formulas. This gives p − 1 families of polynomial noncocommutative deformations of a restricted enveloping algebra of a simple Lie algebra in charactristic p (of Cartan type). In particular, this yields new families of noncommutative noncocommutative Hopf algebras of dimension pp in char p.