  We quantize the generalized-Witt algebra in characteristic 0 with its Lie
bialgebra structures discovered by Song-Su (\cite{GY}). Via a modulo p
reduction and a modulo "p-restrictedness" reduction process, we get 2^n{-}1
families of truncated p-polynomial noncocommutative deformations of the
restricted universal enveloping algebra of the Jacobson-Witt algebra
\mathbf{W}(n;\underline{1}) (for the Cartan type simple modular restricted Lie
algebra of W type). They are new families of noncommutative and
noncocommutative Hopf algebras of dimension p^{1+np^n} in characteristic p. Our
results generalize a work of Grunspan (J. Algebra 280 (2004), 145--161,
\cite{CG}) in rank n=1 case in characteristic 0. In the modular case, the
argument for a refined version follows from the modular reduction approach
(different from \cite{CG}) with some techniques from the modular Lie algebra
theory.
