  A modular tensor category is a non-degenerate ribbon finite tensor category.
And a ribbon factorizable Hopf algebra is exactly the Hopf algebra whose
finite-dimensional representations form a modular tensor category. The goal of
this paper is to construct both semisimple and non-semisimple modular
categories with Hopf algebras. In particular, we study central extensions of
Hopf algebras and characterize some conditions for a quotient Hopf algebra to
be a factorizable one. Then we give a special way to obtain ribbon elements. As
a result, we apply to some finite dimensional pointed Hopf algebras introduced
by Andruskiewitsch etc and obtain two families of ribbon factorizable Hopf
algebras. With little restrictions to their parameters, we prove that their
representation categories are prime modular tensor categories which are not
tensor equivalent to representation categories of small quantum groups. Lastly,
we construct a family of semisimple ribbon factorizable Hopf algebas A(p,q),
where p,q are prime numbers satisfying some conditions. And we decompose
Rep(A(p,q)) into prime modular tensor categories and prove that A(p,q) can't be
obtained by some obvious ways, i.e. it's not tensor product of trivial Hopf
algebras (group algebras or their dual) and Drinfeld doubles.
