  An example of a finite dimensional factorizable ribbon Hopf C-algebra is
given by a quotient H=u_q(g) of the quantized universal enveloping algebra
U_q(g) at a root of unity q of odd degree. The mapping class group M_{g,1} of a
surface of genus g with one hole projectively acts by automorphisms in the
H-module H^{*\otimes g}, if H^* is endowed with the coadjoint H-module
structure. There exists a projective representation of the mapping class group
M_{g,n} of a surface of genus g with n holes labelled by finite dimensional
H-modules X_1,...,X_n in the vector space Hom_H(X_1\otimes...\otimes
X_n,H^{*\otimes g}). An invariant of closed oriented 3-manifolds is
constructed. Modifications of these constructions for a class of ribbon Hopf
algebras satisfying weaker conditions than factorizability (including most of
u_q(g) at roots of unity q of even degree) are described. The results are
motivated by CFT.
