  We study representations of the mapping class group of the punctured torus on
the double of a finite dimensional possibly non-semisimple Hopf algebra that
arise in the construction of universal, extended topological field theories. We
discuss how for doubles the degeneracy problem of TQFT's is circumvented. We
find compact formulae for the ${\cal S}^{\pm 1}$-matrices using the canonical,
non degenerate forms of Hopf algebras and the bicrossed structure of doubles
rather than monodromy matrices. A rigorous proof of the modular relations and
the computation of the projective phases is supplied using Radford's relations
between the canonical forms and the moduli of integrals. We analyze the
projective $SL(2, Z)$-action on the center of $U_q(sl_2)$ for $q$ an
$l=2m+1$-st root of unity. It appears that the $3m+1$-dimensional
representation decomposes into an $m+1$-dimensional finite representation and a
$2m$-dimensional, irreducible representation. The latter is the tensor product
of the two dimensional, standard representation of $SL(2, Z)$ and the finite,
$m$-dimensional representation, obtained from the truncated TQFT of the
semisimplified representation category of $U_q(sl_2)\,$.
