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We study representations of the mapping class group of the punctured torus onthe double of a finite dimensional possibly non-semisimple Hopf algebra thatarise in the construction of universal, extended topological field theories. Wediscuss how for doubles the degeneracy problem of TQFT's is circumvented. Wefind compact formulae for the ${\cal S}^{\pm 1}$-matrices using the canonical,non degenerate forms of Hopf algebras and the bicrossed structure of doublesrather than monodromy matrices. A rigorous proof of the modular relations andthe computation of the projective phases is supplied using Radford's relationsbetween the canonical forms and the moduli of integrals. We analyze theprojective $SL(2, Z)$-action on the center of $U_q(sl_2)$ for $q$ an$l=2m+1$-st root of unity. It appears that the $3m+1$-dimensionalrepresentation decomposes into an $m+1$-dimensional finite representation and a$2m$-dimensional, irreducible representation. The latter is the tensor productof the two dimensional, standard representation of $SL(2, Z)$ and the finite,$m$-dimensional representation, obtained from the truncated TQFT of thesemisimplified representation category of $U_q(sl_2)\,$.Comment: 45 page

Mappin class group actions on quantum doubles