SL(2,Z)-action for ribbon quasi-Hopf algebras

We study the universal Hopf algebra L of Majid and Lyubashenko in the case that the underlying ribbon category is the category of representations of a finite dimensional ribbon quasi-Hopf algebra A. We show that L=A* with coadjoint action and compute the Hopf algebra structure morphisms of L in terms of the defining data of A. We give explicitly the condition on A which makes Rep(A) factorisable and compute Lyubashenko's projective SL(2,Z)-action on the centre of A in this case. The point of this exercise is to provide the groundwork for the applications to ribbon categories arising in logarithmic conformal field theories - in particular symplectic fermions and W_p-models - and to test a conjectural non-semisimple Verlinde formula.
Comment: 62 pages; v2: corrected inaccuracies in section 4.4 and proposition 5.3, references changed, v3: minor changes, inaccuracies in Prop 4.5, Cor 5.5 fixed, improvements in Sec 7.6 and Rem 7.10, refs updated