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

International audience; 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.