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

Abstract : 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.
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Journal articles

https://hal.archives-ouvertes.fr/hal-02148033
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### Citation

Vanda Farsad, Azat M. Gainutdinov, Ingo Runkel. SL(2,$\mathbb Z$)-action for ribbon quasi-Hopf algebras. J.Algebra, 2019, 522, pp.243-308. ⟨10.1016/j.jalgebra.2018.12.012⟩. ⟨hal-02148033⟩

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