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

Vanda Farsad , Azat M. Gainutdinov (1) , Ingo Runkel

1 IDP - Institut Denis Poisson

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.

Mots clés: Quasi-Hopf algebras; Braided tensor categories; Mapping class group representations; algebra: Hopf; category: representation; category: tensor; group: representation; structure; SL(2; Z)

Domaines: Physique mathématique [math-ph]

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⟩