SL(2,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. 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.

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

Cited by in corpus (17)

• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• A quasi-Hopf algebra for the triplet vertex operator algebra
• Braided Tensor Categories related to ${\mathcal{B}}_p$ Vertex Algebras
• Invertible braided tensor categories
• Mapping Class Group Representations From Non-Semisimple TQFTs
• Logarithmic conformal field theories of type $B_n,\ell =4$ and symplectic fermions
• Projective objects and the modified trace in factorisable finite tensor categories
• Modular Categories and TQFTs Beyond Semisimplicity
• The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre
• Modified traces for quasi-Hopf algebras
• On the Feigin-Tipunin conjecture
• Monadic cointegrals and applications to quasi-Hopf algebras
• Further results on the structure of (co)ends in finite tensor categories
• Recent developments of the categorical Verlinde formula
• Homological Construction of Quantum Representations of Mapping Class Groups
• Kerler-Lyubashenko Functors on 4-Dimensional 2-Handlebodies
• Non-semisimple link and manifold invariants for symplectic fermions