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

TL;DR

This paper investigates the structure and SL(2,Z)-action of the universal Hopf algebra associated with finite-dimensional ribbon quasi-Hopf algebras, aiming to support applications in logarithmic conformal field theories.

Contribution

It explicitly characterizes the Hopf algebra structure of L=A* for ribbon quasi-Hopf algebras and computes the SL(2,Z)-action on the center, providing foundational results for conformal field theory applications.

Findings

L=A* with coadjoint action is the universal Hopf algebra for the category

Explicit conditions for A to be factorisable are derived

Lyubashenko's SL(2,Z)-action on the center is computed

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.