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arXiv · 1702.01086

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

Vanda Farsad·Azat M. Gainutdinov·Ingo Runkel
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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.

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Vanda Farsad, Azat M. Gainutdinov, Ingo Runkel. 2017-02-03. SL(2,Z)-action for ribbon quasi-Hopf algebras. https://doi.org/10.1016/j.jalgebra.2018.12.012

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