Non-degenerate ribbon categories from unrolled quantum groups

There exist several versions of non-semisimple analogues of the modular tensor categories: (i) transparent objects in C are trivial; (ii) the Drinfeld centre of C is equivalent to the Deligne tensor product of C with C^{op}; (iii) the Hopf pairing on the Lyubashenko's coend is non-degenerate. In the semi-simple case, all these conditions lead to the non-degeneracy of the Hopf link or the S-matrix. It was proven recently (by Shimizu) that all three conditions (i)-(iii) are equivalent and we can thus talk about ``the non-degenerate ribbon categories'' as the replacement for ``the modular tensor categories''. I will present a large class of new non-degenerate (finite) ribbon categories that comefrom representation theory of the unrolled quantum group and will discuss mapping class group representations on their Hom spaces. The motivation for our new construction comes from conformal extensions of vertex-operator algebras that appear in Logarithmic CFT