Renormalized Hennings Invariants and 2 + 1-TQFTs

Abstract

We construct non-semisimple 2 + 1-TQFTs yielding mapping class group representations in Lyubashenko’s spaces. In order to do this, we first generalize Beliakova, Blanchet and Geer’s logarithmic Hennings invariants based on quantum sl2 to the setting of finite-dimensional non-degenerate unimodular ribbon Hopf algebras. The tools used for this construction are a Hennings-augmented Reshetikhin–Turaev functor and modified traces. When the Hopf algebra is factorizable, we further show that the universal construction of Blanchet, Habegger, Masbaum and Vogel produces a 2 + 1-TQFT on a not completely rigid monoidal subcategory of cobordisms.