For any finite-dimensional factorizable ribbon Hopf algebra H and any ribbon automorphism of H, we establish the existence of the following structure: an H-bimodule F ω and a bimodule morphism Z ω from Lyubashenko's Hopf algebra object K for the bimodule category to F ω. This morphism is invariant under the natural action of the mapping class group of the one-punctured torus on the space of bimodule morphisms from K to F ω. We further show that the bimodule F ω can be endowed with a natural structure of a commutative symmetric Frobenius algebra in the monoidal category of H-bimodules, and that it is a special Frobenius algebra iff H is semisimple.The bimodules K and F ω can both be characterized as coends of suitable bifunctors. The morphism Z ω is obtained by applying a monodromy operation to the coproduct of F ω; a similar construction for the product of F ω exists as well.Our results are motivated by the quest to understand the bulk state space and the bulk partition function in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple.