  For any finite-dimensional factorizable ribbon Hopf algebra H and any ribbon
automorphism omega of H, we establish the existence of the following structure:
an H-bimodule F_omega and a bimodule morphism Z_omega from Lyubashenko's Hopf
algebra object K for the bimodule category to F_omega. 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_omega. We
further show that the bimodule F_omega 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_omega can both be characterized as coends of suitable
bifunctors. The morphism Z_omega is obtained by applying a monodromy operation
to the coproduct of F_omega; a similar construction for the product of F_omega
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.
