Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms

Understand

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.