Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms

TL;DR

This paper constructs modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms, providing insights into non-semisimple conformal field theories and their bulk state spaces.

Contribution

It establishes the existence of a natural Frobenius algebra structure in the bimodule category derived from ribbon Hopf algebra automorphisms, extending previous frameworks.

Findings

F_omega is a commutative symmetric Frobenius algebra in H-bimodules.

The bimodule F_omega is a special Frobenius algebra iff H is semisimple.

The morphism Z_omega is invariant under the mapping class group action.

Abstract

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.