Basic Hopf algebras and symmetric bimodules

TL;DR

This paper explores specific classes of bicategories related to Hopf algebras, demonstrating their structures, classifying simple transitive birepresentations, and analyzing their properties in the context of fusion categories.

Contribution

It introduces and classifies non-semisimple bicategories associated with finite-dimensional Hopf algebras, revealing their structure and representation theory.

Findings

H_0-simple quasi fiab bicategories with a unique H-cell are fusion categories.

Classified simple transitive birepresentations of certain bicategories.

Number of equivalence classes of birepresentations is finite for $ ext{G}_A$, but not necessarily for $ ext{H}_A$.

Abstract

Motivated by the so-called H-cell reduction theorems, we investigate certain classes of bicategories which have only one H-cell apart from possibly the identity. We show that H_0-simple quasi fiab bicategories with unique H-cell H_0 are fusion categories. We further study two classes of non-semisimple quasi-fiab bicategories with a single H-cell apart from the identity. The first is $\cH_A$, indexed by a finite-dimensional radically graded basic Hopf algebra A, and the second is $\cG_A$, consisting of symmetric projective A-A-bimodules. We show that $\cH_A$ can be viewed as a 1-full subbicategory of $\cG_A$ and classify simple transitive birepresentations for $\cG_A$. We point out that the number of equivalence classes of the latter is finite, while that for $\cH_A$ is generally not.