Basic Hopf algebras and symmetric bimodules

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 H0-simple quasi fiab bicategories with unique H-cell H0 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 ▪, indexed by a finite-dimensional radically graded basic Hopf algebra A, and the second is ▪, consisting of symmetric projective A-A-bimodules. We show that ▪ can be viewed as a 1-full subbicategory of ▪ and classify simple transitive birepresentations for ▪. We point out that the number of equivalence classes of the latter is finite, while that for ▪ is generally not.