  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.
