Hicas of length $\leq 4$

Vanessa Miemietz and Will Turner

A \emph{hica} is a highest weight, homogeneous, indecomposable,
Calabi-Yau category of dimension $0$. A hica has length $l$ if its
objects have Loewy length $l$ and smaller. We classify hicas of
length $\leq 4$, up to equivalence, and study their properties. Over
a fixed field $F$, we prove that hicas of length $4$ are in one-one
correspondence with bipartite graphs. We prove that an algebra
$A_\Gamma$ controlling the hica associated to a bipartite graph
$\Gamma$ is Koszul, if and only if $\Gamma$ is not a simply laced
Dynkin graph, if and only if the quadratic dual of $A_\Gamma$ is
Calabi-Yau of dimension $3$.