# Hicas of length $≤4$.

### Vanessa Miemietz

School of Mathematics Department of Mathematics University of East Anglia University of Aberdeen Norwich, NR4 7TJ, UK Fraser Noble Building### Will Turner

King's College Aberdeen AB24 3UE, UK

## Abstract

A 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 $≤4$, 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 $A_{Γ}$ controlling the hica associated to a bipartite graph $Γ$ is Koszul, if and only if $Γ$ is not a simply laced Dynkin graph, if and only if the quadratic dual of $A_{Γ}$ is Calabi–Yau of dimension 3.

## Cite this article

Vanessa Miemietz, Will Turner, Hicas of length $≤4$.. Doc. Math. 15 (2010), pp. 177–205

DOI 10.4171/DM/294