Hicas of length $\le 4$.

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 $\le 4$, 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.