Hicas of length \(\leq 4\). (Q967980)

Summary: 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 \(\leq 4\), 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 \(3\).