Hicas of Length <= 4

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 $A_\Gamma$ 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 $3$.

2010 Mathematics Subject Classification: 05. Combinatorics, 14. Algebraic geometry, 16. Associative rings and algebras, 18. Category theory, homological algebra

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