DOCUMENTA MATHEMATICA, Vol. 15 (2010), 177-205

Vanessa Miemietz and Will Turner

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

Keywords and Phrases:

Full text: dvi.gz 80 k, dvi 249 k, ps.gz 742 k, pdf 298 k.