Hicas of Length ≤ 4
نویسندگان
چکیده
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 AΓ 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 AΓ 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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