Hicas of length \(\leq 4\).
type: Item
instance of: scholarly article
title: Hicas of length \(\leq 4\).
zbMATH Open document ID: 1215.16013
author: Vanessa Miemietz
author: Will Turner
published in: Documenta Mathematica
publication date: 2010-05-03T00:00:00Z
full work available at URL: https://eudml.org/doc/228245
full work available at URL: 04.html
review text: 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 \(3\).
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 5703302
zbMATH Keywords: indecomposable Calabi-Yau categories
zbMATH Keywords: Loewy lengths
zbMATH Keywords: hicas
zbMATH Keywords: bipartite graphs
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5703302