Cluster categories for algebras of global dimension 2 and quivers with potential

2008/05/07 by Claire Amiot, Amiot, Claire · 14 citations
Mathematics · #16E45 #16G20 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:16E45 #msc:16G20

paper · pdf · doi:10.48550/arxiv.0805.1035

46 pages, small typos as it will appear in Annales de l'Institut Fourier

arxiv created 2009/07/03 · arxiv updated 2009/12/01

Abstract

Let k be a field and A a finite-dimensional k-algebra of global dimension ≤ 2. We construct a triangulated category \Cc_A associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When \Cc_A is \Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluster category \Cc_(Q,W) associated to a quiver with potential (Q,W). When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra \Jj(Q,W).

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