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Pré-Publication, Document De Travail Année : 2008

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

Claire Amiot

Résumé

Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\leq 2$. 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-Schr{ö}er 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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Dates et versions

hal-00278033 , version 1 (07-05-2008)
hal-00278033 , version 2 (03-07-2009)

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Claire Amiot. Cluster categories for algebras of global dimension 2 and quivers with potential. 2008. ⟨hal-00278033v2⟩
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