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hal-00278033, version 2

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

Claire Amiot () 1

(05/2008)

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 $\Jj(Q,W)$.

• 1 :  Institut de Mathématiques de Jussieu (IMJ)
• CNRS : UMR7586 – Université Pierre et Marie Curie (UPMC) - Paris VI – Université Paris VII - Paris Diderot
• Domaine : Mathématiques/Théorie des représentations
• Mots-clés : Cluster category – Calabi-Yau category – cluster-tilting – quiver with potential – dg algebra – preprojective algebra
• Commentaire : 46 pages – small typos as it will appear in Annales de l'Institut Fourier.
• Versions disponibles :  v1 (07-05-2008) v2 (03-07-2009)

• hal-00278033, version 2
• oai:hal.archives-ouvertes.fr:hal-00278033
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• Soumis le : Vendredi 3 Juillet 2009, 17:13:07
• Dernière modification le : Vendredi 3 Juillet 2009, 17:48:07