Cluster categories for algebras of global dimension 2 and quivers with potential
[Catégorie amassée pour des algèbres de dimension globale 2 et des carquois à potentiel]
Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2525-2590.

Soient k un corps et A une k-algèbre de dimension finie et de dimension globale 2. On construit une catégorie triangulée 𝒞 A associée à A, qui est triangle-équivalente à la catégorie amassée 𝒞 A si A est héréditaire. Lorsque 𝒞 A est Hom-finie, on prouve qu’elle est 2-Calabi-Yau et munie d’un objet amas-basculant canonique. Cette nouvelle classe de catégories contient certaines sous-catégories stables de modules sur une algèbre préprojective introduite par Geiss-Leclerc-Schröer et par Buan-Iyama-Reiten-Scott. Ces résultats s’appliquent aussi aux carquois à potentiel. Plus précisément, on introduit une catégorie amassée 𝒞(Q,W) associée à un carquois à potentiel (Q,W). Quand il est Jacobi-fini, on prouve que cette catégorie est munie d’un objet amas-basculant dont l’algèbre d’endomorphismes est isomorphe à l’algèbre jacobienne.

Let k be a field and A a finite-dimensional k-algebra of global dimension 2. We construct a triangulated category 𝒞 A associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When 𝒞 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 𝒞 (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 𝒥(Q,W).

DOI : 10.5802/aif.2499
Classification : 16G20, 16E45
Keywords: Cluster category, Calabi-Yau category, cluster-tilting, quiver with potential, preprojective algebra
Mot clés : catégorie amassée, catégorie de Calabi-Yau, amas-basculant, carquois à potentiel, algèbre préprojective
Amiot, Claire 1

1 Université Paris 7 Institut de Mathématiques de Jussieu Théorie des groupes et des représentations Case 7012 2 Place Jussieu 75251 Paris Cedex 05 (France)
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Amiot, Claire. Cluster categories for algebras of global dimension 2 and quivers with potential. Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2525-2590. doi : 10.5802/aif.2499. http://www.numdam.org/articles/10.5802/aif.2499/

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