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

Amiot, Claire

doi:10.5802/aif.2499

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]

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)

Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2525-2590.

Résumé
Abstract

Soient $k$ un corps et $A$ une $k$ -algèbre de dimension finie et de dimension globale $\leq 2$ . 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 $\leq 2$ . 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

Affiliations des auteurs :

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. https://aif.centre-mersenne.org/articles/10.5802/aif.2499/

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