Cluster categories for algebras of global dimension 2 and quivers with potential
Annales de l'Institut Fourier, Volume 59 (2009) no. 6, p. 2525-2590
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).
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.
DOI : https://doi.org/10.5802/aif.2499
Classification:  16G20,  16E45
Keywords: Cluster category, Calabi-Yau category, cluster-tilting, quiver with potential, preprojective algebra
@article{AIF_2009__59_6_2525_0,
     author = {Amiot, Claire},
     title = {Cluster categories for algebras of global dimension 2 and quivers with potential},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {6},
     year = {2009},
     pages = {2525-2590},
     doi = {10.5802/aif.2499},
     zbl = {pre05673905},
     mrnumber = {2640929},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2009__59_6_2525_0}
}
Cluster categories for algebras of global dimension 2 and quivers with potential. Annales de l'Institut Fourier, Volume 59 (2009) no. 6, pp. 2525-2590. doi : 10.5802/aif.2499. https://aif.centre-mersenne.org/item/AIF_2009__59_6_2525_0/

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