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

Amiot, Claire

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 $\le 2$. 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 $\le 2$. 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 $𝒥(Q,W)$.

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

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