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

Annales de l'Institut Fourier, Tome 59 (2009), p. 2525-2590 / Harvested from Numdam

Résumé

Soient $k$ un corps et $A$ une $k$-algèbre de dimension finie et de dimension globale $\le 2$. On construit une catégorie triangulée ${\mathcal{C}}_A$ associée à $A$, qui est triangle-équivalente à la catégorie amassée ${\mathcal{C}}_A$ si $A$ est héréditaire. Lorsque ${\mathcal{C}}_A$ 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 ${\mathcal{C}}(Q,W )$ 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.

Abstract

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

Détail

Publié le : 2009-01-01

MR 2640929 | Zbl pre05673905

DOI : https://doi.org/10.5802/aif.2499
Classification:  16G20,  16E45
Mots clés: catégorie amassée, catégorie de Calabi-Yau, amas-basculant, carquois à potentiel, algèbre préprojective

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