Soient un corps et une -algèbre de dimension finie et de dimension globale . On construit une catégorie triangulée associée à , qui est triangle-équivalente à la catégorie amassée si est héréditaire. Lorsque 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 associée à un carquois à potentiel . 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 be a field and a finite-dimensional -algebra of global dimension . We construct a triangulated category associated to which, if is hereditary, is triangle equivalent to the cluster category of . When 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 associated to a quiver with potential . 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 .
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
@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}, pages = {2525--2590}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {6}, year = {2009}, doi = {10.5802/aif.2499}, mrnumber = {2640929}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2499/} }
TY - JOUR AU - Amiot, Claire TI - Cluster categories for algebras of global dimension 2 and quivers with potential JO - Annales de l'Institut Fourier PY - 2009 SP - 2525 EP - 2590 VL - 59 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2499/ DO - 10.5802/aif.2499 LA - en ID - AIF_2009__59_6_2525_0 ER -
%0 Journal Article %A Amiot, Claire %T Cluster categories for algebras of global dimension 2 and quivers with potential %J Annales de l'Institut Fourier %D 2009 %P 2525-2590 %V 59 %N 6 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2499/ %R 10.5802/aif.2499 %G en %F AIF_2009__59_6_2525_0
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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