Cluster characters for 2-Calabi–Yau triangulated categories

Palu, Yann

Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2221-2248.

Etant donné un objet amas-basculant $T$ quelconque dans une catégorie triangulée 2-Calabi–Yau sur un corps algébriquement clos (comme dans le cadre de Keller et Reiten), il est possible de définir, pour chaque objet $L$, une fraction rationnelle $X(T,L)$, en utilisant une formule proposée par Caldero et Keller. On montre, de plus, que l’application associant $X(T,L)$ à $L$ est un caractère amassé  ; c’est-à-dire qu’elle vérifie une certaine formule de multiplication. Cela permet de prouver qu’elle induit, dans les cas fini et acyclique, une bijection entre objets rigides indécomposables de la catégorie amassée et variables d’amas de l’algèbre amassée correspondante, confirmant ainsi une conjecture de Caldero et Keller.

Starting from an arbitrary cluster-tilting object $T$ in a 2-Calabi–Yau triangulated category over an algebraically closed field, as in the setting of Keller and Reiten, we define, for each object $L$, a fraction $X(T,L)$ using a formula proposed by Caldero and Keller. We show that the map taking $L$ to $X(T,L)$ is a cluster character, i.e. that it satisfies a certain multiplication formula. We deduce that it induces a bijection, in the finite and the acyclic case, between the indecomposable rigid objects of the cluster category and the cluster variables, which confirms a conjecture of Caldero and Keller.

Classification : 16G20, 18E30
Keywords: Calabi–Yau triangulated category, cluster algebra, cluster category, cluster-tilting object
Mot clés : catégorie triangulée 2-Calabi–Yau, algèbre amassée, catégorie amassée, objet amas-basculant

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