Cluster characters for 2-Calabi–Yau triangulated categories

Abstract

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.