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Pré-Publication, Document De Travail Année : 2007

Cluster characters for triangulated 2-Calabi--Yau categories

Yann Palu

Résumé

Starting from an arbitrary cluster-tilting object $T$ in a 2-Calabi--Yau 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.
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Dates et versions

hal-00136968 , version 1 (18-03-2007)
hal-00136968 , version 2 (23-06-2007)
hal-00136968 , version 3 (10-04-2010)

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Yann Palu. Cluster characters for triangulated 2-Calabi--Yau categories. 2007. ⟨hal-00136968v1⟩
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