Cluster characters for triangulated 2-Calabi--Yau categories

Abstract

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.

21 pages. The numberings now coincide with those of the published version

References in corpus (2)

• On triangulated orbit categories
• Acyclic Calabi-Yau categories

Cited by in corpus (15)

• Triangular bases in quantum cluster algebras and monoidal categorification conjectures
• Cluster categories for marked surfaces: punctured case
• Sortable elements in infinite Coxeter groups
• Positivity for cluster algebras from surfaces
• Internally Calabi-Yau algebras and cluster-tilting objects
• Denominators in cluster algebras of affine type
• On cluster algebras with coefficients and 2-Calabi-Yau categories
• On cluster algebras arising from unpunctured surfaces II
• On the combinatorics of rigid objects in 2-Calabi-Yau categories
• Cluster expansion formulas and perfect matchings
• Quantum cluster algebras from unpunctured triangulated surfaces: arbitrary coefficients and quantization
• Positivity for quantum cluster algebras from unpunctured orbifolds
• Modules determined by their composition factors in higher homological algebra
• On the cluster multiplication theorem for acyclic cluster algebras
• Lifting to cluster-tilting objects in 2-Calabi-Yau triangulated categories