Caldero-Keller approach to the denominators of cluster variables
نویسنده
چکیده
Buan, Marsh and Reiten proved that if a cluster-tilting object T in a cluster category C associated to an acyclic quiver Q satisfies certain conditions with respect to the exchange pairs in C, then the denominator in its reduced form of every cluster variable in the cluster algebra associated to Q has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T . In this paper, we give an alternative proof of this result using the Caldero-Keller approach to acyclic cluster algebras and the work of Palu on cluster characters.
منابع مشابه
Notes on the Cluster Multiplication Formulas for 2-calabi-yau Categories
Y. Palu has generalized the cluster multiplication formulas to 2Calabi-Yau categories with cluster tilting objects ([Pa2]). The aim of this note is to construct a variant of Y. Palu’s formula and deduce a new version of the cluster multiplication formula ([XX]) for acyclic quivers in the context of cluster categories. Introduction Cluster algebras were introduced by S. Fomin and A. Zelevinsky [...
متن کاملA Multiplication Formula for Algebras with 2-calabi-yau Properties
We associate to any object in the nilpotent module category of an algebra with the 2-Calabi-Yau property a character (in the sense of [11]) and prove a multiplication formula for the characters. This formula extends a multiplication formula for the evaluation forms (in particular, dual semicanonical basis) associated to modules over a preprojective algebra given by Geiss, Leclerc and Schröer [6...
متن کاملGeneralised Friezes and a Modified Caldero-chapoton Map Depending on a Rigid Object
The (usual) Caldero-Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps “reachable” indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalises the idea that the cluster category is a “categorification” of the cluster algebra. The definition of the Caldero-C...
متن کاملOn the Cluster Multiplication Theorem for Acyclic Cluster Algebras
In [3] and [13], the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the properties of 2–Calabi–Yau (Auslander–Reiten formula) and high order associativity. Introduction Cluster algebras were introduced by Fomin and Zelevinsky in [9]. By definition, the cluste...
متن کاملCLUSTER ALGEBRAS AND CLUSTER CATEGORIES
These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...
متن کامل