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 [FZ] in order to develop a combinatorial approach to study problems of total positivity in algebraic groups and canonical bases in quantum groups. The link between acyclic cluster algebras and representation theory of quivers were first revealed in [MRZ]. In [BMRRT], the authors introduced the cluster categories as the categorification of acyclic cluster algebras. In [CC], the authors introduced a certain structure of Hall algebra involving the cluster category by associating its objects to some variables given by an explicit map X? called the Caldero-Chapoton map. The images of the map are called generalized cluster variables. For simply laced Dynkin quivers, P. Caldero and B. Keller constructed a cluster multiplication formula (of finite type) between two generalized cluster variables ([CK]). The Caldero-Chapoton map and the Caldero -Keller cluster multiplication theorem open a way to construct cluster algebras from 2-Calabi-Yau categories. The cluster multiplication formula of finite type was generalized to affine type in [Hu] and any type in [XX] and [Xu]. Y. Palu [Pa2] further extended the formula to 2-Calabi-Yau categories with cluster tilting objects. The aim this note is twofold. One is to simplify the cluster multiplication formula in [XX] and [Xu]. In practice, the formula is useful for constructing Z-bases of cluster algebras of affine type ( [DXX], [Du]). However, the formula is given in the context of module categories and a bit complicate. We construct a more ‘unified’ version in the context of cluster categories (Theorem 3.3). The other aim is to construct a variant of Y.Palu’s formula (Theorem 1.2). In particular, when applied to acyclic quivers, the variant is exactly the cluster multiplication formula in [XX] and [Xu]. Since the variant implies Y.Palu’s formula, we can view it as a refinement of Y.Palu’s formula. Date: January 28, 2010. 2000 Mathematics Subject Classification. Primary 16G20, 16G70; Secondary 14M99, 18E30.