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...

Motivated by a recent conjecture by Hernandez and Leclerc [30], we embed a Fomin-Zelevinsky cluster algebra [20] into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties [48]. Graded quiver varieties controlling the image can be identified wi...

Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type. Using the standard definition, the problem is infeasible since it uses mutations that can lead to an infinite process. Barot, Geiss and Zelevinsky (2006) present...

In this paper, we prove some combinatorial results on generalized cluster algebras . To be more precise, that (i) the seeds of a algebra A ( S ) whose clusters contain particular variables form connected subgraph exchange graph ; (ii) there exists bijection from set to another algebra, if their initial matrices satisfy mild condition. Moreover, preserves these two algebras. As applications seco...

Using recursion formulas for vertex operator algebra higher genus characters with formal parameters identified local coordinates around marked points on a Riemann surface of arbitrary genus, we introduce the notion cluster structure. Cluster elements and mutation rules are explicitly defined, simplest example is presented.

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 generators of c...

We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of “tagged triangulations” of the surface, and determine its homotopy type and its growth rate.

This thesis is concerned with higher cluster tilting objects in generalized higher cluster categories and tropical friezes associated with Dynkin diagrams. The generalized cluster category arising from a suitable 3-Calabi-Yau differential graded algebra was introduced by C. Amiot. It is Hom-finite, 2-Calabi-Yau and admits a canonical cluster-tilting object. In this thesis, we extend these resul...

We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2−Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a maximal rigid object T in the cluster tube Γn of rank n (n > 1). For any indecomposable rigid object M in Γn, we define an analogous XM of Caldero-Chapton’s formula (or Palu’s cluster c...

For each Dynkin diagram $D$, we define a ''cluster configuration space'' ${\mathcal{M}}_D$ and partial compactification ${\widetilde {\mathcal{M}}}_D$. $D = A_{n-3}$, have ${\mathcal{M}}_{A_{n-3}} {\mathcal{M}}_{0,n}$, the space of $n$ points on ${\mathbb P}^1$, {\mathcal{M}}}_{A_{n-3}}$ was studied in this case by Brown. The {\mathcal{M}}}_D$ is smooth affine algebraic variety with stratificat...