Quivers with relations arising from clusters (𝐴_{𝑛} case)

Abstract

Abstract. Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A n . We associate to each cluster C of U an abelian category C C such that the indecomposable objects of C C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of C C . Then, we generalize the "denominator theorem" of Fomin and Zelevinsky to any cluster. IntroductionCluster algebras were introduced in the work of A. Berenstein, S. Fomin, and A. Zelevinsky [2,5,6,7]. This theory appeared in the context of dual canonical basis and more particularly in the study of the Berenstein-Zelevinsky conjecture. Cluster algebras are now connected with many topics: double Bruhat cells, Poisson varieties, total positivity, Teichmüller spaces. The main results on cluster algebras are on one hand the classification of finite cluster algebras by root systems and on the other hand the realization of algebras of regular functions on double Bruhat cells in terms of cluster algebras.Recall some facts about cluster algebras. Cluster algebras U of rank n form a class of algebras defined axiomatically in terms of a distinguished set of generators {u 1 , . . . , u n }. A cluster is a set of "cluster variables" {w 1 , . . . , w n } obtained combinatorially from {u 1 , . . . , u n }. The so-called Laurent phenomenon asserts that each cluster variable is a Laurent polynomial in the set of variables given by a cluster. For each cluster C, one can define combinatorially an oriented quiver Q C .Suppose from now on that U is a finite cluster algebra, i.e. there only exists a finite number of cluster variables. It is known that U can be described by the data of a root system X n . Moreover, there exists a cluster Σ such that Q Σ is the alternating quiver on X n . S. Fomin and A. Zelevinsky give in [6] a more precise description of the Laurent phenomenon: there exists a one-to-one correspondence α → w α between the set of almost positive roots of X n , i.e. positive roots and simple negative roots, and the set of cluster variables, such that the denominator of w α as a Laurent polynomial in Σ is given by the decomposition of α in the basis of simple roots. Via Gabriel's Theorem, this property suggests a link between cluster algebras and representation theory of artinian rings. This relation was already investigated in [10] and is the core subject of the recent papers [3,4].

Differential graded categories

“…For d = 2, one obtains the cluster category associated with the Dynkin graph. This category was introduced in [27] for type A and in [6] in the general case. It serves in the representation-theoretic approach (cf.…”

Combinatorial model for the cluster categories of type E

“…Following the spirit of [5] we now define minimal rotations between diagonals of r,s,t . Let k, l be non-neighbouring vertices of and let c ∈ {R, B, P}.…”

“…4.1. Let be the stable translation quiver of unoriented diagonals of , as defined in CalderoChapoton-Schiffler's paper [5]. Then there is a projection π 2 : 7 1,2 → , which maps…”

“…Moreover, let be the stable translation quiver having as vertices the unoriented diagonals of and arrows given by minimal clockwise rotations, see [5] for details. 6 The support of Hom(τ −1 X, −) in C Q is called the front Ext-hammock of X .…”

Categorification of Acyclic Cluster Algebras: An Introduction

“…The definition of the cluster category is due to BuanMarsh-Reineke-Reiten-Todorov [5] (for arbitrary Q without oriented cycles) and, independently and with a very different, more geometric description, to Caldero-Chapoton-Schiffler [13] (for Q of type A n ).…”