Cluster categories of type $${\mathbb {A}}_\infty ^\infty $$ A ∞ ∞ and triangulations of the infinite strip

Abstract

Abstract. We first study the (canonical) orbit category of the bounded derived category of finite dimensional representations of a quiver with no infinite path, and we pay more attention on the case where the quiver is of infinite Dynkin type. In particular, its Auslander-Reiten components are explicitly described. When the quiver is of type A∞ or A ∞ ∞ , we show that this orbit category is a cluster category, that is, its cluster-tilting subcategories form a cluster structure as defined in [3]. When the quiver is of type A ∞ ∞ , we shall give a geometrical description of the cluster structure of the cluster category by using triangulations of the infinite strip in the plane. In particular, we shall show that the cluster-tilting subcategories are precisely given by compact triangulations.

Cotorsion pairs in cluster categories of typeA∞∞

“…Theorem 5.3 and Corollary 5.4 in[19]). There is a bijection from the set of (isoclasses of ) indecomposable objects in C to the set of (isotopy classes of ) arcs in B∞.…”

“…for any b < a < p < f < e and d < c (see the lower picture in Figure 3). Note that this order is not compatible with that given in [19]. A large class of Ptolemy diagrams can be obtained in the following way.…”

“…In this subsection, we recall from [19] a geometric description of a cluster category of type A ∞ ∞ . Let Q be a quiver of type A ∞ ∞ without infinite path, and rep(Q) the category of finite dimensional k-linear representations of Q.…”

“…Following [19], denote by B ∞ the infinite strip in the plane of the marked points (x, y) with 0 ≤ y ≤ 1. The points l i = (i, 1), i ∈ Z, are called upper marked points, and the points r i = (−i, 0), i ∈ Z, are called lower marked points.…”

“…Notice that the works above only deal with 2-Calabi-Yau categories having cluster tilting subcategories or maximal rigid subcategories, which contain finitely many indecomposable objects except Ng's work. Recently, Liu and Paquette [19] introduced another 2-Calabi-Yau category, the cluster category C of type A ∞ ∞ , which admits cluster categories having infinitely many indecomposable objects. They gave a geometric realization of C , via an infinite strip with marked points B ∞ in the plane.…”