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

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 ${\mathbb{A}}_{\infty }$ or ${\mathbb{A}}_{\infty }^{\infty }$, we show that this orbit category is a cluster category, that is, its cluster-tilting subcategories form a cluster structure. When the quiver is of type ${\mathbb{A}}_{\infty }^{\infty }$, 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.

41 pages

References in corpus (1)

• On triangulated orbit categories