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

TL;DR

This paper explores the structure of cluster categories derived from infinite Dynkin quivers, providing explicit descriptions of their Auslander-Reiten components and a geometric interpretation via triangulations of an infinite strip.

Contribution

It establishes that certain orbit categories of derived categories of infinite Dynkin quivers are cluster categories and describes their cluster-tilting subcategories geometrically.

Findings

Explicit description of Auslander-Reiten components for these categories.

Identification of cluster-tilting subcategories with compact triangulations.

Geometric realization of cluster structures using 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.