A geometric interpretation of the triangulated structure of m-cluster categories

TL;DR

This paper provides a geometric interpretation of the triangulated structure of m-cluster categories of type A_n and characterizes components from the m-th power of their Auslander-Reiten quivers.

Contribution

It offers a new geometric perspective on the triangulated structure of m-cluster categories and characterizes components from the m-th power of their Auslander-Reiten quivers.

Findings

Geometric interpretation of the triangulated structure of m-cluster categories

Characterization of connected components from the m-th power of Auslander-Reiten quivers

Clarification of open problems in geometric descriptions of m-cluster categories

Abstract

The aim of this note is to answer several open problems arising from the geometric description of the $m$-cluster categories of type $A_n$ and their realization in terms of the $m$-th power of a translation quiver. In particular, we give a geometric interpretation of the triangulated structure of $m$-cluster categories. Furthermore, we characterize all the connected components arising from a cluster category when taking the $m$-th power of its Auslander-Reiten quiver.