A geometric description of $m$-cluster categories

TL;DR

This paper provides a geometric model for m-cluster categories of type A_{n-1} using diagonals of a regular polygon, generalizing previous results and employing translation quivers.

Contribution

It introduces a geometric description of m-cluster categories via diagonals of polygons and develops the concept of the mth power of a translation quiver.

Findings

m-cluster category of type A_{n-1} is equivalent to diagonals of a regular polygon

Introduces the notion of the mth power of a translation quiver

Generalizes previous results for m=1

Abstract

We show that the $m$-cluster category of type $A_{n-1}$ is equivalent to a certain geometrically-defined category of diagonals of a regular $nm+2$-gon. This generalises a result of Caldero, Chapoton and Schiffler for $m=1$. The approach uses the theory of translation quivers and their corresponding mesh categories. We also introduce the notion of the $m$th power of a translation quiver and show how it can be used to realise the $m$-cluster category in terms of the cluster category.