A geometric description of $m$-cluster categories

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.

14 pages, 5 figures

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• On triangulated orbit categories
• Generalized cluster complexes and Coxeter combinatorics
• Tilting theory and cluster combinatorics
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Cited by in corpus (2)

• A geometric model for cluster categories of type $D_n$
• $m$-cluster categories and $m$-replicated algebras