A geometric description of 𝑚-cluster categories

Baur, Karin; Marsh, Bethany

doi:10.1090/S0002-9947-08-04441-3

A geometric description of $m$-cluster categories
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by Karin Baur and Bethany R. Marsh PDF

Trans. Amer. Math. Soc. 360 (2008), 5789-5803

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.

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