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A geometric description of $ m$-cluster categories

Author(s): Karin Baur; Robert J. Marsh
Journal: Trans. Amer. Math. Soc. 360 (2008), 5789-5803.
MSC (2000): Primary 16G20, 16G70, 18E30; Secondary 05E15, 17B37
Posted: May 28, 2008
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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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Additional Information:

Karin Baur
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England
Address at time of publication: Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland
Email: k.baur@mcs.le.ac.uk

Robert J. Marsh
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England
Email: marsh@maths.leeds.ac.uk

DOI: 10.1090/S0002-9947-08-04441-3
PII: S 0002-9947(08)04441-3
Keywords: Cluster category, $m$-cluster category, polygon dissection, $m$-divisible, cluster algebra, simplicial complex, mesh category, diagonal, Auslander-Reiten quiver, derived category, triangulated category
Received by editor(s): July 26, 2006
Posted: May 28, 2008
Additional Notes: This research was supported by Engineering and Physical Sciences Research Council grant GR/S35387/01.
Copyright of article: Copyright 2008, Karin Baur and Robert J. Marsh

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