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Cluster categories of type \({\mathbb {A}}_\infty ^\infty \) and triangulations of the infinite strip

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Abstract

We call a 2-Calabi–Yau triangulated category a cluster category if its cluster-tilting subcategories form a cluster structure as defined in Buan et al. (Compos Math 145:1035–1079, 2009). In this paper, we show that the canonical orbit category of the bounded derived category of finite dimensional representations of a quiver without infinite paths of type \({\mathbb {A}}_\infty \) or \({\mathbb {A}}_\infty ^\infty \) is a cluster category. Moreover, for a cluster category of type \({\mathbb {A}}_\infty ^\infty ,\) we give a geometrical description of its cluster structure in terms of triangulations of an infinite strip with marked points in the plane.

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Authors and Affiliations

  1. Département de mathématiques, Université de Sherbrooke, Sherbrooke, Québec, Canada

    Shiping Liu

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-1009, USA

    Charles Paquette

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  1. Shiping Liu
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  2. Charles Paquette
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Correspondence to Charles Paquette.

Additional information

Both authors were supported in part by the Natural Science and Engineering Research Council of Canada, while the second-named author was also supported in part by the Atlantic Association for Research in the Mathematical Sciences. They are grateful to the referee for drawing their attention to the references [11, 13].

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Liu, S., Paquette, C. Cluster categories of type \({\mathbb {A}}_\infty ^\infty \) and triangulations of the infinite strip. Math. Z. 286, 197–222 (2017). https://doi.org/10.1007/s00209-016-1760-4

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