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Cluster structures from 2-Calabi–Yau categories with loops

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Abstract

We generalise the notion of cluster structures from the work of Buan–Iyama–Reiten–Scott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Hom-finite 2-Calabi–Yau category, the set of maximal rigid objects satisfies these axioms whenever there are no 2-cycles in the quivers of their endomorphism rings. We apply this result to the cluster category of a tube, and show that this category forms a good model for the combinatorics of a type B cluster algebra.

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Authors and Affiliations

  1. Institutt for Matematiske Fag, Norges Teknisk-Naturvitenskapelige Universitet, 7491, Trondheim, Norway

    Aslak Bakke Buan & Dagfinn F. Vatne

  2. Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Bethany R. Marsh

Authors
  1. Aslak Bakke Buan
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  2. Bethany R. Marsh
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  3. Dagfinn F. Vatne
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Correspondence to Dagfinn F. Vatne.

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Buan, A.B., Marsh, B.R. & Vatne, D.F. Cluster structures from 2-Calabi–Yau categories with loops. Math. Z. 265, 951–970 (2010). https://doi.org/10.1007/s00209-009-0549-0

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  • DOI: https://doi.org/10.1007/s00209-009-0549-0

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