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Cluster algebras arising from cluster tubes

Yu Zhou Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, China, Current address:, Fakultät für Mathematik, Universität Bielefeld, D-33501, Bielefeld, Germany Bin Zhu Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, China, bzhu@math.tsinghua.edu.cn

Combinatorics Representation Theory Rings and Algebras mathscidoc:2204.06003

Journal of the London Mathematical Society, 89, (3), 703-723, 2014.6
We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2−Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object T in the cluster tube C_n of rank n. For any indecomposable rigid object M in C_n, we define an analogous X_M of Caldero-Chapton's formula (or Palu's cluster character formula) by using the geometric information of M. We show that X_M,X_{M′} satisfy the mutation formula when M,M′ form an exchange pair, and that X_? : M↦X_M gives a bijection from the set of indecomposable rigid objects in C_n to the set of cluster variables of cluster algebra of type C_{n−1}, which induces a bijection between the set of basic maximal rigid objects in C_n and the set of clusters. This strengths a surprising result proved recently by Buan-Marsh-Vatne that the combinatorics of maximal rigid objects in the cluster tube Cn encode the combinatorics of the cluster algebra of type B_{n−1} since the combinatorics of cluster algebras of type B_{n−1} or of type C_{n−1} are the same by a result of Fomin and Zelevinsky. As a consequence, we give a categorification of cluster algebras of type C.
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@inproceedings{yu2014cluster,
  title={Cluster algebras arising from cluster tubes},
  author={Yu Zhou, and Bin Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220429170423066703179},
  booktitle={Journal of the London Mathematical Society},
  volume={89},
  number={3},
  pages={703-723},
  year={2014},
}
Yu Zhou, and Bin Zhu. Cluster algebras arising from cluster tubes. 2014. Vol. 89. In Journal of the London Mathematical Society. pp.703-723. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220429170423066703179.
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