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Cluster automorphism groups of cluster algebras with coefficients

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Abstract

We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. We introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra (i.e., the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients, cluster algebras with universal geometric coefficients, and cluster algebras from surfaces (except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, China

    Wen Chang

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Wen Chang & Bin Zhu

Authors
  1. Wen Chang
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  2. Bin Zhu
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Correspondence to Wen Chang.

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Chang, W., Zhu, B. Cluster automorphism groups of cluster algebras with coefficients. Sci. China Math. 59, 1919–1936 (2016). https://doi.org/10.1007/s11425-016-5148-z

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  • DOI: https://doi.org/10.1007/s11425-016-5148-z

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