A conjecture on cluster automorphisms of cluster algebras

  • Received: 01 July 2019 Revised: 01 August 2019
  • Primary: 13F60

  • A cluster automorphism is a $ \mathbb{Z} $-algebra automorphism of a cluster algebra $ \mathcal A $ satisfying that it sends a cluster to another and commutes with mutations. Chang and Schiffler conjectured that a cluster automorphism of $ \mathcal A $ is just a $ \mathbb{Z} $-algebra homomorphism of a cluster algebra sending a cluster to another. The aim of this article is to prove this conjecture.

    Citation: Peigen Cao, Fang Li, Siyang Liu, Jie Pan. A conjecture on cluster automorphisms of cluster algebras[J]. Electronic Research Archive, 2019, 27: 1-6. doi: 10.3934/era.2019006

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  • A cluster automorphism is a $ \mathbb{Z} $-algebra automorphism of a cluster algebra $ \mathcal A $ satisfying that it sends a cluster to another and commutes with mutations. Chang and Schiffler conjectured that a cluster automorphism of $ \mathcal A $ is just a $ \mathbb{Z} $-algebra homomorphism of a cluster algebra sending a cluster to another. The aim of this article is to prove this conjecture.



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    [1] Assem I., Schiffler R., Shamchenko V. (2012) Cluster automorphisms. Proc. Lond. Math. Soc. 104: 1271-1302.
    [2] Berenstein A., Fomin S., Zelevinsky A. (2005) Cluster algebras Ⅲ: Upper bounds and double Bruhat cells. Duke Math. J. 126: 1-52.
    [3] P. Cao and F. Li, The enough g-pairs property and denominator vectors of cluster algebras, preprint, arXiv: 1803.05281 [math.RT].
    [4] P. Cao and F. Li, Unistructurality of cluster algebras, preprint, arXiv: 1809.05116 [math.RT].
    [5] W. Chang and R. Schiffler, A note on cluster automorphism groups, preprint, arXiv: 1812.05034 [math.RT].
    [6] Chang W., Zhu B. (2016) Cluster automorphism groups of cluster algebras with coefficients. Sci. China Math. 59: 1919-1936.
    [7] W. Chang and B. Zhu, Cluster automorphism groups and automorphism groups of exchange graphs, preprint, arXiv: 1506.02029 [math.RT].
    [8] Chang W., Zhu B. (2016) Cluster automorphism groups of cluster algebras of finite type. J. Algebra 447: 490-515.
    [9] Fomin S., Zelevinsky A. (2002) Cluster algebras Ⅰ: Foundations. J. Amer. Math. Soc. 15: 497-529.
    [10] Fomin S., Zelevinsky A. (2003) Cluster algebras Ⅱ: Finite type classification. Invent. Math. 154: 63-121.
    [11] Fomin S., Zelevinsky A. (2007) Cluster algebras Ⅳ: Coefficients. Compos. Math. 143: 112-164.
    [12] Gross M., Hacking P., Keel S., Kontsevich M. (2018) Canonical bases for cluster algebras. J. Amer. Math. Soc. 31: 497-608.
    [13] Huang M., Li F., Yang Y. (2018) On structure of sign-skew-symmetric cluster algebras of geometric type, Ⅰ: In view of sub-seeds and seed homomorphisms. Sci. China Math. 61: 831-854.
    [14] F. Li and S. Liu, Periodicities in cluster algebras and cluster automorphism groups, preprint, arXiv: 1903.00893 [math.RT].
    [15] Lee K., Schiffler R. (2015) Positivity for cluster algebras. Ann. of Math. 182: 73-125.
    [16] Saleh I. (2014) Exchange maps of cluster algebras. Int. Electron. J. Algebra 16: 1-15.
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  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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