2019, 27: 1-6. doi: 10.3934/era.2019006

A conjecture on cluster automorphisms of cluster algebras

Department of Mathematics, Zhejiang University (Yuquan Campus), Hangzhou, Zhejiang, 310027, China

* Corresponding authors: Fang Li, Siyang Liu

Received  July 2019 Revised  August 2019 Published  August 2019

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. Electronic Research Archive, 2019, 27: 1-6. doi: 10.3934/era.2019006
References:
[1]

I. AssemR. Schiffler and V. Shamchenko, Cluster automorphisms, Proc. Lond. Math. Soc., 104 (2012), 1271-1302.  doi: 10.1112/plms/pdr049.  Google Scholar

[2]

A. BerensteinS. Fomin and A. Zelevinsky, Cluster algebras Ⅲ: Upper bounds and double Bruhat cells, Duke Math. J., 126 (2005), 1-52.  doi: 10.1215/S0012-7094-04-12611-9.  Google Scholar

[3]

P. Cao and F. Li, The enough g-pairs property and denominator vectors of cluster algebras, preprint, arXiv: 1803.05281 [math.RT]. Google Scholar

[4]

P. Cao and F. Li, Unistructurality of cluster algebras, preprint, arXiv: 1809.05116 [math.RT]. Google Scholar

[5]

W. Chang and R. Schiffler, A note on cluster automorphism groups, preprint, arXiv: 1812.05034 [math.RT]. Google Scholar

[6]

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

[7]

W. Chang and B. Zhu, Cluster automorphism groups and automorphism groups of exchange graphs, preprint, arXiv: 1506.02029 [math.RT]. Google Scholar

[8]

W. Chang and B. Zhu, Cluster automorphism groups of cluster algebras of finite type, J. Algebra, 447 (2016), 490-515.  doi: 10.1016/j.jalgebra.2015.09.045.  Google Scholar

[9]

S. Fomin and A. Zelevinsky, Cluster algebras Ⅰ: Foundations, J. Amer. Math. Soc., 15 (2002), 497-529.  doi: 10.1090/S0894-0347-01-00385-X.  Google Scholar

[10]

S. Fomin and A. Zelevinsky, Cluster algebras Ⅱ: Finite type classification, Invent. Math., 154 (2003), 63-121.  doi: 10.1007/s00222-003-0302-y.  Google Scholar

[11]

S. Fomin and A. Zelevinsky, Cluster algebras Ⅳ: Coefficients, Compos. Math., 143 (2007), 112-164.  doi: 10.1112/S0010437X06002521.  Google Scholar

[12]

M. GrossP. HackingS. Keel and M. Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc., 31 (2018), 497-608.  doi: 10.1090/jams/890.  Google Scholar

[13]

M. HuangF. Li and Y. Yang, On structure of sign-skew-symmetric cluster algebras of geometric type, Ⅰ: In view of sub-seeds and seed homomorphisms, Sci. China Math., 61 (2018), 831-854.  doi: 10.1007/s11425-016-9100-8.  Google Scholar

[14]

F. Li and S. Liu, Periodicities in cluster algebras and cluster automorphism groups, preprint, arXiv: 1903.00893 [math.RT]. Google Scholar

[15]

K. Lee and R. Schiffler, Positivity for cluster algebras, Ann. of Math., 182 (2015), 73-125.  doi: 10.4007/annals.2015.182.1.2.  Google Scholar

[16]

I. Saleh, Exchange maps of cluster algebras, Int. Electron. J. Algebra, 16 (2014), 1-15.  doi: 10.24330/ieja.266223.  Google Scholar

show all references

References:
[1]

I. AssemR. Schiffler and V. Shamchenko, Cluster automorphisms, Proc. Lond. Math. Soc., 104 (2012), 1271-1302.  doi: 10.1112/plms/pdr049.  Google Scholar

[2]

A. BerensteinS. Fomin and A. Zelevinsky, Cluster algebras Ⅲ: Upper bounds and double Bruhat cells, Duke Math. J., 126 (2005), 1-52.  doi: 10.1215/S0012-7094-04-12611-9.  Google Scholar

[3]

P. Cao and F. Li, The enough g-pairs property and denominator vectors of cluster algebras, preprint, arXiv: 1803.05281 [math.RT]. Google Scholar

[4]

P. Cao and F. Li, Unistructurality of cluster algebras, preprint, arXiv: 1809.05116 [math.RT]. Google Scholar

[5]

W. Chang and R. Schiffler, A note on cluster automorphism groups, preprint, arXiv: 1812.05034 [math.RT]. Google Scholar

[6]

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

[7]

W. Chang and B. Zhu, Cluster automorphism groups and automorphism groups of exchange graphs, preprint, arXiv: 1506.02029 [math.RT]. Google Scholar

[8]

W. Chang and B. Zhu, Cluster automorphism groups of cluster algebras of finite type, J. Algebra, 447 (2016), 490-515.  doi: 10.1016/j.jalgebra.2015.09.045.  Google Scholar

[9]

S. Fomin and A. Zelevinsky, Cluster algebras Ⅰ: Foundations, J. Amer. Math. Soc., 15 (2002), 497-529.  doi: 10.1090/S0894-0347-01-00385-X.  Google Scholar

[10]

S. Fomin and A. Zelevinsky, Cluster algebras Ⅱ: Finite type classification, Invent. Math., 154 (2003), 63-121.  doi: 10.1007/s00222-003-0302-y.  Google Scholar

[11]

S. Fomin and A. Zelevinsky, Cluster algebras Ⅳ: Coefficients, Compos. Math., 143 (2007), 112-164.  doi: 10.1112/S0010437X06002521.  Google Scholar

[12]

M. GrossP. HackingS. Keel and M. Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc., 31 (2018), 497-608.  doi: 10.1090/jams/890.  Google Scholar

[13]

M. HuangF. Li and Y. Yang, On structure of sign-skew-symmetric cluster algebras of geometric type, Ⅰ: In view of sub-seeds and seed homomorphisms, Sci. China Math., 61 (2018), 831-854.  doi: 10.1007/s11425-016-9100-8.  Google Scholar

[14]

F. Li and S. Liu, Periodicities in cluster algebras and cluster automorphism groups, preprint, arXiv: 1903.00893 [math.RT]. Google Scholar

[15]

K. Lee and R. Schiffler, Positivity for cluster algebras, Ann. of Math., 182 (2015), 73-125.  doi: 10.4007/annals.2015.182.1.2.  Google Scholar

[16]

I. Saleh, Exchange maps of cluster algebras, Int. Electron. J. Algebra, 16 (2014), 1-15.  doi: 10.24330/ieja.266223.  Google Scholar

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