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

[1]

Inês Cruz, M. Esmeralda Sousa-Dias. Reduction of cluster iteration maps. Journal of Geometric Mechanics, 2014, 6 (3) : 297-318. doi: 10.3934/jgm.2014.6.297

[2]

Valentin Ovsienko, MichaeL Shapiro. Cluster algebras with Grassmann variables. Electronic Research Announcements, 2019, 26: 1-15. doi: 10.3934/era.2019.26.001

[3]

Octav Cornea and Francois Lalonde. Cluster homology: An overview of the construction and results. Electronic Research Announcements, 2006, 12: 1-12.

[4]

Gerhard Keller, Carlangelo Liverani. Coupled map lattices without cluster expansion. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 325-335. doi: 10.3934/dcds.2004.11.325

[5]

Takashi Hara and Gordon Slade. The incipient infinite cluster in high-dimensional percolation. Electronic Research Announcements, 1998, 4: 48-55.

[6]

Shuping Li, Zhen Jin. Impacts of cluster on network topology structure and epidemic spreading. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3749-3770. doi: 10.3934/dcdsb.2017187

[7]

Xiwei Liu, Tianping Chen, Wenlian Lu. Cluster synchronization for linearly coupled complex networks. Journal of Industrial & Management Optimization, 2011, 7 (1) : 87-101. doi: 10.3934/jimo.2011.7.87

[8]

Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein. Higher pentagram maps, weighted directed networks, and cluster dynamics. Electronic Research Announcements, 2012, 19: 1-17. doi: 10.3934/era.2012.19.1

[9]

A. Procacci, Benedetto Scoppola. Convergent expansions for random cluster model with $q>0$ on infinite graphs. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1145-1178. doi: 10.3934/cpaa.2008.7.1145

[10]

Feyza Gürbüz, Panos M. Pardalos. A decision making process application for the slurry production in ceramics via fuzzy cluster and data mining. Journal of Industrial & Management Optimization, 2012, 8 (2) : 285-297. doi: 10.3934/jimo.2012.8.285

[11]

Van Cyr, John Franks, Bryna Kra, Samuel Petite. Distortion and the automorphism group of a shift. Journal of Modern Dynamics, 2018, 13: 147-161. doi: 10.3934/jmd.2018015

[12]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-15. doi: 10.3934/dcdss.2020066

[13]

Arvind Ayyer, Carlangelo Liverani, Mikko Stenlund. Quenched CLT for random toral automorphism. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 331-348. doi: 10.3934/dcds.2009.24.331

[14]

Hari Bercovici, Viorel Niţică. A Banach algebra version of the Livsic theorem. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 523-534. doi: 10.3934/dcds.1998.4.523

[15]

Van Cyr, Bryna Kra. The automorphism group of a minimal shift of stretched exponential growth. Journal of Modern Dynamics, 2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483

[16]

Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047

[17]

Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420

[18]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[19]

Viktor Levandovskyy, Gerhard Pfister, Valery G. Romanovski. Evaluating cyclicity of cubic systems with algorithms of computational algebra. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2023-2035. doi: 10.3934/cpaa.2012.11.2023

[20]

Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399

2018 Impact Factor: 0.263

Article outline

[Back to Top]