# American Institute of Mathematical Sciences

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:

show all references

##### References:
 [1] Fang Li, Jie Pan. On inner Poisson structures of a quantum cluster algebra without coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021021 [2] 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 [3] Valentin Ovsienko, MichaeL Shapiro. Cluster algebras with Grassmann variables. Electronic Research Announcements, 2019, 26: 1-15. doi: 10.3934/era.2019.26.001 [4] Octav Cornea and Francois Lalonde. Cluster homology: An overview of the construction and results. Electronic Research Announcements, 2006, 12: 1-12. [5] Gerhard Keller, Carlangelo Liverani. Coupled map lattices without cluster expansion. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 325-335. doi: 10.3934/dcds.2004.11.325 [6] Takashi Hara and Gordon Slade. The incipient infinite cluster in high-dimensional percolation. Electronic Research Announcements, 1998, 4: 48-55. [7] 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 [8] 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 [9] David E. Bernholdt, Mark R. Cianciosa, Clement Etienam, David L. Green, Kody J. H. Law, Jin M. Park. Corrigendum to "Cluster, classify, regress: A general method for learning discontinuous functions [1]". Foundations of Data Science, 2020, 2 (1) : 81-81. doi: 10.3934/fods.2020005 [10] 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 [11] 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 [12] David E. Bernholdt, Mark R. Cianciosa, David L. Green, Jin M. Park, Kody J. H. Law, Clement Etienam. Cluster, classify, regress: A general method for learning discontinuous functions. Foundations of Data Science, 2019, 1 (4) : 491-506. doi: 10.3934/fods.2019020 [13] Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248 [14] Gillala Rekha, V Krishna Reddy, Amit Kumar Tyagi. A novel approach for solving skewed classification problem using cluster based ensemble method. Mathematical Foundations of Computing, 2020, 3 (1) : 1-9. doi: 10.3934/mfc.2020001 [15] 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 [16] Hongyan Guo. Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra. Electronic Research Archive, 2021, 29 (4) : 2673-2685. doi: 10.3934/era.2021008 [17] 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 [18] Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017 [19] Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1115-1129. doi: 10.3934/dcdss.2020066 [20] Arvind Ayyer, Carlangelo Liverani, Mikko Stenlund. Quenched CLT for random toral automorphism. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 331-348. doi: 10.3934/dcds.2009.24.331

2020 Impact Factor: 1.833