A conjecture on cluster automorphisms of cluster algebras

Peigen Cao,Jie Pan,Fang Li,Siyang Liu

doi:10.3934/era.2019006

Abstract

<abstract><p>A cluster automorphism is a <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-algebra automorphism of a cluster algebra <inline-formula><tex-math id="M2">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> satisfying that it sends a cluster to another and commutes with mutations. Chang and Schiffler conjectured that a cluster automorphism of <inline-formula><tex-math id="M3">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> is just a <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-algebra homomorphism of a cluster algebra sending a cluster to another. The aim of this article is to prove this conjecture.</p></abstract>

Highlights

Cluster algebras were invented by Fomin and Zelevinsky in a series of papers [9, 2, 10, 11]
A cluster algebra is a Z-subalgebra of an ambient field F = Q(u1, · · ·, un) generated by certain combinatorially defined generators, which are grouped into overlapping clusters
We first recall the definition of cluster automorphisms, which were introduced by Assem, Schiffler and Shamchenko in [1]

Summary

Introduction

Cluster algebras were invented by Fomin and Zelevinsky in a series of papers [9, 2, 10, 11]. F is called a cluster automorphism of A if there exists another seed (z, B′) of A such that (1) f (x) = z; (2) f (μx(x)) = μf(x)(z) for any x ∈ x. The following very insightful conjecture on cluster automorphisms is by Chang and Schiffler, which suggests that we can weaken the conditions in Definition 1.1. It suggests that the second condition in Definition 1.1 can be obtained from the first one and the assumption that f is a Z-algebra homomorphism. [5, Conjecture 1] Let A be a cluster algebra, and f : A → A be a Z-algebra homomorphism. Mathematics Subject Classification(2010): 13F60 Keywords: cluster algebra, cluster automorphism Date: version of August 9, 2019

Preliminaries

The proof of main result