Pseudo grading on cluster automorphism group with application to cluster algebras of rank 3

2025/03/28 by Fu, Changjian, Liang, Zhanhong
#13F60 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2503.22101

Abstract

We introduce a pseudo ℕ-grading on the cluster auotmorphism group Aut(A) with respect to an initial seed of A, which consists of a family of subsets \G_i\_{i∈ ℕ} of Aut(A) such that Aut(A)=\bigcup_{i∈ ℕ}G_i and G_k⋅ G_l⊂ \bigcup_i=0^k+lG_i. We prove that Aut(A) is generated by G₀∪ G₁, leading to an elementary approach for calculating cluster automorphism groups of certain cluster algebras. As an application, we completely determined the cluster automorphism groups of cluster algebras of rank 3 with indecomposable exchange matrices.

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