标题： 在簇自同构群上的伪分级及其在秩$3$的簇代数中的应用 显示英文标题

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

摘要： 我们引入了一个关于初始种子$\mathcal{A}$的簇自同构群$\operatorname{Aut}(\mathcal{A})$的伪$\mathbb{N}$分级，该分级由$\operatorname{Aut}(\mathcal{A})$的子集族$\{G_i\}_{i\in \mathbb{N}}$组成，使得$\operatorname{Aut}(\mathcal{A})=\bigcup_{i\in \mathbb{N}}G_i$且$G_k\cdot G_l\subset \bigcup_{i=0}^{k+l}G_i$。 我们证明了$\operatorname{Aut}(\mathcal{A})$由$G_0\cup G_1$生成，从而为计算某些簇代数的簇自同构群提供了一种基本方法。 作为应用，我们完全确定了秩为$3$且不可分解交换矩阵的簇代数的簇自同构群。 显示英文摘要

摘要： We introduce a pseudo $\mathbb{N}$-grading on the cluster auotmorphism group $\operatorname{Aut}(\mathcal{A})$ with respect to an initial seed of $\mathcal{A}$, which consists of a family of subsets $\{G_i\}_{i\in \mathbb{N}}$ of $\operatorname{Aut}(\mathcal{A})$ such that $\operatorname{Aut}(\mathcal{A})=\bigcup_{i\in \mathbb{N}}G_i$ and $G_k\cdot G_l\subset \bigcup_{i=0}^{k+l}G_i$. We prove that $\operatorname{Aut}(\mathcal{A})$ is generated by $G_0\cup G_1$, 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.