Pseudo grading on cluster automorphism group with application to cluster algebras of rank $3$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913763647553536 |
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| author | Fu, Changjian Liang, Zhanhong |
| author_facet | Fu, Changjian Liang, Zhanhong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22101 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pseudo grading on cluster automorphism group with application to cluster algebras of rank $3$ Fu, Changjian Liang, Zhanhong Rings and Algebras 13F60 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. |
| title | Pseudo grading on cluster automorphism group with application to cluster algebras of rank $3$ |
| topic | Rings and Algebras 13F60 |
| url | https://arxiv.org/abs/2503.22101 |