Symmetric mutations algebras in the context of sub-cluster algebras
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866916149739913216 |
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| author | Saleh, Ibrahim |
| author_facet | Saleh, Ibrahim |
| contents | For a rooted cluster algebra $\mathcal{A}(Q)$ over a valued quiver $Q$, a \emph{symmetric cluster variable} is any cluster variable belonging to a cluster associated with a quiver $σ(Q)$, for some permutation $σ$. The subalgebra of $\mathcal{A}(Q)$ generated by all symmetric cluster variables is called the \emph{symmetric mutation subalgebra} and is denoted by $\mathcal{B}(Q)$. In this paper we identify the class of cluster algebras that satisfy $\mathcal{B}(Q)=\mathcal{A}(Q)$, which contains almost every quiver of finite mutation type. In the process of proving the main theorem, we provide a classification of quivers mutation classes based on their weights. Some properties of symmetric mutation subalgebras are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_08211 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Symmetric mutations algebras in the context of sub-cluster algebras Saleh, Ibrahim Representation Theory 13F60 For a rooted cluster algebra $\mathcal{A}(Q)$ over a valued quiver $Q$, a \emph{symmetric cluster variable} is any cluster variable belonging to a cluster associated with a quiver $σ(Q)$, for some permutation $σ$. The subalgebra of $\mathcal{A}(Q)$ generated by all symmetric cluster variables is called the \emph{symmetric mutation subalgebra} and is denoted by $\mathcal{B}(Q)$. In this paper we identify the class of cluster algebras that satisfy $\mathcal{B}(Q)=\mathcal{A}(Q)$, which contains almost every quiver of finite mutation type. In the process of proving the main theorem, we provide a classification of quivers mutation classes based on their weights. Some properties of symmetric mutation subalgebras are given. |
| title | Symmetric mutations algebras in the context of sub-cluster algebras |
| topic | Representation Theory 13F60 |
| url | https://arxiv.org/abs/2105.08211 |