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Rooted mutation groups and finite type cluster algebras.

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Τίτλος: Rooted mutation groups and finite type cluster algebras.
Συγγραφείς: Saleh, Ibrahim (AUTHOR) salehi@uww.edu
Πηγή: Discussiones Mathematicae: General Algebra & Applications. 2025, Vol. 45 Issue 2, p261-279. 19p.
Θεματικοί όροι: *Cluster algebras, *Mutations (Algebra), *Isomorphism (Mathematics)
Περίληψη: For a fixed seed $(X, Q)$, a rooted mutation loop is a sequence of mutations that preserves $(X, Q)$. The group generated by all rooted mutation loops is called rooted mutation group and will be denoted by $\mathcal{M}(Q)$. The global mutation group of $(X, Q)$, denoted $\mathcal{M}$, is the group of all mutation sequences subject to the relations on the cluster structure of $(X, Q)$. In this article, we show that two finite type cluster algebras $\mathcal{A}(Q)$ and $\mathcal{A}(Q')$ are isomorphic if and only if their rooted mutation groups are isomorphic and the sets $\mathcal{M}/\mathcal{M}(Q)$ and $\mathcal{M'}/\mathcal{M}(Q')$ are in one to one correspondence. The second main result shows that the group $\mathcal{M}(Q)$ and the set $\mathcal{M}/\mathcal{M}(Q)$ determine the finiteness of the cluster algebra $\mathcal{A}(Q)$ and vice versa. [ABSTRACT FROM AUTHOR]
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  Data: Rooted mutation groups and finite type cluster algebras.
– Name: Author
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  Data: <searchLink fieldCode="AR" term="%22Saleh%2C+Ibrahim%22">Saleh, Ibrahim</searchLink> (AUTHOR)<i> salehi@uww.edu</i>
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  Data: <searchLink fieldCode="JN" term="%22Discussiones+Mathematicae%3A+General+Algebra+%26+Applications%22">Discussiones Mathematicae: General Algebra & Applications</searchLink>. 2025, Vol. 45 Issue 2, p261-279. 19p.
– Name: Subject
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  Group: Su
  Data: *<searchLink fieldCode="DE" term="%22Cluster+algebras%22">Cluster algebras</searchLink><br />*<searchLink fieldCode="DE" term="%22Mutations+%28Algebra%29%22">Mutations (Algebra)</searchLink><br />*<searchLink fieldCode="DE" term="%22Isomorphism+%28Mathematics%29%22">Isomorphism (Mathematics)</searchLink>
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  Label: Abstract
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  Data: For a fixed seed $(X, Q)$, a <span style="font-style: italic;">rooted mutation loop</span> is a sequence of mutations that preserves $(X, Q)$. The group generated by all rooted mutation loops is called <span style="font-style: italic;">rooted mutation group</span> and will be denoted by $\mathcal{M}(Q)$. The <span style="font-style: italic;">global mutation group</span> of $(X, Q)$, denoted $\mathcal{M}$, is the group of all mutation sequences subject to the relations on the cluster structure of $(X, Q)$. In this article, we show that two finite type cluster algebras $\mathcal{A}(Q)$ and $\mathcal{A}(Q')$ are isomorphic if and only if their rooted mutation groups are isomorphic and the sets $\mathcal{M}/\mathcal{M}(Q)$ and $\mathcal{M'}/\mathcal{M}(Q')$ are in one to one correspondence. The second main result shows that the group $\mathcal{M}(Q)$ and the set $\mathcal{M}/\mathcal{M}(Q)$ determine the finiteness of the cluster algebra $\mathcal{A}(Q)$ and vice versa. [ABSTRACT FROM AUTHOR]
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        Value: 10.7151/dmgaa.1477
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 19
        StartPage: 261
    Subjects:
      – SubjectFull: Cluster algebras
        Type: general
      – SubjectFull: Mutations (Algebra)
        Type: general
      – SubjectFull: Isomorphism (Mathematics)
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      – TitleFull: Rooted mutation groups and finite type cluster algebras.
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            – D: 01
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              Text: 2025
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              Y: 2025
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