DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score (2023): 0.6

SJR (2023): 0.214

SNIP (2023): 0.604

Index Copernicus (2022): 121.02

H-Index: 5

Discussiones Mathematicae - General Algebra and Applications

Article in press

Authors:

I. Saleh

Ibrahim Saleh

University of Wisconsin Whitewater

email: salehi@uww.edu

Title:

Rooted mutation groups and finite type cluster algebras

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Source:

Discussiones Mathematicae - General Algebra and Applications

Received: 2024-02-29 , Revised: 2024-08-19 , Accepted: 2024-08-20 , Available online: 2025-05-19 , https://doi.org/10.7151/dmgaa.1477

Abstract:

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.

Keywords:

cluster algebras, subseeds, rooted mutation loops

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