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On class groups of upper cluster algebras

Part of: Arithmetic rings and other special rings

Published online by Cambridge University Press:  24 January 2025

Mara Pompili Open the ORCID record for Mara Pompili [Opens in a new window]
Mara Pompili*
Affiliation:
Department of Mathematics and Scientific Computing, University of Graz, NAWI Graz, Heinrichstrasse 36, Graz 8010, Austria
*
e-mail: mara.pompili@uni-graz.at
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Abstract

We compute the class groups of full rank upper cluster algebras in terms of the exchange polynomials. This characterizes the UFDs among these algebras. Our results simultaneously generalize theorems of Garcia Elsener, Lampe, and Smertnig from 2019 and of Cao, Keller, and Qin from 2023. Furthermore, we show that every (upper) cluster algebra is a finite factorization domain.

Keywords

Cluster algebrasupper cluster algebrasfactorization theoryclass groups

MSC classification

Primary: 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
Secondary: 13F60: Cluster algebras 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 21
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The author was partially supported by the Austrian Science Fund (FWF), project 10.55776/DOC-183-N

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