Generalized cluster algebras that are Krull domains can have any finitely generated abelian group as their class group, including finite ones with torsion.
Factorial cluster algebras
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abstract
We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a cluster algebra to be a factorial algebra. This can be used to construct cluster algebras, which are isomorphic to polynomial rings. We also study various kinds of upper bounds for cluster algebras, and we prove that factorial cluster algebras coincide with their upper bounds.
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2024 1verdicts
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Every finitely generated abelian group is the class group of a generalized cluster algebra
Generalized cluster algebras that are Krull domains can have any finitely generated abelian group as their class group, including finite ones with torsion.