Every finitely generated abelian group is the class group of a generalized cluster algebra

TL;DR

This paper characterizes the class groups of generalized cluster algebras, showing they can realize any finitely generated abelian group and exploring their factorization properties and UFD criteria.

Contribution

It establishes that every finitely generated abelian group is realizable as a class group of a generalized cluster algebra and analyzes their algebraic properties.

Findings

Any finitely generated abelian group can be realized as a class group.

Generalized cluster algebras are FF-domains.

Cluster variables are strong atoms.

Abstract

We determine the class group of those generalized cluster algebras that are Krull domains. In particular, this provides a criterion for determining whether or not a generalized cluster algebra is a UFD. In fact, any finitely generated abelian group can be realized as the class group of a generalized cluster algebra. Additionally, we show that generalized cluster algebras are FF-domains and that their cluster variables are strong atoms. Finally, we examine the factorization and ring-theoretic properties of Laurent phenomenon algebras.