Every finitely generated abelian group is the class group of a generalized cluster algebra
Mara Pompili
TL;DR
The work analyzes factorization and class-group properties of generalized cluster algebras, proving that every finitely generated abelian group $G$ can be realized as the class group of a Krull generalized cluster algebra and that these algebras may exhibit torsion in their class groups, unlike classical cases. It develops the necessary machinery—generalized seeds, mutation, and the Starfish Lemma—to establish when upper and lower bounds coincide and to characterize factoriality via exchange polynomials. A key contribution is a constructive realization (Theorem 4.2) showing how to encode any $G$ into a carefully designed acyclic, coprime generalized seed, yielding $\mathcal C(A) \cong G$ with each class containing $|R|$ height-1 prime divisors. The LP-algebra section connects these ideas to Laurent phenomenon algebras, demonstrating analogous FF-domain and Krull-class-group behavior and highlighting cases like the Markov LP-algebra that illustrate deviations from cluster-algebra intuition. Overall, the paper extends the understanding of factorization in generalized cluster frameworks and provides a versatile blueprint for embedding arbitrary finitely generated abelian groups into class groups of Krull domains.
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.