Factoriality and class groups of cluster algebras

Philipp Lampe, Durham University. Part of the Algebra Seminar Series

We report on a joint work with Ana Garcia Elsener and Daniel Smertnig about the factorization theory of cluster algebras. The talk begins with an introduction to class groups and cluster algebras. Next, we give an explicit description of class groups of acyclic cluster algebras in terms of partner sets. As a Krull domain is factorial if and only its class group is zero, this constitutes a generalization of the study of factoriality of cluster algebras started by Geiß-Leclerc-Schröer. As a corollary, we reprove and extend a classification of factoriality for cluster algebras of Dynkin type.