# Mara Pompili: On the arithmetic of upper cluster algebras

# On the arithmetic of upper cluster algebras

Mara Pompili, University of Graz, Austria

Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky (2002) endowed with a family of distinguished generators, which are constructed recursively using mutations. Cluster algebras have been the focus of intense research since, thanks to the many links that have been discovered with a wide range of subjects, although their ring-theoretic properties are not so well-explored. The main focus of the talk is on factorization-theoretical properties of upper cluster algebras, an upper bound for cluster algebras given by the Laurent phenomenon. We give a full description of the class group of full rank upper cluster algebras in term of the exchange polynomials. Understanding class groups allows us to study unique factorization properties of Krull domains. This extends results of Garcia-Elsener–Lampe–Smertnig (2019) on acyclic cluster algebras and of Cao–Keller–Qin (2022).

Vljudno vabljeni. Roman Drnovšek in Primož Moravec