Factorial cluster algebras
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.
Cite
@article{arxiv.1110.1199,
title = {Factorial cluster algebras},
author = {Christof Geiß and Bernard Leclerc and Jan Schröer},
journal= {arXiv preprint arXiv:1110.1199},
year = {2013}
}
Comments
24 pages. v2: Small improvements. Added more examples of non-factorial cluster algebras. v3: Fixed an inaccuracy in the proof of Theorem 1.4. Generalized Proposition 6.3. Some further small improvements. v3: small corrections and updates, final version to appear in Documenta Math