Factorial cluster algebras

Details Factorial cluster algebras Factorial cluster algebras

2011-10-06

arxiv.org/abs/1110.1199v3

Mathematics

Rings and Algebras

24 pages. v2: Small improvements. Added more examples of non-factorial cluster algebras. v3: Fixed an inaccuracy in the proof

Scientific paper

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.

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