We develop a new approach to cluster algebras, based on the notion of an upper cluster algebra defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in , we show that under an assumption of ``acyclicity,'' a cluster algebra coincides with its upper counterpart and is finitely generated; in this case, we also describe its defining ideal and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to an upper cluster algebra explicitly defined in terms of relevant combinatorial data.
Arkady Berenstein. Sergey Fomin. Andrei Zelevinsky. "Cluster algebras III: Upper bounds and double Bruhat cells." Duke Math. J. 126 (1) 1 - 52, 15 January 2005. https://doi.org/10.1215/S0012-7094-04-12611-9