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Cluster algebras III: Upper bounds and double Bruhat cells

2005; Duke University Press; Volume: 126; Issue: 1 Linguagem: Inglês

10.1215/s0012-7094-04-12611-9

1547-7398

Arkady Berenstein, Sergey Fomin, Andrei Zelevinsky,

Advanced Algebra and Geometry

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 [7], 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.