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Cluster algebras. III: Upper bounds and double Bruhat cells. (English) Zbl 1135.16013
Summary: 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 part I [S. Fomin and A. Zelevinsky, J. Am. Math. Soc. 15, No. 2, 497-529 (2002; Zbl 1021.16017)], 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.

MSC:
16G20 Representations of quivers and partially ordered sets
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
17B20 Simple, semisimple, reductive (super)algebras
05E15 Combinatorial aspects of groups and algebras (MSC2010)
14M17 Homogeneous spaces and generalizations
22E46 Semisimple Lie groups and their representations
20G05 Representation theory for linear algebraic groups
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