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15 January 2005 Cluster algebras III: Upper bounds and double Bruhat cells
Arkady Berenstein, Sergey Fomin, Andrei Zelevinsky
Duke Math. J. 126(1): 1-52 (15 January 2005). DOI: 10.1215/S0012-7094-04-12611-9

Abstract

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.

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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

Information

Published: 15 January 2005
First available in Project Euclid: 15 December 2004

zbMATH: 1135.16013
MathSciNet: MR2110627
Digital Object Identifier: 10.1215/S0012-7094-04-12611-9

Subjects:
Primary: 16S99
Secondary: 05E15, 14M17, 22E46

Rights: Copyright © 2005 Duke University Press

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Vol.126 • No. 1 • 15 January 2005
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