Cluster algebras III: Upper bounds and double Bruhat cells
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Cluster algebras III: Upper bounds and double Bruhat cells

Arkady Berenstein, Sergey Fomin and Andrei Zelevinsky

Source: Duke Math. J. Volume 126, Number 1 (2005), 1-52.

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.

Related Works:

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

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1103136474
Mathematical Reviews number (MathSciNet): MR2110627
Digital Object Identifier: doi:10.1215/S0012-7094-04-12611-9

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