Duke Mathematical Journal

Cluster algebras III: Upper bounds and double Bruhat cells

Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky

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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.

Article information

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

First available in Project Euclid: 15 December 2004

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Zentralblatt MATH identifier

Primary: 16S99: None of the above, but in this section
Secondary: 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80] 14M17: Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15] 22E46: Semisimple Lie groups and their representations


Berenstein, Arkady; Fomin, Sergey; Zelevinsky, Andrei. Cluster algebras III: Upper bounds and double Bruhat cells. Duke Math. J. 126 (2005), no. 1, 1--52. doi:10.1215/S0012-7094-04-12611-9. http://projecteuclid.org/euclid.dmj/1103136474.

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See also

  • First article in series: S. Fomin, A. Zelevinsky. Cluster Algebras I: Foundations. J. Amer. Math. Soc. 154 (2002), pp. 497-529.
  • Second article in series: S. Fomin, A. Zelevinsky. Cluster Algebras II: Finite Type Classification. Invent. Math. 154 (2003), pp. 63-121.