Duke Mathematical Journal
- Duke Math. J.
- Volume 126, Number 1 (2005), 1-52.
Cluster algebras III: Upper bounds and double Bruhat cells
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 , 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.
Duke Math. J., Volume 126, Number 1 (2005), 1-52.
First available in Project Euclid: 15 December 2004
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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. https://projecteuclid.org/euclid.dmj/1103136474
- 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.