Cluster algebras III: Upper bounds and double Bruhat cells
Abstract
We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon from math.RT/0104151, 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 the upper cluster algebra explicitly defined in terms of relevant combinatorial data.
Cite
@article{arxiv.math/0305434,
title = {Cluster algebras III: Upper bounds and double Bruhat cells},
author = {Arkady Berenstein and Sergey Fomin and Andrei Zelevinsky},
journal= {arXiv preprint arXiv:math/0305434},
year = {2007}
}
Comments
39 pages. Minor editorial changes, a reference added. This is the final version, to appear in Duke Mathematical Journal