Cluster algebras III: Upper bounds and double Bruhat cells

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

2003-05-30

arxiv.org/abs/math/0305434v3

Mathematics

Representation Theory

39 pages. Minor editorial changes, a reference added. This is the final version, to appear in Duke Mathematical Journal

Scientific paper

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.

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