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.

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

Cited by in corpus (6)

• Planar Matrices and Arrays of Feynman Diagrams
• Augmentations, Fillings, and Clusters
• A categorification of acyclic principal coefficient cluster algebras
• Fock-Goncharov conjecture and polyhedral cones for $U\subset S{L}_n$ and base affine space $S{L}_n/U$
• Donaldson-Thomas Transformation of Double Bruhat Cells in General Linear Groups
• From frieze patterns to cluster categories