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An Infinitely Generated Upper Cluster Algebra

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arxiv 1305.6867 v1 pith:W4AXP4GG submitted 2013-05-29 math.RT math.AG

David E Speyer This is my paper · ORCID
classification math.RTmath.AG
keywords clusteralgebrageneratedpmatrixupperalgebrasansweringb-matrix
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abstract

We show that upper cluster algebras need not be finitely generated, answering a question of Berenstein, Fomin and Zelevinsky. Our counter-example is a cluster algebra with B-matrix $\begin{pmatrix} 0 & 3 & -3 \\ -3 & 0 & 3 \\ 3 & -3 & 0 \end{pmatrix}$ and coefficients obeying a genericity condition.

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  1. Cluster structures on Cox rings

    math.AG 2024-12 conditional novelty 6.0 of 10

    A general lifting procedure produces graded upper cluster algebra structures, or the unique candidates for them, on Cox rings, with applications to flag varieties and a new diagonal partial compactification.

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