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.
An Infinitely Generated Upper Cluster Algebra
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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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Cluster structures on Cox rings
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.