An Infinitely Generated Upper Cluster Algebra

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 $\left(\begin{array}{ccc}0& 3& -3\\ -3& 0& 3\\ 3& -3& 0\end{array}\right)$ and coefficients obeying a genericity condition.

3 pages

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