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Quivers with relations arising from clusters (A_n case)

Philippe Caldero·Frederic Chapoton·Ralf Schiffler
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Abstract

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A_n. We associate to each cluster C of U an abelian category Cat_C such that the indecomposable objects of Cat_C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of Cat_C. Then, we generalize the ``denominator Theorem'' of Fomin and Zelevinsky to any cluster.

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Philippe Caldero, Frederic Chapoton, Ralf Schiffler. 2004-01-23. Quivers with relations arising from clusters (A_n case). https://doi.org/10.1090/s0002-9947-05-03753-0

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