Quivers with relations arising from clusters (An case)
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 An. We associate to each cluster C of U an abelian category CatC such that the indecomposable objects of CatC 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 CatC. Then, we generalize the ``denominator Theorem'' of Fomin and Zelevinsky to any cluster.
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