Quivers with relations arising from clusters (A_n case)

2004/01/23 by Philippe Caldero, Frederic Chapoton, Ralf Schiffler · 1 citation
Mathematics · #math.RT #math.RA #msc:16G20 #msc:16G70 #msc:05E15 #msc:20F55

paper · pdf · doi:10.1090/s0002-9947-05-03753-0

published as Transactions of the A.M.S. 358 no 3 (2006), pages 1347-1364 · 18 pages, 6 figures

arxiv created 2004/01/23 · arxiv updated 2014/04/09

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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