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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Quivers with relations arising from clusters $(A_n$ case)


Authors: P. Caldero, F. Chapoton and R. Schiffler
Journal: Trans. Amer. Math. Soc. 358 (2006), 1347-1364
MSC (2000): Primary 16G20, 16G70, 05E15, 20F55
DOI: https://doi.org/10.1090/S0002-9947-05-03753-0
Published electronically: May 26, 2005
MathSciNet review: 2187656
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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 $\mathcal{C}_C$ such that the indecomposable objects of $\mathcal{C}_C$ 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 $\mathcal{C}_C$. Then, we generalize the ``denominator theorem'' of Fomin and Zelevinsky to any cluster.


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Additional Information

P. Caldero
Affiliation: Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622 Villeurbanne Cedex, France

F. Chapoton
Affiliation: Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622 Villeurbanne Cedex, France

R. Schiffler
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6

DOI: https://doi.org/10.1090/S0002-9947-05-03753-0
Received by editor(s): February 25, 2004
Received by editor(s) in revised form: May 24, 2004
Published electronically: May 26, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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