Quivers with relations arising from clusters (𝐴_{𝑛} case)
Top Cited Papers
Open Access
- 26 May 2005
- journal article
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 358 (3) , 1347-1364
- https://doi.org/10.1090/s0002-9947-05-03753-0
Abstract
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U U be a cluster algebra of type A n A_n . We associate to each cluster C C of U U an abelian category C C \mathcal {C}_C such that the indecomposable objects of C C \mathcal {C}_C are in natural correspondence with the cluster variables of U U which are not in C C . We give an algebraic realization and a geometric realization of C C \mathcal {C}_C . Then, we generalize the “denominator theorem” of Fomin and Zelevinsky to any cluster.Keywords
All Related Versions
This publication has 7 references indexed in Scilit:
- Cluster algebras III: Upper bounds and double Bruhat cellsDuke Mathematical Journal, 2005
- Y -systems and generalized associahedraAnnals of Mathematics, 2003
- Generalized associahedra via quiver representationsTransactions of the American Mathematical Society, 2003
- Cluster algebras II: Finite type classificationInventiones Mathematicae, 2003
- Cluster algebras I: FoundationsJournal of the American Mathematical Society, 2001
- Representation Theory of Artin AlgebrasPublished by Cambridge University Press (CUP) ,1995
- Auslander-Reiten sequences and representation-finite algebrasPublished by Springer Nature ,1980