Quivers with relations arising from clusters (𝐴_{𝑛} case)-Reference-Cited by-同舟云学术

Quivers with relations arising from clusters (𝐴_{𝑛} case)

Published:2005-05-26 Issue:3 Volume:358 Page:1347-1364
ISSN:0002-9947
Container-title:Transactions of the American Mathematical Society
language:en
Short-container-title:Trans. Amer. Math. Soc.

Author:

Caldero P.,Chapoton F.,Schiffler R.

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.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Link

http://www.ams.org/tran/2006-358-03/S0002-9947-05-03753-0/S0002-9947-05-03753-0.pdf

Reference10 articles.

1. Cambridge Studies in Advanced Mathematics;Auslander, Maurice,1997

2. Cluster algebras. III. Upper bounds and double Bruhat cells;Berenstein, Arkady;Duke Math. J.,2005

3. A. Buan, R. J. Marsh, M. Reineke, I. Reiten, and G. Todorov, Tilting theory and cluster combinatorics, arXiv:math.RT/0402054.

4. A. Buan, R. J. Marsh, and I. Reiten, Cluster-tilted algebras, arXiv:math.RT/0402075.

5. Cluster algebras. I. Foundations;Fomin, Sergey;J. Amer. Math. Soc.,2002

Cited by 254 articles. 订阅此论文施引文献订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献

1. On Gorenstein algebras of finite Cohen-Macaulay type: Dimer tree algebras and their skew group algebras;Journal of Algebra;2024-12

2. Categorifications of Non-Integer Quivers: Types ₄, ₃ and ₂(2+1);Representation Theory of the American Mathematical Society;2024-08-29

3. Mutation commutativity between geometric realizations and related quivers of classical cluster algebras and applications;Journal of Algebra and Its Applications;2024-08-14

4. The coloured mutation class of $$\mathbb {A}_n$$-quivers;Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas;2024-08-09

5. A cluster algebra approach to presentations of the monoid of uniform block permutations;Semigroup Forum;2024-07-29