DMCA
Clusters and seeds for acyclic cluster algebras (2007)
| Citations: | 38 - 9 self |
BibTeX
@MISC{Buan07clustersand,
author = {Aslak Bakke Buan and Robert J. Marsh and Idun Reiten and Gordana Todorov},
title = { Clusters and seeds for acyclic cluster algebras },
year = {2007}
}
Share
 |
 |
 |
 |
||
OpenURL
Abstract
Cluster algebras are commutative algebras that were introduced by Fomin and Zelevinsky in order to model the dual canonical basis of a quantum group and total positivity in algebraic groups. Cluster categories were introduced as a representation-theoretic model for cluster algebras. In this article we use this representation-theoretic approach to prove a conjecture of Fomin and Zelevinsky, that for cluster algebras with no coefficients associated to quivers with no oriented cycles, a seed is determined by its cluster. We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.
Keyphrases
acyclic cluster algebra cluster algebra representation-theoretic model representation-theoretic approach algebraic group total positivity indecomposable exceptional module commutative algebra dual canonical basis non-polynomial cluster variable associated hereditary algebra quantum group composition factor cluster category oriented cycle
