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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Clusters and seeds in acyclic cluster algebras


Authors: Aslak Bakke Buan, Robert J. Marsh, Idun Reiten and Gordana Todorov; \protect\break with an Appendix coauthored in addition by P. Caldero; B. Keller
Journal: Proc. Amer. Math. Soc. 135 (2007), 3049-3060
MSC (2000): Primary 16G20, 16G70
Published electronically: June 19, 2007
MathSciNet review: 2322734
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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.


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Additional Information

Aslak Bakke Buan
Affiliation: Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway
Email: aslakb@math.ntnu.no

Robert J. Marsh
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England
Email: rjm25@mcs.le.ac.uk

Idun Reiten
Affiliation: Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway
Email: idunr@math.ntnu.no

Gordana Todorov
Affiliation: Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, Massachusetts 02115
Email: todorov@neu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08801-6
PII: S 0002-9939(07)08801-6
Received by editor(s): December 1, 2005
Received by editor(s) in revised form: June 4, 2006
Published electronically: June 19, 2007
Communicated by: Martin Lorenz
Article copyright: © Copyright 2007 by the authors