Clusters and seeds in acyclic cluster algebras
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- by Aslak Bakke Buan, Bethany R. Marsh, Idun Reiten and Gordana Todorov; with an Appendix coauthored in addition by P. Caldero; B. Keller
- Proc. Amer. Math. Soc. 135 (2007), 3049-3060
- DOI: https://doi.org/10.1090/S0002-9939-07-08801-6
- Published electronically: June 19, 2007
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.References
- Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no. 1, 1–52. MR 2110627, DOI 10.1215/S0012-7094-04-12611-9
- Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), no. 1, 323–332. MR 2247893, DOI 10.1090/S0002-9947-06-03879-7
- Buan A., Marsh R., Reiten I. Cluster mutation via quiver representations, preprint arxiv:math.RT/0412077, to appear in Commentarii Mathematici Helvetici.
- Buan A., Marsh R., Reineke M., Reiten I., Todorov G. Tilting theory and cluster combinatorics, Advances in Mathematics, 204 (2), (2006), 572–618.
- Caldero P., Chapoton F. Cluster algebras as Hall algebras of quiver representations, Commentarii Mathematici Helvetici, 81, (2006), 595–616.
- P. Caldero, F. Chapoton, and R. Schiffler, Quivers with relations arising from clusters ($A_n$ case), Trans. Amer. Math. Soc. 358 (2006), no. 3, 1347–1364. MR 2187656, DOI 10.1090/S0002-9947-05-03753-0
- Caldero P., Chapoton F., Schiffler R., Quivers with relations and cluster tilted algebras. Algebras and Representation Theory, 9, (2006), no. 4, 359–376.
- Caldero P., Keller B. From triangulated categories to cluster algebras, to appear in Inventiones Math.
- Caldero P., Keller B. From triangulated categories to cluster algebras II, Annales Scientifiques de l’Ecole Normale Supérieure, 4eme serie, 39, (2006), 983–1009.
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. MR 1887642, DOI 10.1090/S0894-0347-01-00385-X
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63–121. MR 2004457, DOI 10.1007/s00222-003-0302-y
- Bernhard Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551–581. MR 2184464
- Otto Kerner, Representations of wild quivers, Representation theory of algebras and related topics (Mexico City, 1994) CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 65–107. MR 1388560
- Robert Marsh, Markus Reineke, and Andrei Zelevinsky, Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), no. 10, 4171–4186. MR 1990581, DOI 10.1090/S0002-9947-03-03320-8
- Reiten I., Todorov G. unpublished.
Bibliographic Information
- Aslak Bakke Buan
- Affiliation: Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway
- Email: aslakb@math.ntnu.no
- Bethany R. Marsh
- Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England
- MR Author ID: 614298
- ORCID: 0000-0002-4268-8937
- 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
- B. Keller
- MR Author ID: 99940
- ORCID: 0000-0002-4493-2040
- 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
- © Copyright 2007 by the authors
- Journal: Proc. Amer. Math. Soc. 135 (2007), 3049-3060
- MSC (2000): Primary 16G20, 16G70
- DOI: https://doi.org/10.1090/S0002-9939-07-08801-6
- MathSciNet review: 2322734