Quivers with potentials and their representations II: Applications to cluster algebras
Authors:
Harm Derksen, Jerzy Weyman and Andrei Zelevinsky
Journal:
J. Amer. Math. Soc. 23 (2010), 749790
MSC (2010):
Primary 16G10; Secondary 16G20, 16S38, 16D90
Published electronically:
February 8, 2010
MathSciNet review:
2629987
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Abstract: We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the ``Cluster algebras IV'' paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called vectors, and a family of integer polynomials called polynomials. In the case of skewsymmetric exchange matrices we find an interpretation of these vectors and polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about vectors and polynomials made in loc. cit.
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Additional Information
Harm Derksen
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email:
hderksen@umich.edu
Jerzy Weyman
Affiliation:
Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email:
j.weyman@neu.edu
Andrei Zelevinsky
Affiliation:
Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email:
andrei@neu.edu
DOI:
http://dx.doi.org/10.1090/S0894034710006624
Received by editor(s):
April 16, 2009
Received by editor(s) in revised form:
November 13, 2009
Published electronically:
February 8, 2010
Additional Notes:
The first author was supported by the NSF grants DMS0349019 and DMS0901298.
The second author was supported by the NSF grant DMS0600229.
The third author was supported by the NSF grants DMS0500534 and DMS0801187, and by a Humboldt Research Award.
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© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
