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), 749-790

MSC (2010):
Primary 16G10; Secondary 16G20, 16S38, 16D90

DOI:
https://doi.org/10.1090/S0894-0347-10-00662-4

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 skew-symmetric 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:
https://doi.org/10.1090/S0894-0347-10-00662-4

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 DMS-0349019 and DMS-0901298.

The second author was supported by the NSF grant DMS-0600229.

The third author was supported by the NSF grants DMS-0500534 and DMS-0801187, and by a Humboldt Research Award.

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.