Quivers with potentials and their representations II: Applications to cluster algebras

Derksen, Harm; Weyman, Jerzy; Zelevinsky, Andrei

doi:10.1090/S0894-0347-10-00662-4

Quivers with potentials and their representations II: Applications to cluster algebras
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by Harm Derksen, Jerzy Weyman and Andrei Zelevinsky

J. Amer. Math. Soc. 23 (2010), 749-790

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

Published electronically: February 8, 2010

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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 $\mathbf {g}$-vectors, and a family of integer polynomials called $F$-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these $\mathbf {g}$-vectors and $F$-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about $\mathbf {g}$-vectors and $F$-polynomials made in loc. cit.

References

Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. Techniques of representation theory. MR 2197389, DOI 10.1017/CBO9780511614309
A. Białynicki-Birula, On fixed point schemes of actions of multiplicative and additive groups, Topology 12 (1973), 99–103. MR 313261, DOI 10.1016/0040-9383(73)90024-4
A. Buan, O. Iyama, I. Reiten, D. Smith, Mutation of cluster-tilting objects and potentials, arXiv:0804.3813, to appear in Compositio Math.
Aslak Bakke Buan, Bethany R. Marsh, and Idun Reiten, Denominators of cluster variables, J. Lond. Math. Soc. (2) 79 (2009), no. 3, 589–611. MR 2506688, DOI 10.1112/jlms/jdn082
M. C. R. Butler and A. D. King, Minimal resolutions of algebras, J. Algebra 212 (1999), no. 1, 323–362. MR 1670674, DOI 10.1006/jabr.1998.7599
Philippe Caldero and Frédéric Chapoton, Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv. 81 (2006), no. 3, 595–616. MR 2250855, DOI 10.4171/CMH/65
Philippe Caldero and Bernhard Keller, From triangulated categories to cluster algebras. II, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 6, 983–1009 (English, with English and French summaries). MR 2316979, DOI 10.1016/j.ansens.2006.09.003
Philippe Caldero and Markus Reineke, On the quiver Grassmannian in the acyclic case, J. Pure Appl. Algebra 212 (2008), no. 11, 2369–2380. MR 2440252, DOI 10.1016/j.jpaa.2008.03.025
Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky, Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (2008), no. 1, 59–119. MR 2480710, DOI 10.1007/s00029-008-0057-9
V. V. Fock, A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Annales Sci. de. l’École Norm. Sup. 42 (2009), 865–930.
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. IV. Coefficients, Compos. Math. 143 (2007), no. 1, 112–164. MR 2295199, DOI 10.1112/S0010437X06002521
Changjian Fu and Bernhard Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories, Trans. Amer. Math. Soc. 362 (2010), no. 2, 859–895. MR 2551509, DOI 10.1090/S0002-9947-09-04979-4
William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
A. Hubery, Acyclic cluster algebras via Ringel-Hall algebras, preprint.
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
H. Nakajima, Quiver varieties and cluster algebras, arXiv:0905.0002.
Aidan Schofield, General representations of quivers, Proc. London Math. Soc. (3) 65 (1992), no. 1, 46–64. MR 1162487, DOI 10.1112/plms/s3-65.1.46
Horst Schubert, Categories, Springer-Verlag, New York-Heidelberg, 1972. Translated from the German by Eva Gray. MR 0349793, DOI 10.1007/978-3-642-65364-3
Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833