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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Cluster algebras and semipositive symmetrizable matrices

Author: Ahmet I. Seven
Journal: Trans. Amer. Math. Soc. 363 (2011), 2733-2762
MSC (2010): Primary 05E15; Secondary 13F60, 05C50, 15B36, 17B67
Published electronically: December 10, 2010
MathSciNet review: 2763735
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Abstract: There is a particular analogy between combinatorial aspects of cluster algebras and Kac-Moody algebras: roughly speaking, cluster algebras are associated with skew-symmetrizable matrices while Kac-Moody algebras correspond to (symmetrizable) generalized Cartan matrices. Both classes of algebras and the associated matrices have the same classification of finite type objects by the well-known Cartan-Killing types. In this paper, we study an extension of this correspondence to the affine type. In particular, we establish the cluster algebras which are determined by the generalized Cartan matrices of affine type.

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  • 1. M. Barot, C. Geiss and A. Zelevinsky, Cluster algebras of finite type and positive symmetrizable matrices. J. London Math. Soc. (2) 73 (2006), no. 3, 545-564. MR 2241966 (2007i:05190)
  • 2. M. Barot and A. Seven, Cluster algebras of finite mutation type, in preparation.
  • 3. J. Bastian, Mutation classes of $ \tilde{A}_n$-quivers and derived equivalence classification of cluster tilted algebras of type $ \tilde{A}_n$, arXiv:0901.1515
  • 4. A. Buan and I. Reiten, Acyclic quivers of finite mutation type, Int. Math. Res. Not. 2006, Art. ID 12804, 10 pp. MR 2249997 (2007f:16034)
  • 5. A. Buan, I. Reiten and A. Seven, Tame concealed algebras and cluster quivers of minimal infinite type. J. Pure Appl. Algebra 211 (2007), no. 1, 71-82. MR 2333764 (2008f:16039)
  • 6. P. Caldero and B. Keller, From triangulated categories to cluster algebras II, Ann. Sci. Ecole Norm. Sup. (4) 39 (2006), 983-1009. MR 2316979 (2008m:16031)
  • 7. A. Felikson, M. Shapiro and P. Tumarkin, Skew-symmetric cluster algebras of finite mutation type, Arxiv:0811.1703.
  • 8. S. Fomin, M. Shapiro and D. Thurston, Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta. Math. 201 (2008), no.1, 83-146. MR 2448067 (2010b:57032)
  • 9. S. Fomin and A. Zelevinsky, Cluster Algebras II. Finite type classification, Invent Math. 154 (2003), 63-121. MR 2004457 (2004m:17011)
  • 10. D. Happel and D. Vossieck, Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42 (1983), no. 2-3, 221-243. MR 701205 (84m:16022)
  • 11. V. Kac, Infinite dimensional Lie algebras, Cambridge University Press (1991). MR 1104219 (92k:17038)
  • 12. C.M. Ringel, Tame algebras and integral quadratic forms, Springer Lecture Notes in Mathematics, vol. 1099, 1984. MR 774589 (87f:16027)
  • 13. A. Seven, Recognizing cluster algebras of finite type, Electron. J. Combin. 14 (1) (2007). MR 2285804 (2008c:05192)

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Additional Information

Ahmet I. Seven
Affiliation: Department of Mathematics, Middle East Technical University, 06531, Ankara, Turkey

Received by editor(s): June 26, 2009
Received by editor(s) in revised form: November 5, 2009, and November 18, 2009
Published electronically: December 10, 2010
Additional Notes: The author’s research was supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK) grant #107T050
Article copyright: © Copyright 2010 American Mathematical Society

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