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Transactions of the American Mathematical Society
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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Additional Information

Ahmet I. Seven
Affiliation: Department of Mathematics, Middle East Technical University, 06531, Ankara, Turkey
Email: aseven@metu.edu.tr

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05255-9
PII: S 0002-9947(2010)05255-9
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