Mutation Classes of 3×3 Generalized Cartan Matrices
- Ahmet I. Seven
- … show all 1 hide
Purchase on Springer.com
$29.95 / €24.95 / £19.95*
* Final gross prices may vary according to local VAT.
One of the recent developments in representation theory has been the introduction of cluster algebras by Fomin and Zelevinsky. It is now well known that these algebras are closely related with different areas of mathematics. A particular analogy exists 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. In this paper, we describe an interplay between these two classes of matrices in size 3. In particular, we give a characterization of the mutation classes associated with the generalized Cartan matrices of size 3, generalizing results of Beineke–Bruestle–Hille.
- A. Beineke, T. Bruestle, and L. Hille, Cluster-cyclic quivers with three vertices and the Markov equation, arXiv:math/0612213.
- M. Barot, C. Geiss, and A. Zelevinsky, Cluster algebras of finite type and positive symmetrizable matrices. J. Lond. Math. Soc. (2) 73 (2006), no. 3, 545–564. CrossRef
- P. Caldero and B. Keller, From triangulated categories to cluster algebras II, Ann. Sci. Éc. Norm. Super. (4) 39 (2006), 983–1009.
- S. Fomin and A. Zelevinsky, Cluster Algebras II, Invent. Math. 12 (2003), 335–380.
- V. Kac, Infinite dimensional Lie algebras, Cambridge University Press, Cambridge (1991).
- B. Keller, Cluster algebras, quiver representations and triangulated categories, arXiv:0807.1960.
- S. Parter and J. Youngs, The symmetrization of matrices with diagonal matrices, J. Math. Anal. Appl., 4, 1962, 102–110. CrossRef
- A. N. Rudakov, Markov numbers and exceptional bundles on P 2. Izv. Akad. Nauk SSSR, Ser. Mat. 52 (1988), no. 1, 100–112, 240; (Russian) translation in Math. USSR-Izv. 32 (1989), no. 1, 99–112.
- A. Seven, Cluster algebras and semipositive symmetrizable matrices. Trans. Am. Math. Soc. 363 (2011), no. 5, 2733–2762. CrossRef
- Mutation Classes of 3×3 Generalized Cartan Matrices
- Book Title
- Highlights in Lie Algebraic Methods
- pp 205-211
- Print ISBN
- Online ISBN
- Series Title
- Progress in Mathematics
- Series Volume
- Birkhäuser Boston
- Copyright Holder
- Springer Science+Business Media, LLC
- Additional Links
- Skew-symmetrizable matrix
- Generalized Cartan matrix
- Industry Sectors
- eBook Packages
- Editor Affiliations
- ID1. , Department of Mathematics, Weizmann Institute
- ID2. , Department of Mathematics, Univeristy of Haifa
- ID3. School of Engineering and Science, Jacobs University
- Ahmet I. Seven (1)
- Author Affiliations
- 1. Middle East Technical University, 06800, Ankara, Turkey
To view the rest of this content please follow the download PDF link above.