Frontiers of Mathematics in China >
Cluster characters for cyclic quivers
Received date: 10 May 2010
Accepted date: 29 May 2011
Published date: 01 Aug 2012
Copyright
We define an analogue of the Caldero-Chapoton map for the cluster category of finite-dimensional nilpotent representations over a cyclic quiver. We prove that it is a cluster character and satisfies some inductive formulas for the multiplication between the generalized cluster variables (the images of objects of the cluster category under this map). Moreover, we construct a -basis for the algebra generated by all generalized cluster variables.
Key words:
Cyclic quiver; cluster algebra;
Ming DING , Fan XU . Cluster characters for cyclic quivers[J]. Frontiers of Mathematics in China, 2012 , 7(4) : 679 -693 . DOI: 10.1007/s11464-011-0146-5
1 |
Buan A, Marsh R, Reineke M, Reiten I, Todorov G. Tilting theory and cluster combinatorics. Adv Math, 2006, 204: 572-618
|
2 |
Buan A, Marsh R, Vatne D. Cluster structure from 2-Calabi-Yau categories with loops. Math Z, 2010, 265(4): 951-970
|
3 |
Caldero P, Chapoton F. Cluster algebras as Hall algebras of quiver representations. Comm Math Helv, 2006, 81: 595-616
|
4 |
Caldero P, Keller B. From triangulated categories to cluster algebras. Invent Math, 2008, 172(1): 169-211
|
5 |
Ding M, Xiao J, Xu F. Integral bases of cluster algebras and representations of tame quivers. Algebr Represent Theor (to appear). ArXiv: 0901.1937 [math.RT]
|
6 |
Fomin S, Zelevinsky A. Cluster algebras. I. Foundations. J Amer Math Soc, 2002, 15(2): 497-529
|
7 |
Keller B. On triangulated orbit categories. Documenta Math, 2005, 10: 551-581
|
8 |
Marsh R, Reineke M, Zelevinsky A. Generalized associahedra via quiver representations. Trans Amer Math Soc, 2003, 355(1): 4171-4186
|
9 |
Palu Y. Cluster characters for 2-Calabi–Yau triangulated categories. Ann Inst Fourier, 2008, 58(6): 2221-2248
|
10 |
Ringel C M. Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics, Vol 1099. Berlin: Springer, 1984
|
11 |
Xiao J, Xu F. Green’s formula with
|
12 |
Zhou Y, Zhu B. Cluster algebras of type C via cluster tubes. ArXiv: 1008.3444v1 [math.RT]
|
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