Frontiers of Mathematics in China >

2012 , Vol. 7 >Issue 4: 679 - 693

DOI: https://doi.org/10.1007/s11464-011-0146-5

RESEARCH ARTICLE

Cluster characters for cyclic quivers

  • Ming DING 1 ,
  • Fan XU , 2
Expand
  • 1. School of Mathematical Sciences, Nankai University, Tianjin 300071, China
  • 2. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received date: 10 May 2010

Accepted date: 29 May 2011

Published date: 01 Aug 2012

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
Fold

Abstract

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 ℤ-basis for the algebra generated by all generalized cluster variables.

Key words： Cyclic quiver; cluster algebra; ℤ-basis

Cite this article

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

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
Buan A, Marsh R, Reineke M, Reiten I, Todorov G. Tilting theory and cluster combinatorics. Adv Math, 2006, 204: 572-618

DOI

2
Buan A, Marsh R, Vatne D. Cluster structure from 2-Calabi-Yau categories with loops. Math Z, 2010, 265(4): 951-970

DOI

3
Caldero P, Chapoton F. Cluster algebras as Hall algebras of quiver representations. Comm Math Helv, 2006, 81: 595-616

DOI

4
Caldero P, Keller B. From triangulated categories to cluster algebras. Invent Math, 2008, 172(1): 169-211

DOI

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]

DOI

6
Fomin S, Zelevinsky A. Cluster algebras. I. Foundations. J Amer Math Soc, 2002, 15(2): 497-529

DOI

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

DOI

9
Palu Y. Cluster characters for 2-Calabi–Yau triangulated categories. Ann Inst Fourier, 2008, 58(6): 2221-2248

DOI

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 ℂ*-action and Caldero-Keller’s formula. Prog Math (to appear)

12
Zhou Y, Zhu B. Cluster algebras of type C via cluster tubes. ArXiv: 1008.3444v1 [math.RT]

Options
Outlines

/

〈 〉