Cluster algebras as Hall algebras of quiver representations

Commentarii Mathematici Helvetici

Volumn 81, Issue 3, 2006, Pages 595-616

  Caldero, Philippe ^a   Chapoton, Frédéric ^a  

^a UNIVERSITÉ LYON 1   (France)

Author keywords

Cluster algebra; Grassmannian of submodules; Quiver representation

Indexed keywords

EID: 33750855116     PISSN: 00102571     EISSN: 14208946     Source Type: Journal    
DOI: 10.4171/CMH/65     Document Type: Article

References (17)

1. [ARS95] M. Auslander, I. Reiten, and S. O. Smalø, Representation theory of Artin algebras. Cambridge Stud. Adv. Math. 36, Cambridge University Press, Cambridge 1995. Zbl 0834.16001 MR 1314422
2. [BFZ05] A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. 126 (2005), 1-52. Zbl 02147024 MR 2110627
3. +] A. Buan, R. J. Marsh, M. Reineke, I. Reiten, and G. Todorov, Tilting theory and cluster combinatorics. arXiv:math.RT/0402054, 2004.
4. [CC73] J. H. Conway and H. S. M. Coxeter, Triangulated polygons and frieze patterns. Math. Gaz. 57 (1973), 87-94. Zbl 0285.05028 MR 0461269
5. n case). Trans. Amer. Math. Soc. 358 (2006), 1347-1364. Zbl 02237892
6. [CK] P. Caldero and B. Keller, From triangulated categories to cluster algebras. Preprint, math.RT/0506018, 2005.
7. [DG80] M. Demazure and P. Gabriel, Introduction to algebraic geometry and algebraic groups. Translated from the French by J. Bell, North-Holland Math. Stud. 39, North-Holland Publishing Co., Amsterdam 1980. Zbl 0431.14015 MR 0563524
8. [FG03] V. V. Fock and A. B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory. Preprint, math.AG/0311149, 2003.
9. [FZ02] S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15 (2002), 497-529 (electronic). Zbl 1021.16017 MR 1887642
10. [FZ03a] S. Fomin and A. Zelevinsky, Cluster algebras. II. Finite type classification. Invent. Math. 154 (2003), 63-121. Zbl 1054.17024 MR 2004457
11. [FZ03b] S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra. Ann. of Math. (2) 158 (2003), 977-1018. Zbl 1057.52003 MR 2031858
12. [Gab80] P. Gabriel, Auslander-Reiten sequences and representation-finite algebras. In Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math. 831, Springer-Verlag, Berlin 1980, 1-71. Zbl 0445.16023 MR 0607140
13. [Kel] B. Keller, On triangulated orbit categories. Doc. Math., J. DMV 10 (2005), 551-581. Zbl 02238184
14. [MRZ03] R. Marsh, M. Reineke, and A. Zelevinsky, Generalized associahedra via quiver representations. Trans. Amer. Math. Soc. 355 (2003), 4171-4186. Zbl 1042.52007 MR 1990581
15. [Rei] M. Reineke, Counting rational points of quiver moduli. Preprint, math. AG/0505389, 2005.
16. [Rin90] C. M. Ringel, Hall algebras and quantum groups. Invent. Math. 101 (1990), 583-591. Zbl 0735.16009 MR 1062796
17. [Sco06] J. S. Scott, Grassmannians and Cluster Algebras. Proc. London Math. Soc. 92 (2006), 345-380. Zbl 05014382 MR 2205721