From triangulated categories to cluster algebras

Inventiones Mathematicae

Volumn 172, Issue 1, 2008, Pages 169-211

  Caldero, Philippe ^a   Keller, Bernhard ^b  

^a UNIVERSITÉ LYON 1   (France)

^b UNIVERSITÉ PARIS DESCARTES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 41249083675     PISSN: 00209910     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00222-008-0111-4     Document Type: Article

References (36)

1. Berenstein, A., Fomin, S., Zelevinsky, A.: Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. 126(1), 1-52 (2005)
2. Brenner, S., Butler, M.C.R.: The equivalence of certain functors occurring in the representation theory of Artin algebras and species. J. Lond. Math. Soc., II. Ser. 14(1), 183-187 (1976)
3. Buan, A.B., Marsh, R.J., Reiten, I.: Cluster tilted algebras. Trans. Am. Math. Soc. 359(1), 323-332 (2007)
4. Buan, A.B., Marsh, R.J., Reiten, I.: Cluster mutation via quiver representations. Comment. Math. Helv. (to appear). math.RT/0412077
5. Buan, A.B., Marsh, R.J., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572-618 (2006)
6. Caldero, P.: Toric degenerations of Schubert varieties. Transform. Groups 7(1), 51-60 (2002)
7. Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81(3), 595-616 (2006)
8. n case). Trans. Am. Math. Soc. 358(3), 1347-1364 (2006)
9. Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations and cluster tilted algebras. Algebr. Represent. Theory 9(4), 359-376 (2006)
10. Caldero, P., Schiffler, R.: Rational smoothness of varieties of representations for quivers of Dynkin type. Ann. Inst. Fourier 54(2), 295-315 (2004)
11. Fock, V.V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. math.AG/0311245
12. Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497-529 (2002)
13. Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63-121 (2003)
14. Gabriel, P.: Auslander-Reiten sequences and representation-finite algebras. Representation Theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979). Lect. Notes Math., vol. 831. Springer (1980)
15. Geiss, C., Leclerc, B., Schröer, J.: Rigid modules over preprojective algebras. Invent. Math. 165(3), 589-632 (2006)
16. Geiss, C., Leclerc, B., Schröer, J.: Semicanonical bases and preprojective algebras II: A multiplication formula. Ann. Sci. Éc. Norm. Supér, IV Sér. 38(2), 193-253 (2005)
17. Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. Mosc. Math. J. 3(3), 899-934, 1199 (2003)
18. Green, J.A.: Hall algebras, hereditary algebras and quantum groups. Invent. Math. 120(2), 361-377 (1995)
19. Happel, D.: Triangulated categories in the representation theory of finite-dimensional algebras. Lond. Math. Soc. Lect. Note Ser., vol. 119. Cambridge University Press, Cambridge (1988)
20. Hubery, A.: Hall Polynomials for Affine Quivers. arXiv:math/0703178
21. Kac, V.G.: Infinite root systems, representations of graphs and invariant theory II. J. Algebra 78, 141-162 (1982)
22. Keller, B.: Triangulated orbit categories. Doc. Math. 10, 551-581 (2005)
23. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3(2), 447-498 (1990)
24. Lusztig, G.: Introduction to Quantum Groups. Prog. Math., vol. 110. Birkhäuser, Boston (1993)
25. Marsh, R., Reineke, M., Zelevinsky, A.: Generalized associahedra via quiver representations. Trans. Am. Math. Soc. 355(10), 4171-4186 (2003)
26. Peng, L., Xiao, J.: Triangulated categories and Kac-Moody algebras. Invent. Math. 140(3), 563-603 (2000)
27. Reineke, M.: Counting rational points of quiver moduli. Int. Math. Res. Not. 2006, Art.ID 70456, 19 pp.
28. Riedtmann, C.: Lie algebras generated by indecomposables. J. Algebra 170, 526-546 (1994)
29. Ringel, C.M.: Hall polynomials for the representation-finite hereditary algebras. Adv. Math. 84, 137-178 (1990)
30. Ringel, C.M.: Hall algebras. Banach Cent. Publ. 26, 433-447 (1990)
31. Scott, J.: Grassmannians and Cluster Algebras. Ph.D. thesis. Northeastern University (2003)
32. Serre, J.P.: Espaces fibrés algébriques. In: Séminaire C. Chevalley, pp. 1-37 (1958)
33. Toën, B.: Derived Hall algebras. Duke Math. J. 135(3), 587-615 (2006)
34. Toën, B., Vaquié, M.: Moduli of objects in dg-categories. Ann. Sci. Éc. Norm. Supér., IV. Sér. 40(2) (2007)
35. Verdier, J.-L.: Catégories dérivées, état 0. In: SGA 4.5, 1977. Lect. Notes, vol. 569, pp. 262-308. Springer (1977)
36. Xiao, J., Zhu, B.: Locally finite triangulated categories. J. Algebra 290, 473-490 (2005)