From triangulated categories to cluster algebras. (English) Zbl 1141.18012

Let \(Q\) be a simply laced Dynkin quiver and let \(k\) be a field. In the paper under review the authors give a realisation of the cluster algebra as a Hall algebra coming from a triangulated category. This gives at once also a geometrical interpretation for the multiplication of two cluster variables.
The underlying geometrical object is the Grassmannian of a module for a quiver algebra \(kQ\). For this purpose it is proved that there is a unique way to associate to any object of the cluster category of \(Q\) a cluster variable satisfying certain naturality properties, as described by P. Caldero and F. Chapoton [Comment. Math. Helv. 81, No. 3, 595–616 (2006; Zbl 1119.16013)]. The product of the cluster variables \(X_M\) and \(X_N\) of two objects \(M\) and \(N\) of the cluster category multiplied by the Euler characteristic of the projectivisation of \(\text{Ext}^1(M,N)\) equals to the sum over all objects \(Y\) of the cluster category which occur as middle term of a triangle starting at \(M\) and ending at \(N\), or vice versa, of the cluster variable corresponding to \(Y\) multiplied with the Euler characteristic of the projectivisation of the subset of the \(\text{Ext}^1(N,M)\) having \(Y\) as middle term, plus the corresponding coefficient of \(\text{Ext}^1(M,N)\). The proof of this most astonishing result is rather complicate and involved and uses a variety of methods.
The statement has various corollaries. First, it realises the cluster algebra as a Hall algebra. Then, it settles the positivity conjecture. A third nice application is given in terms of degenerations of objects. The paper ends with four conjectures.


18E30 Derived categories, triangulated categories (MSC2010)
16G20 Representations of quivers and partially ordered sets
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
17B35 Universal enveloping (super)algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers


Zbl 1119.16013
Full Text: DOI arXiv


[1] Berenstein, A., Fomin, S., Zelevinsky, A.: Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. 126(1), 1–52 (2005) · Zbl 1135.16013 · doi:10.1215/S0012-7094-04-12611-9
[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) · Zbl 0351.16011 · doi:10.1112/jlms/s2-14.1.183
[3] Buan, A.B., Marsh, R.J., Reiten, I.: Cluster tilted algebras. Trans. Am. Math. Soc. 359(1), 323–332 (2007) · Zbl 1123.16009 · doi:10.1090/S0002-9947-06-03879-7
[4] Buan, A.B., Marsh, R.J., Reiten, I.: Cluster mutation via quiver representations. Comment. Math. Helv. (to appear). math.RT/0412077 · Zbl 1193.16016
[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) · Zbl 1127.16011 · doi:10.1016/j.aim.2005.06.003
[6] Caldero, P.: Toric degenerations of Schubert varieties. Transform. Groups 7(1), 51–60 (2002) · Zbl 1050.14040 · doi:10.1007/BF01253464
[7] Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81(3), 595–616 (2006) · Zbl 1119.16013 · doi:10.4171/CMH/65
[8] Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (An case). Trans. Am. Math. Soc. 358(3), 1347–1364 (2006) · Zbl 1137.16020 · doi:10.1090/S0002-9947-05-03753-0
[9] Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations and cluster tilted algebras. Algebr. Represent. Theory 9(4), 359–376 (2006) · Zbl 1127.16013 · doi:10.1007/s10468-006-9018-1
[10] Caldero, P., Schiffler, R.: Rational smoothness of varieties of representations for quivers of Dynkin type. Ann. Inst. Fourier 54(2), 295–315 (2004) · Zbl 1126.17013
[11] Fock, V.V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. math.AG/0311245 · Zbl 1225.53070
[12] Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002) · Zbl 1021.16017 · doi:10.1090/S0894-0347-01-00385-X
[13] Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003) · Zbl 1054.17024 · doi:10.1007/s00222-003-0302-y
[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) · Zbl 1167.16009 · doi:10.1007/s00222-006-0507-y
[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) · Zbl 1131.17006
[17] Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. Mosc. Math. J. 3(3), 899–934, 1199 (2003) · Zbl 1057.53064
[18] Green, J.A.: Hall algebras, hereditary algebras and quantum groups. Invent. Math. 120(2), 361–377 (1995) · Zbl 0836.16021 · doi:10.1007/BF01241133
[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) · Zbl 0635.16017
[20] Hubery, A.: Hall Polynomials for Affine Quivers. arXiv:math/0703178 · Zbl 1241.16011
[21] Kac, V.G.: Infinite root systems, representations of graphs and invariant theory II. J. Algebra 78, 141–162 (1982) · Zbl 0497.17007 · doi:10.1016/0021-8693(82)90105-3
[22] Keller, B.: Triangulated orbit categories. Doc. Math. 10, 551–581 (2005) · Zbl 1086.18006
[23] Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3(2), 447–498 (1990) · Zbl 0703.17008 · doi:10.1090/S0894-0347-1990-1035415-6
[24] Lusztig, G.: Introduction to Quantum Groups. Prog. Math., vol. 110. Birkhäuser, Boston (1993) · Zbl 0788.17010
[25] Marsh, R., Reineke, M., Zelevinsky, A.: Generalized associahedra via quiver representations. Trans. Am. Math. Soc. 355(10), 4171–4186 (2003) · Zbl 1042.52007 · doi:10.1090/S0002-9947-03-03320-8
[26] Peng, L., Xiao, J.: Triangulated categories and Kac–Moody algebras. Invent. Math. 140(3), 563–603 (2000) · Zbl 0966.16006 · doi:10.1007/s002220000062
[27] Reineke, M.: Counting rational points of quiver moduli. Int. Math. Res. Not. 2006, Art.ID 70456, 19 pp. · Zbl 1113.14018
[28] Riedtmann, C.: Lie algebras generated by indecomposables. J. Algebra 170, 526–546 (1994) · Zbl 0841.16018 · doi:10.1006/jabr.1994.1351
[29] Ringel, C.M.: Hall polynomials for the representation-finite hereditary algebras. Adv. Math. 84, 137–178 (1990) · Zbl 0799.16013 · doi:10.1016/0001-8708(90)90043-M
[30] Ringel, C.M.: Hall algebras. Banach Cent. Publ. 26, 433–447 (1990) · Zbl 0778.16004
[31] Scott, J.: Grassmannians and Cluster Algebras. Ph.D. thesis. Northeastern University (2003) · Zbl 1088.22009
[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) · Zbl 1117.18011 · doi:10.1215/S0012-7094-06-13536-6
[34] Toën, B., Vaquié, M.: Moduli of objects in dg-categories. Ann. Sci. Éc. Norm. Supér., IV. Sér. 40(2) (2007) · Zbl 1140.18005
[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) · Zbl 1110.16013 · doi:10.1016/j.jalgebra.2005.05.011
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