# zbMATH — the first resource for mathematics

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.

##### MSC:
 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:
##### References:
  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  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  Buan, A.B., Marsh, R.J., Reiten, I.: Cluster tilted algebras. Trans. Am. Math. Soc. 359(1), 323–332 (2007) · Zbl 1123.16009  Buan, A.B., Marsh, R.J., Reiten, I.: Cluster mutation via quiver representations. Comment. Math. Helv. (to appear). math.RT/0412077 · Zbl 1193.16016  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  Caldero, P.: Toric degenerations of Schubert varieties. Transform. Groups 7(1), 51–60 (2002) · Zbl 1050.14040  Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81(3), 595–616 (2006) · Zbl 1119.16013  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  Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations and cluster tilted algebras. Algebr. Represent. Theory 9(4), 359–376 (2006) · Zbl 1127.16013  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  Fock, V.V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. math.AG/0311245 · Zbl 1225.53070  Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002) · Zbl 1021.16017  Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003) · Zbl 1054.17024  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)  Geiss, C., Leclerc, B., Schröer, J.: Rigid modules over preprojective algebras. Invent. Math. 165(3), 589–632 (2006) · Zbl 1167.16009  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  Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. Mosc. Math. J. 3(3), 899–934, 1199 (2003) · Zbl 1057.53064  Green, J.A.: Hall algebras, hereditary algebras and quantum groups. Invent. Math. 120(2), 361–377 (1995) · Zbl 0836.16021  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  Hubery, A.: Hall Polynomials for Affine Quivers. arXiv:math/0703178 · Zbl 1241.16011  Kac, V.G.: Infinite root systems, representations of graphs and invariant theory II. J. Algebra 78, 141–162 (1982) · Zbl 0497.17007  Keller, B.: Triangulated orbit categories. Doc. Math. 10, 551–581 (2005) · Zbl 1086.18006  Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3(2), 447–498 (1990) · Zbl 0703.17008  Lusztig, G.: Introduction to Quantum Groups. Prog. Math., vol. 110. Birkhäuser, Boston (1993) · Zbl 0788.17010  Marsh, R., Reineke, M., Zelevinsky, A.: Generalized associahedra via quiver representations. Trans. Am. Math. Soc. 355(10), 4171–4186 (2003) · Zbl 1042.52007  Peng, L., Xiao, J.: Triangulated categories and Kac–Moody algebras. Invent. Math. 140(3), 563–603 (2000) · Zbl 0966.16006  Reineke, M.: Counting rational points of quiver moduli. Int. Math. Res. Not. 2006, Art.ID 70456, 19 pp. · Zbl 1113.14018  Riedtmann, C.: Lie algebras generated by indecomposables. J. Algebra 170, 526–546 (1994) · Zbl 0841.16018  Ringel, C.M.: Hall polynomials for the representation-finite hereditary algebras. Adv. Math. 84, 137–178 (1990) · Zbl 0799.16013  Ringel, C.M.: Hall algebras. Banach Cent. Publ. 26, 433–447 (1990) · Zbl 0778.16004  Scott, J.: Grassmannians and Cluster Algebras. Ph.D. thesis. Northeastern University (2003) · Zbl 1088.22009  Serre, J.P.: Espaces fibrés algébriques. In: Séminaire C. Chevalley, pp. 1–37 (1958)  Toën, B.: Derived Hall algebras. Duke Math. J. 135(3), 587–615 (2006) · Zbl 1117.18011  Toën, B., Vaquié, M.: Moduli of objects in dg-categories. Ann. Sci. Éc. Norm. Supér., IV. Sér. 40(2) (2007) · Zbl 1140.18005  Verdier, J.-L.: Catégories dérivées, état 0. In: SGA 4.5, 1977. Lect. Notes, vol. 569, pp. 262–308. Springer (1977)  Xiao, J., Zhu, B.: Locally finite triangulated categories. J. Algebra 290, 473–490 (2005) · Zbl 1110.16013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.