Abstract
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra \(\mathcal{A}\) of finite type can be realized as a Hall algebra, called exceptional Hall algebra, of the cluster category. This realization provides a natural basis for \(\mathcal{A}\). We prove new results and formulate conjectures on ‘good basis’ properties, positivity, denominator theorems and toric degenerations.
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Berenstein, A., Fomin, S., Zelevinsky, A.: Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. 126(1), 1–52 (2005)
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)
Buan, A.B., Marsh, R.J., Reiten, I.: Cluster tilted algebras. Trans. Am. Math. Soc. 359(1), 323–332 (2007)
Buan, A.B., Marsh, R.J., Reiten, I.: Cluster mutation via quiver representations. Comment. Math. Helv. (to appear). math.RT/0412077
Buan, A.B., Marsh, R.J., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)
Caldero, P.: Toric degenerations of Schubert varieties. Transform. Groups 7(1), 51–60 (2002)
Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81(3), 595–616 (2006)
Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (An case). Trans. Am. Math. Soc. 358(3), 1347–1364 (2006)
Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations and cluster tilted algebras. Algebr. Represent. Theory 9(4), 359–376 (2006)
Caldero, P., Schiffler, R.: Rational smoothness of varieties of representations for quivers of Dynkin type. Ann. Inst. Fourier 54(2), 295–315 (2004)
Fock, V.V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. math.AG/0311245
Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002)
Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003)
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)
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)
Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. Mosc. Math. J. 3(3), 899–934, 1199 (2003)
Green, J.A.: Hall algebras, hereditary algebras and quantum groups. Invent. Math. 120(2), 361–377 (1995)
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)
Hubery, A.: Hall Polynomials for Affine Quivers. arXiv:math/0703178
Kac, V.G.: Infinite root systems, representations of graphs and invariant theory II. J. Algebra 78, 141–162 (1982)
Keller, B.: Triangulated orbit categories. Doc. Math. 10, 551–581 (2005)
Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3(2), 447–498 (1990)
Lusztig, G.: Introduction to Quantum Groups. Prog. Math., vol. 110. Birkhäuser, Boston (1993)
Marsh, R., Reineke, M., Zelevinsky, A.: Generalized associahedra via quiver representations. Trans. Am. Math. Soc. 355(10), 4171–4186 (2003)
Peng, L., Xiao, J.: Triangulated categories and Kac–Moody algebras. Invent. Math. 140(3), 563–603 (2000)
Reineke, M.: Counting rational points of quiver moduli. Int. Math. Res. Not. 2006, Art.ID 70456, 19 pp.
Riedtmann, C.: Lie algebras generated by indecomposables. J. Algebra 170, 526–546 (1994)
Ringel, C.M.: Hall polynomials for the representation-finite hereditary algebras. Adv. Math. 84, 137–178 (1990)
Ringel, C.M.: Hall algebras. Banach Cent. Publ. 26, 433–447 (1990)
Scott, J.: Grassmannians and Cluster Algebras. Ph.D. thesis. Northeastern University (2003)
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)
Toën, B., Vaquié, M.: Moduli of objects in dg-categories. Ann. Sci. Éc. Norm. Supér., IV. Sér. 40(2) (2007)
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)
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Caldero, P., Keller, B. From triangulated categories to cluster algebras. Invent. math. 172, 169–211 (2008). https://doi.org/10.1007/s00222-008-0111-4
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DOI: https://doi.org/10.1007/s00222-008-0111-4