Cluster algebras. II: Finite type classification. (English) Zbl 1054.17024

For part I see the authors’ paper in J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017).
The authors introduced cluster algebras with a motivation to understand the structure of “dual canonical bases” for the homogeneous coordinate rings of algebraic varieties related to semi-simple algebraic groups. The cluster algebra structure describes these rings inside an ambient field by a distinguished family of generators called “cluster variables”. A cluster algebra is said to be of finite type if it has a finite number of cluster variables. In this paper, the authors provide a complete classification of cluster algebras of finite type.


17B20 Simple, semisimple, reductive (super)algebras
05E15 Combinatorial aspects of groups and algebras (MSC2010)
16S99 Associative rings and algebras arising under various constructions
52B12 Special polytopes (linear programming, centrally symmetric, etc.)


Zbl 1021.16017
Full Text: DOI arXiv


[1] Adler, I.: Abstract polytopes, Ph.D. thesis, Department of Operations Research, Stanford University, 1971
[2] Aschbacher, M.: Finite groups acting on homology manifolds. Pac. J. Math. 181, Special Issue, 3–36 (1997) · Zbl 0912.57021
[3] Barvinok, A.: A course in convexity. Am. Math. Soc. 2002 · Zbl 1014.52001
[4] Bolker, E., Guillemin, V., Holm, T.: How is a graph like a manifold? math.CO/0206103
[5] Bott, R., Taubes, C.: On the self-linking of knots. Topology and physics. J. Math. Phys. 35, 5247–5287 (1994) · Zbl 0863.57004
[6] Bourbaki, N.: Groupes et algèbres de Lie. Ch. IV–VI. Paris: Hermann 1968 · Zbl 0186.33001
[7] Chapoton, F., Fomin, S., Zelevinsky, A.: Polytopal realizations of generalized associahedra, Can. Math. Bull. 45, 537–566 (2002) · Zbl 1018.52007
[8] Fomin, S., Zelevinsky, A.: Double Bruhat cells and total positivity. J. Am. Math. Soc. 12, 335–380 (1999) · Zbl 0913.22011
[9] Fomin, S., Zelevinsky, A.: Cluster algebras I: Foundations. J. Am. Math. Soc. 15, 497–529 (2002) · Zbl 1021.16017
[10] Fomin, S., Zelevinsky, A.: The Laurent phenomenon. Adv. Appl. Math. 28, 119–144 (2002) · Zbl 1012.05012
[11] Fomin, S., Zelevinsky, A.: Y-systems and generalized associahedra. To appear in Ann. Math. (2) · Zbl 1057.52003
[12] Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. math.QA/0208033 · Zbl 1217.13001
[13] Gelfand, I., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants and Multidimensional Determinants, 523 pp. Boston: Birkhäuser 1994 · Zbl 0827.14036
[14] Guillemin, V., Zara, C.: Equivariant de Rham theory and graphs. Asian J. Math. 3, 49–76 (1999) · Zbl 0971.58001
[15] Kac, V.: Infinite dimensional Lie algebras, 3rd edition. Cambridge: Cambridge University Press 1990 · Zbl 0716.17022
[16] Kung, J., Rota, G.-C.: The invariant theory of binary forms. Bull. Am. Math. Soc., New. Ser. 10, 27–85 (1984) · Zbl 0577.15020
[17] Lee, C.W.: The associahedron and triangulations of the n-gon. Eur. J. Comb. 10, 551–560 (1989) · Zbl 0682.52004
[18] Lusztig, G.: Total positivity in reductive groups. In: Lie theory and geometry: in honor of Bertram Kostant. Progress in Mathematics 123. Birkhäuser 1994 · Zbl 0845.20034
[19] Lusztig, G.: Introduction to total positivity. In: Positivity in Lie theory: open problems. de Gruyter Exp. Math. 26, 133–145. Berlin: de Gruyter 1998 · Zbl 0929.20035
[20] Markl, M.: Simplex, associahedron, and cyclohedron. Contemp. Math. 227, 235–265 (1999) · Zbl 0919.18003
[21] Marsh, R., Reineke, M., Zelevinsky, A.: Generalized associahedra via quiver representations. To appear in Trans. Am. Math. Soc. · Zbl 1042.52007
[22] Massey, W.S.: Algebraic topology: an introduction. Springer-Verlag 1977 · Zbl 0361.55002
[23] McMullen, P., Schulte, E.: Abstract Regular Polytopes. Cambridge: Cambridge University Press 2003 · Zbl 1039.52011
[24] Simion, R.: A type-B associahedron. Adv. Appl. Math. 30, 2–25 (2003) · Zbl 1047.52006
[25] Stasheff, J.D.: Homotopy associativity of H-spaces. I, II. Trans. Am. Math. Soc. 108, 275–292, 293–312 (1963) · Zbl 0114.39402
[26] Sturmfels, B.: Algorithms in invariant theory. Springer-Verlag 1993 · Zbl 0802.13002
[27] Zelevinsky, A.: From Littlewood-Richardson coefficients to cluster algebras in three lectures. In: Symmetric Functions 2001: Surveys of Developments and Perspectives, S. Fomin (Ed.), NATO Science Series II, vol. 74. Kluwer Academic Publishers 2002 · Zbl 1155.17303
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