Introduction to Cluster Algebras

Part of the CRM Series in Mathematical Physics book series (CRM)


These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians \(\mathit{Gr}_{2}(\mathbb{C}^{n})\), and the appearance of double Bruhat cells in the theory of total positivity. Once the definition of cluster algebras is introduced in several stages of increasing generality, proofs of fundamental results are sketched in the rank 2 case. From these foundations we build up the notion of Poisson structures compatible with a cluster algebra structure and indicate how this leads to a quantization of cluster algebras. Finally we give applications of these ideas to integrable systems in the form of Zamolodchikov periodicity and the pentagram map.



These notes are based on lectures we presented during the 2016 ASIDE summer school in Montreal. We thank the ASIDE organizers for inviting us and also for coordinating the efforts to compile the lecture notes. We thank Sophie Morier-Genoud and the anonymous referee for several helpful comments.


  1. 1.
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov, J. Trnka, Scattering amplitudes and the positive Grassmannian. arXiv:1212.5605Google Scholar
  2. 2.
    A. Berenstein, S. Fomin, A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. 126(1), 1–52 (2005)MATHGoogle Scholar
  3. 3.
    A. Berenstein, D. Rupel, Quantum cluster characters of Hall algebras. Sel. Math. N. Ser. 21(4), 1121–1176 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    A. Berenstein, A. Zelevinsky, Quantum cluster algebras. Adv. Math. 195(2), 405–455 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    A.B. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    P. Caldero, F. Chapoton, Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81(3), 595–616 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    P. Caldero, B. Keller, From triangulated categories to cluster algebras. II. Ann. Sci. École Norm. Sup. (4) 39(6), 983–1009 (2006)Google Scholar
  8. 8.
    P. Di Francesco, R. Kedem, Positivity of the T-system cluster algebra. Electron. J. Comb. 16(1), 140 (2009)Google Scholar
  9. 9.
    P. Di Francesco, R. Kedem, Q-systems, heaps, paths and cluster positivity. Commun. Math. Phys. 293(3), 727–802 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    R. Eager, S. Franco, Colored BPS pyramid partition functions, quivers and cluster transformations. J. High Energy Phys. 2012(9), 038 (2012)Google Scholar
  11. 11.
    V. Fock, A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. 103, 1–211 (2006)CrossRefMATHGoogle Scholar
  12. 12.
    S. Fomin, M. Shapiro, D. Thurston, Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math. 201(1), 83–146 (2008)MathSciNetMATHGoogle Scholar
  13. 13.
    S. Fomin, A. Zelevinsky, Double Bruhat cells and total positivity. J. Am. Math. Soc. 12(2), 335–380 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    S. Fomin, A. Zelevinsky, Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (electronic) (2002)Google Scholar
  15. 15.
    S. Fomin, A. Zelevinsky, The Laurent phenomenon. Adv. Appl. Math. 28(2), 119–144 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    S. Fomin, A. Zelevinsky, Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003)MATHGoogle Scholar
  17. 17.
    S. Fomin, A. Zelevinsky, Cluster algebras. IV. Coefficients. Compos. Math. 143(1), 112–164 (2007)CrossRefMATHGoogle Scholar
  18. 18.
    A.P. Fordy, R.J. Marsh, Cluster mutation-periodic quivers and associated Laurent sequences. J. Algebraic Comb. 34(1), 19–66 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    C. Geiß, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras. Invent. Math. 165(3), 589–632 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    C. Geiß, B. Leclerc, J. Schröer, Cluster algebras in algebraic Lie theory. Transform. Groups 18(1), 149–178 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    C. Geiß, B. Leclerc, J. Schröer, Cluster structures on quantum coordinate rings. Sel. Math. N. Ser. 19(2), 337–397 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    M. Gekhtman, T. Nakanishi, D. Rupel, Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities. arXiv:16118.02813Google Scholar
  23. 23.
    M. Gekhtman, M. Shapiro, S. Tabachnikov, A. Vainshtein, Higher pentagram maps, weighted directed networks, and cluster dynamics. Electron. Res. Announc. Math. Sci. 19, 1–17 (2012)MathSciNetMATHGoogle Scholar
  24. 24.
    M. Gekhtman, M. Shapiro, A. Vainshtein, Cluster algebras and Poisson geometry. Mosc. Math. J. 3(3), 899–934 (2003)MathSciNetMATHGoogle Scholar
  25. 25.
    M. Gekhtman, M. Shapiro, A. Vainshtein, Cluster Algebras and Poisson Geometry. Mathematical Surveys and Monographs, vol. 167 (American Mathematical Society, Providence, 2010)Google Scholar
  26. 26.
    M. Glick, The pentagram map and Y -patterns. Adv. Math. 227(2), 1019–1045 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    A.B. Goncharov, R. Kenyon, Dimers and cluster integrable systems. Ann. Sci. Éc. Norm. Supér. (4) 46(5), 747–813 (2013)Google Scholar
  28. 28.
    M. Gross, P. Hacking, S. Keel, M. Kontsevich, Canonical bases for cluster algebras. arXiv:1411.1394Google Scholar
  29. 29.
    A. Hatcher, On triangulations of surfaces. Topol. Appl. 40(2), 189–194 (1991)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    A.N.W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences. Trans. Am. Math. Soc. 359(10), 5019–5034 (2007)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    B. Keller, Categorification of acyclic cluster algebras: an introduction. arXiv:0801.3103Google Scholar
  32. 32.
    Y. Kimura, Quantum unipotent subgroup and dual canonical basis. Kyoto J. Math. 52(2), 277–331 (2012)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Y. Kimura, F. Qin, Graded quiver varieties, quantum cluster algebras and dual canonical basis. Adv. Math. 262, 261–312 (2014)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    M. Kogan, A. Zelevinsky, On symplectic leaves and integrable systems in standard complex semisimple Poisson–Lie groups. Int. Math. Res. Not. 2002(32), 1685–1702 (2002)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    K. Lee, L. Li, A. Zelevinsky, Greedy elements in rank 2 cluster algebras. Sel. Math. N. Ser. 20(1), 57–82 (2014)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    K. Lee, R. Schiffler, A combinatorial formula for rank 2 cluster variables. J. Algebraic Comb. 37(1), 67–85 (2013)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    K. Lee, R. Schiffler, Positivity for cluster algebras. Ann. Math. (2) 182(1), 73–125 (2015)Google Scholar
  38. 38.
    G. Musiker, J. Propp, Combinatorial interpretations for rank-two cluster algebras of affine type. Electron. J. Comb. 14(1), 15 (2007)Google Scholar
  39. 39.
    G. Musiker, R. Schiffler, L. Williams, Bases for cluster algebras from surfaces. Compos. Math. 149(2), 217–263 (2013)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    T. Nakanishi, Periodicities in cluster algebras and dilogarithm identities. in Representations of Algebras and Related Topics, ed. by A. Skowroński, K. Yamagata, EMS Series of Congress Reports (European Mathematical Society, Zürich, 2011), pp. 407–443CrossRefGoogle Scholar
  41. 41.
    V. Ovsienko, R. Schwartz, S. Tabachnikov, The pentagram map: a discrete integrable system. Commun. Math. Phys. 299(2), 409–446 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    V. Ovsienko, R.E. Schwartz, S. Tabachnikov, Liouville–Arnold integrability of the pentagram map on closed polygons. Duke Math. J. 162(12), 2149–2196 (2013)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    P. Pylyavskyy, Zamolodchikov integrability via rings of invariants. arXiv:1506.05378Google Scholar
  44. 44.
    F. Qin, Quantum cluster variables via Serre polynomials. J. Reine Angew. Math. 668, 149–190 (2012)MathSciNetMATHGoogle Scholar
  45. 45.
    D. Rupel, On a quantum analog of the Caldero–Chapoton formula. Int. Math. Res. Not. 2011(14), 3207–3236 (2011)MathSciNetMATHGoogle Scholar
  46. 46.
    D. Rupel, Quantum cluster characters for valued quivers. Trans. Am. Math. Soc. 367(10), 7061–7102 (2015)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    R. Schwartz, The pentagram map. Exp. Math. 1(1), 71–81 (1992)MathSciNetMATHGoogle Scholar
  48. 48.
    R.E. Schwartz, Discrete monodromy, pentagrams, and the method of condensation. J. Fixed Point Theory Appl. 3(2), 379–409 (2008)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    F. Soloviev, Integrability of the pentagram map. Duke Math. J. 162(15), 2815–2853 (2013)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    A.Y. Volkov, On the periodicity conjecture for Y -systems. Commun. Math. Phys. 276(2), 509–517 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    L. Williams, Cluster algebras: an introduction. arXiv:1212.6263Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA
  2. 2.Department of MathematicsUniversity of Notre DameNotre DameUSA

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