Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1495–1523 | Cite as

A 2-Calabi–Yau realization of finite-type cluster algebras with universal coefficients

  • Alfredo Nájera ChávezEmail author


We categorify various finite-type cluster algebras with coefficients using completed orbit categories associated to Frobenius categories. Namely, the Frobenius categories we consider are the categories of finitely generated Gorenstein projective modules over the singular Nakajima category associated to a Dynkin diagram and their standard Frobenius quotients. In particular, we are able to categorify all finite-type skew-symmetric cluster algebras with universal coefficients and finite-type Grassmannian cluster algebras. Along the way, we classify the standard Frobenius models of a certain family of triangulated orbit categories which include all finite-type n-cluster categories, for all integers \(n\ge 1\).



  1. 1.
    Amiot, C.: Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier 59(6), 2525–2590 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Graduate Texts in Mathematics, vol. 13, 2nd edn. Springer, New York (1974)CrossRefzbMATHGoogle Scholar
  3. 3.
    Asashiba, H.: A generalization of Gabriel’s Galois covering functors and derived equivalences. J. Algebra 334(1), 109–149 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bernstein, I.N., Gelfand, I.M., Ponomarev, V.A.: Coxeter functors and Gabriel’s theorem. Uspechi mat. Nauk. 28, 19–33 (1973)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bondal, A.I., Kapranov, M.M.: Enhanced triangulated categories. Mat. Sb. 181(5), 669–683 (1990). Translation in Math. USSR-Sb. 70 no. 1, 93107zbMATHGoogle Scholar
  7. 7.
    Bongartz, K.: Algebras and quadratic forms. J. Lond. Math. Soc. (2) 28, 461–469 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buan, A.B., Marsh, R.J., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buan, A.B., Iyama, O., Reiten, I., Scott, J.: Cluster structures for 2-Calabi–Yau categories and unipotent groups. Compos. Math. 145, 1035–1079 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (\(A_n\)-case). Trans. Am. Math. Soc. 358, 1347–1364 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Demonet, L., Iyama, O.: Lifting preprojective algebras to orders and categorifying partial flag varieties. Algebra Number Theory 10(7), 1527–1580 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Demonet, L., Luo, X.: Ice quivers with potentials associated with triangulations and Cohen–Macaulay modules over orders. Trans. Am. Math. Soc. 368(6), 4257–4293 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Drinfeld, V.: DG quotients of DG categories. J. Algebra 272(2), 643–691 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15, 497–529 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fomin, S., Zelevinsky, A.: Clusters algebras IV: coefficients. Compos. Math. 143, 112–164 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fomin, S., Zelevinsky, A.: Y-systems and generalized associahedra. Ann. Math. 158, 977–1018 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fu, C., Keller, B.: On cluster algebras with coefficients and 2-Calabi–Yau categories. Trans. Am. Math. Soc. 362(2), 859–895 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gabriel, P.: Auslander–Reiten Sequences and Representation-Finite Algebras, Representation Theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), pp. 1–71. Springer, Berlin (1980)Google Scholar
  19. 19.
    Gabriel, P., Roiter, A.V.: Representations of Finite-Dimensional Algebras. Encyclopaedia of Mathematical Sciences, vol. 73. Springer, Berlin (1992)Google Scholar
  20. 20.
    Geiss, C., Leclerc, B., Schröer, J.: Partial flag varieties and preprojective algebras. Ann. Inst. Fourier (Grenoble) 58(3), 825–876 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Happel, D.: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras. Cambridge University Press, Cambridge (1988)CrossRefzbMATHGoogle Scholar
  22. 22.
    Happel, D.: On the derived category of a finite-dimesional algebra. Comment. Math. Helv. 62(3), 339–389 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Iyama, O., Yoshino, Y.: Mutation in triangulated categories and rigid Cohen–Macaulay modules. Invent. Math. 172, 117–168 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jensen, B., King, A., Su, X.: A categorification of Grassmannian cluster algebras. Proc. Lond. Math. Soc. (3) 113(2), 185–212 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Keller, B.: On differential graded categories. In: International Congress of Mathematicians, vol. 2, pp. 151–190. European Mathematical Society, Zurich (2006)Google Scholar
  26. 26.
    Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551–581 (2005)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi–Yau. Adv. Math. 211(1), 123–151 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Keller, B., Scherotzke, S.: Graded quiver varieties and derived categories. J. Reine Angew. Math. 713, 85–127 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Krause, H.: Krull-Schmidt categories and projective covers. Expo. Math. 33(4), 535–549 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nájera Chávez, A.: On Frobenius (completed) orbit categories. Algebr. Represent. Theory 20(4), 1007–1027 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Nakajima, H.: Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Am. Math. Soc. 14(1), 145–238 (2001). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nakanishi, T., Zelevinsky, A.: On tropical dualities in cluster algebras. In: Proceedings of Representation Theory of Algebraic Groups and Quantum Groups, 10. Contemp. Math. 565, 217–226 (2012)Google Scholar
  33. 33.
    Neeman, A.: The derived category of an exact category. J. Algebra 135(2), 388–394 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Palu, Y.: Cluster characters for 2-Calabi–Yau triangulated categories. Ann. Inst. Fourier (Grenoble) 58(6), 2221–2248 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Qin, F.: Quantum groups via cyclic quiver varieties I. Compos. Math. 152(2), 299–326 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Quillen, D.: Higher Algebraic \({K}\)-Theory. I, Algebraic \(K\)-Theory. I: Higher \(K\)-Theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Mathematics, vol. 341, pp. 85–147. Springer, Berlin (1973)zbMATHGoogle Scholar
  37. 37.
    Reading, N.: Universal geometric cluster algebras. Math. Z. 277, 499–547 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Reading, N.: Universal geometric cluster algebras from surfaces. Trans. Am. Math. Soc. 366, 6647–6685 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Reading, N.: Universal geometric coefficients for the once-punctured torus. Sém. Lothar. Combin. 71, B71e (2013/14)Google Scholar
  40. 40.
    Riedtmann, Ch.: Algebren, Darstellungsköcher, Überlagerungen und Zurück. Comment. Math. Helv. 55(2), 199–224 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Riedtmann, Ch.: Representation-Finite Self-Injective Algebras of Class \(A_n\), Representation Theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979). Lecture Notes in Mathematics, vol. 832, pp. 449–520. Springer, Berlin (1980)Google Scholar
  42. 42.
    Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics, vol. 1099. Springer, Berlin (1984)zbMATHGoogle Scholar
  43. 43.
    Scherotzke, S.: Desingularization of quiver Grassmannians via Nakajima categories. Algebr. Represent. Theory 20(1), 231–243 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Tabuada, G.: Une structure de catégorie de modéles de Quillen sur la catégorie des dg-catégories. C. R. Math. Acad. Sci. Paris 340(1), 15–19 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Tabuada, G.: On Drinfeld’s DG quotient. J. Algebra 323, 1226–1240 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.CONACyT-Instituto de matemáticas UNAMOaxacaMexico

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