A Course on Cluster Tilted Algebras

  • Ibrahim AssemEmail author
Part of the CRM Short Courses book series (CRMSC)


These notes are an expanded version of a mini-course given in the CIMPA School ‘Homological Methods, Representation Theory and Cluster Algebras’, held from the 7th to the 18th of March 2016 in Mar del Plata (Argentina). The aim of the course was to introduce the participants to cluster tilted algebras and their applications in the representation theory of algebras.



The author gratefully acknowledges partial support from the NSERC of Canada, the FRQ-NT of Québec and the Université de Sherbrooke. He is also grateful to all members of the research group in Mar del Plata for their kind hospitality during his stay there.


  1. 1.
    Amiot, C.: Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier (Grenoble) 59(6), 2525–2590 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amiot, C.: On generalized cluster categories. In: A. Skowroński, K. Yamagata (eds.) Representations of Algebras and Related Topics, EMS Ser. Congr. Rep., pp. 1–53. Eur. Math. Soc., Zürich (2011). DOI
  3. 3.
    Assem, I., Brüstle, T., Charbonneau-Jodoin, G., Plamondon, P.-G.: Gentle algebras arising from surface triangulations. Algebra Number Theory 4(2), 201–229 (2010). DOI Scholar
  4. 4.
    Assem, I., Brüstle, T., Schiffler, R.: Cluster-tilted algebras and slices. J. Algebra 319(8), 3464–3479 (2008). DOI
  5. 5.
    Assem, I., Brüstle, T., Schiffler, R.: Cluster-tilted algebras as trivial extensions. Bull. Lond. Math. Soc. 40(1), 151–162 (2008). DOI Scholar
  6. 6.
    Assem, I., Brüstle, T., Schiffler, R.: On the Galois coverings of a cluster-tilted algebra. J. Pure Appl. Algebra 213(7), 1450–1463 (2009). DOI Scholar
  7. 7.
    Assem, I., Bustamante, J.C., Dionne, J., Le Meur, P., Smith, D.: Representation theory of partial relation extensions. Colloquium Math., to appear. arXiv:1604.01269
  8. 8.
    Assem, I., Bustamante, J.C., Igusa, K., Schiffler, R.: The first Hochschild cohomology group of a cluster tilted algebra revisited. Internat. J. Algebra Comput. 23(4), 729–744 (2013). DOI
  9. 9.
    Assem, I., Cappa, J.A., Platzeck, M.I., Verdecchia, M.: Módulos inclinantes y álgebras inclinadas, Notas Álgebra Anál., vol. 21. Univ. Nac. del Sur , Inst. Mat., Bahía Blanca (2008)Google Scholar
  10. 10.
    Assem, I., Coelho, F.U.: Two-sided gluings of tilted algebras. J. Algebra 269(2), 456–479 (2003). DOI Scholar
  11. 11.
    Assem, I., Coelho, F.U., Trepode, S.: The left and the right parts of a module category. J. Algebra 281(2), 518–534 (2004). DOI
  12. 12.
    Assem, I., Coelho, F.U., Trepode, S.: The bound quiver of a split extension. J. Algebra Appl. 7(4), 405–423 (2008). DOI Scholar
  13. 13.
    Assem, I., Dupont, G.: Modules over cluster-tilted algebras determined by their dimension vectors. Comm. Algebra 41(12), 4711–4721 (2013). DOI
  14. 14.
    Assem, I., Gatica, M.A., Schiffler, R., Taillefer, R.: Hochschild cohomology of relation extension algebras. J. Pure Appl. Algebra 220(7), 2471–2499 (2016). DOI
  15. 15.
    Assem, I., Redondo, M.J.: The first Hochschild cohomology group of a Schurian cluster-tilted algebra. Manuscripta Math. 128(3), 373–388 (2009). DOI Scholar
  16. 16.
    Assem, I., Redondo, M.J., Schiffler, R.: On the first Hochschild cohomology group of a cluster-tilted algebra. Algebr. Represent. Theory 18(6), 1547–1576 (2015). DOI Scholar
  17. 17.
    Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory, London Math. Soc. Stud. Texts, vol. 65. Cambridge Univ. Press, Cambridge (2006). DOI
  18. 18.
    Assem, I., Skowroński, A.: Iterated tilted algebras of type \({\tilde{\mathbb{A}}_n}\). Math. Z. 195(2), 269–290 (1987). DOI
  19. 19.
    Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991). DOI
  20. 20.
    Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., vol. 36. Cambridge Univ. Press, Cambridge (1995). DOI
  21. 21.
    Babaei, F., Grimeland, Y.: Special biserial cluster-tilted algebras. Comm. Algebra 42(6), 2740–2758 (2014). DOI
  22. 22.
    Barot, M., Fernández, E., Platzeck, M.I., Pratti, N.I., Trepode, S.: From iterated tilted algebras to cluster-tilted algebras. Adv. Math. 223(4), 1468–1494 (2010). DOI
  23. 23.
    Barot, M., Geiss, C., Zelevinsky, A.: Cluster algebras of finite type and positive symmetrizable matrices. J. London Math. Soc. (2) 73(3), 545–564 (2006). DOI Scholar
  24. 24.
    Barot, M., Trepode, S.: Cluster tilted algebras with a cyclically oriented quiver. Comm. Algebra 41(10), 3613–3628 (2013). DOI Scholar
  25. 25.
    Bastian, J.: Mutation classes of \({\tilde{\mathbb{A}}_n}\)-quivers and derived equivalence classification of cluster tilted algebras of type \({\tilde{\mathbb{A}}_n}\). Algebra Number Theory 5(5), 567–594 (2011). DOI
  26. 26.
    Bastian, J., Holm, T., Ladkani, S.: Derived equivalence classification of the cluster-tilted algebras of Dynkin type \({\mathbb{E}}\). Algebr. Represent. Theory 16(2), 527–551 (2013). DOI
  27. 27.
    Bastian, J., Holm, T., Ladkani, S.: Towards derived equivalence classification of the cluster-tilted algebras of Dynkin type \({\mathbb{D}}\). J. Algebra 410, 277–332 (2014). DOI
  28. 28.
    Beaudet, L., Brüstle, T., Todorov, G.: Projective dimension of modules over cluster-tilted algebras. Algebr. Represent. Theory 17(6), 1797–1807 (2014). DOI Scholar
  29. 29.
    Bertani-Økland, M.A., Oppermann, S., Wrålsen, A.: Constructing tilted algebras from cluster-tilted algebras. J. Algebra 323(9), 2408–2428 (2010). DOI Scholar
  30. 30.
    Bertani-Økland, M.A., Oppermann, S., Wrålsen, A.: Finding a cluster-tilting object for a representation finite cluster-tilted algebra. Colloq. Math. 121(2), 249–263 (2010). DOI Scholar
  31. 31.
    Bobiński, G., Buan, A.B.: The algebras derived equivalent to gentle cluster tilted algebras. J. Algebra Appl. 11(1), 1250,012, 26 (2012). DOI
  32. 32.
    Bongartz, K.: Algebras and quadratic forms. J. London Math. Soc. (2) 28(3), 461–469 (1983). DOI
  33. 33.
    Bongartz, K., Gabriel, P.: Covering spaces in representation-theory. Invent. Math. 65(3), 331–378 (1981/82). DOI Scholar
  34. 34.
    Bordino, N., Fernández, E., Trepode, S.: On the quiver with relations of a quasitilted algebra and applications. Comm. Algebra 45(9), 4050–4061 (2017). DOI Scholar
  35. 35.
    Buan, A.B., Iyama, O., Reiten, I., Smith, D.: Mutation of cluster-tilting objects and potentials. Amer. J. Math. 133(4), 835–887 (2011). DOI
  36. 36.
    Buan, A.B., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006). DOI
  37. 37.
    Buan, A.B., Marsh, R.J., Reiten, I.: Cluster-tilted algebras of finite representation type. J. Algebra 306(2), 412–431 (2006). DOI
  38. 38.
    Buan, A.B., Marsh, R.J., Reiten, I.: Cluster-tilted algebras. Trans. Amer. Math. Soc. 359(1), 323–332 (2007). DOI
  39. 39.
    Buan, A.B., Marsh, R.J., Reiten, I.: Cluster mutation via quiver representations. Comment. Math. Helv. 83(1), 143–177 (2008). DOI
  40. 40.
    Buan, A.B., Vatne, D.F.: Derived equivalence classification for cluster-tilted algebras of type \({{\mathbb{A}}_n}\). J. Algebra 319(7), 2723–2738 (2008). DOI
  41. 41.
    Bustamante, J.C., Gubitosi, V.: Hochschild cohomology and the derived class of \(m\)-cluster tilted algebras of type \({\mathbb{A}}\). Algebr. Represent. Theory 17(2), 445–467 (2014). DOI
  42. 42.
    Butler, M.C.R., Ringel, C.M.: Auslander–Reiten sequences with few middle terms and applications to string algebras. Comm. Algebra 15(1-2), 145–179 (1987). DOI Scholar
  43. 43.
    Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (\({\mathbb{A}_n}\) case). Trans. Amer. Math. Soc. 358(3), 1347–1364 (2006). DOI
  44. 44.
    Cartan, H., Eilenberg, S.: Homological Algebra. Princeton Univ. Press, Princeton, N. J. (1956)zbMATHGoogle Scholar
  45. 45.
    Coelho, F.U., Lanzilotta, M.A.: Algebras with small homological dimensions. Manuscripta Math. 100(1), 1–11 (1999). DOI Scholar
  46. 46.
    Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15(2), 497–529 (2002). DOI Scholar
  47. 47.
    Gabriel, P.: The universal cover of a representation-finite algebra. In: M. Auslander, E. Lluis (eds.) Representations of Algebras (Puebla, 1980), Lecture Notes in Math., vol. 903, pp. 68–105. Springer, Berlin (1981)Google Scholar
  48. 48.
    Ge, W., Lv, H., Zhang, S.: Cluster-tilted algebras of type \({{\mathbb{D}}_n}\). Comm. Algebra 38(7), 2418–2432 (2010). DOI
  49. 49.
    Gelfand, S.I., Manin, Yu.I.: Methods of Homological Algebra. Springer, Berlin (1996). DOI Scholar
  50. 50.
    Grivel, P.-P.: Catégories dérivées et foncteurs dérivés. In: Algebraic \(D\)-Modules, Perspect. Math, vol. 2, pp. 1–108. Academic Press, Boston, MA (1987)Google Scholar
  51. 51.
    Happel, D.: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Math. Soc. Lecture Note Ser., vol. 119. Cambridge Univ. Press, Cambridge (1988). DOI
  52. 52.
    Happel, D.: Hochschild cohomology of finite-dimensional algebras. In: M.-P. Malliavin (ed.) Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, Lecture Notes in Math., vol. 1404, pp. 108–126. Springer, Berlin (1989). DOI Scholar
  53. 53.
    Happel, D.: Hochschild cohomology of piecewise hereditary algebras. Colloq. Math. 78(2), 261–266 (1998)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Amer. Math. Soc. 120(575) (1996). DOI Scholar
  55. 55.
    Happel, D., Unger, L.: Almost complete tilting modules. Proc. Amer. Math. Soc. 107(3), 603–610 (1989). DOI Scholar
  56. 56.
    Keller, B.: Hochschild cohomology and derived Picard groups. J. Pure Appl. Algebra 190(1–3), 177–196 (2004). DOI Scholar
  57. 57.
    Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551–581 (2005)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Keller, B.: Deformed Calabi–Yau completions. J. Reine Angew. Math. 654, 125–180 (2011). DOI
  59. 59.
    Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi–Yau. Adv. Math. 211(1), 123–151 (2007). DOI Scholar
  60. 60.
    Oryu, M., Schiffler, R.: On one-point extensions of cluster-tilted algebras. J. Algebra 357, 168–182 (2012). DOI
  61. 61.
    de la Peña, J.A., Saorín, M.: On the first Hochschild cohomology group of an algebra. Manuscripta Math. 104(4), 431–442 (2001). DOI
  62. 62.
    Ringel, C.M.: The self-injective cluster-tilted algebras. Arch. Math. (Basel) 91(3), 218–225 (2008). DOI
  63. 63.
    Ringel, C.M.: Cluster-concealed algebras. Adv. Math. 226(2), 1513–1537 (2011). DOI Scholar
  64. 64.
    Schiffler, R., Serhiyenko, K.: Induced and coinduced modules over cluster-tilted algebras. J. Algebra 472, 226–258 (2017). DOI Scholar
  65. 65.
    Schröer, J., Zimmermann, A.: Stable endomorphism algebras of modules over special biserial algebras. Math. Z. 244(3), 515–530 (2003)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Skowroński, A.: Simply connected algebras and Hochschild cohomologies. In: V. Dlab, H. Lenzing (eds.) Representations of Algebras (Ottawa, ON, 1992), CMS Conf. Proc., vol. 14, pp. 431–447. Amer. Math. Soc., Providence, RI (1993)Google Scholar
  67. 67.
    Smith, D.: On tilting modules over cluster-tilted algebras. Illinois J. Math. 52(4), 1223–1247 (2008)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Verdier, J.-L.: Catégories dérivées : Quelques résultats (État 0). In: Cohomologie étale, Lecture Notes in Math., vol. 569, pp. 262–311. Springer, Berlin–New York (1977). DOI

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de SherbrookeSherbrookeCanada

Personalised recommendations