Algebras and Representation Theory

, Volume 16, Issue 2, pp 527–551 | Cite as

Derived Equivalence Classification of the Cluster-Tilted Algebras of Dynkin Type E

  • Janine Bastian
  • Thorsten HolmEmail author
  • Sefi Ladkani


We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E 6, E 7 and E 8 which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as supplementary material to this paper. From a structural point of view the remarkable outcome of our classification is that two cluster-tilted algebras of Dynkin type E are derived equivalent if and only if their Cartan matrices represent equivalent bilinear forms over the integers which in turn happens if and only if the two algebras are connected by a sequence of “good” mutations. This is reminiscent of the derived equivalence classification of cluster-tilted algebras of Dynkin type A, but quite different from the situation in Dynkin type D where a far-reaching classification has been obtained using similar methods as in the present paper but some very subtle questions are still open.


Cluster-tilted algebra Derived category Derived equivalences Dynkin diagram Finite representation type Good mutation 

Mathematics Subject Classifications (2010)

Primary 16G10 · 16E35 · 18E30 Secondary 13F60 · 16G60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Auslander, M., Platzeck, M.I., Reiten, I.: Coxeter functors without diagrams. Trans. Am. Math. Soc. 250, 1–46 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bastian, J.: Mutation classes of \(\tilde{A}_n\)-quivers and derived equivalence classification of cluster tilted algebras of type \(\tilde{A}_n\). Algebra Number Theory (to appear). Preprint available at arXiv:0901.1515
  3. 3.
    Bastian, J., Holm, T., Ladkani, S.: Derived equivalences for cluster-tilted algebras of Dynkin type D. Preprint available at arXiv:1012.4661.
  4. 4.
    Bastian, J., Holm, T., Ladkani, S.: Derived equivalence classification of cluster-tilted algebras of Dynkin type E. Preprint versions of the present paper, available at arXiv:0906.3422
  5. 5.
    Bocian, R., Skowroński, A.: Weakly symmetric algebras of Euclidean type. J. Reine Angew. Math. 580, 157–200 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbol. Comput. 24(3–4), 235–265 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Brenner, S., Butler, M.C.R.: Generalizations of the Bernstein–Gelfand–Ponomarev reflection functors. In: Representation Theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979). Lecture Notes in Math., vol. 832, pp. 103–169. Springer, Berlin (1980)CrossRefGoogle Scholar
  8. 8.
    Buan, A.B., Iyama, O., Reiten, I., Smith, D.: Mutation of cluster-tilting objects and potentials. Am. J. Math. 133, 835–887 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Buan, A.B., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Buan, A.B., Marsh, R., Reiten, I.: Cluster-tilted algebras of finite representation type. J. Algebra 306(2), 412–431 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Buan, A.B., Marsh, R., Reiten, I.: Cluster-tilted algebras. Trans. Am. Math. Soc. 359(1), 323–332 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Buan, A.B., Vatne, D.F.: Derived equivalence classification for cluster-tilted algebras of type A n. J. Algebra 319(7), 2723–2738 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (A n case). Trans. Am. Math. Soc. 358(3), 1347–1364 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002) (electronic)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Holm, T.: Cartan determinants for gentle algebras. Arch. Math. (Basel) 85(3), 233–239 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Keller, B.: Quiver mutation in Java. Java applet available at B. Keller’s home pageGoogle Scholar
  18. 18.
    Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi–Yau. Adv. Math. 211(1), 123–151 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Ladkani, S.: On derived equivalences of categories of sheaves over finite posets. J. Pure Appl. Algebra 212, 435–451 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Ladkani, S.: Perverse equivalences, BB-tilting, mutations and applications. Preprint available at arXiv:1001.4765
  21. 21.
    Ladkani, S.: Cluster-tilted algebras of Dynkin type E. Supplementary material to the present paper available at
  22. 22.
    Lenzing, H.: Coxeter transformations associated with finite-dimensional algebras. In: Computational Methods for Representations of Groups and Algebras (Essen, 1997). Progr. Math., vol. 173, pp. 287–308. Birkhäuser, Basel (1999)CrossRefGoogle Scholar
  23. 23.
    Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. 39(3), 436–456 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Vatne, D.F.: The mutation class of D n quivers. Commun. Algebra 38(3), 1137–1146 (2010)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institut für Algebra, Zahlentheorie und Diskrete MathematikLeibniz Universität HannoverHannoverGermany
  2. 2.Max-Planck-Institut für MathematikBonnGermany

Personalised recommendations