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

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Abstract

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.

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References

  1. Auslander, M., Platzeck, M.I., Reiten, I.: Coxeter functors without diagrams. Trans. Am. Math. Soc. 250, 1–46 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  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. Bastian, J., Holm, T., Ladkani, S.: Derived equivalences for cluster-tilted algebras of Dynkin type D. Preprint available at arXiv:1012.4661.

  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. Bocian, R., Skowroński, A.: Weakly symmetric algebras of Euclidean type. J. Reine Angew. Math. 580, 157–200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbol. Comput. 24(3–4), 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  8. Buan, A.B., Iyama, O., Reiten, I., Smith, D.: Mutation of cluster-tilting objects and potentials. Am. J. Math. 133, 835–887 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Buan, A.B., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Buan, A.B., Marsh, R., Reiten, I.: Cluster-tilted algebras of finite representation type. J. Algebra 306(2), 412–431 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Buan, A.B., Marsh, R., Reiten, I.: Cluster-tilted algebras. Trans. Am. Math. Soc. 359(1), 323–332 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Holm, T.: Cartan determinants for gentle algebras. Arch. Math. (Basel) 85(3), 233–239 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Keller, B.: Quiver mutation in Java. Java applet available at B. Keller’s home page

  18. Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi–Yau. Adv. Math. 211(1), 123–151 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ladkani, S.: On derived equivalences of categories of sheaves over finite posets. J. Pure Appl. Algebra 212, 435–451 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ladkani, S.: Perverse equivalences, BB-tilting, mutations and applications. Preprint available at arXiv:1001.4765

  21. Ladkani, S.: Cluster-tilted algebras of Dynkin type E. Supplementary material to the present paper available at http://www.math.uni-bonn.de/people/sefil/CTA/index.html

  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)

    Chapter  Google Scholar 

  23. Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. 39(3), 436–456 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  24. Vatne, D.F.: The mutation class of D n quivers. Commun. Algebra 38(3), 1137–1146 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Author notes

  1. Sefi Ladkani

    Present address: Mathematical Institute of the University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

Authors and Affiliations

  1. Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Janine Bastian & Thorsten Holm

  2. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Sefi Ladkani

Authors
  1. Janine Bastian
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  2. Thorsten Holm
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  3. Sefi Ladkani
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Corresponding author

Correspondence to Thorsten Holm.

Additional information

This work has been carried out in the framework of the priority program SPP 1388 Darstellungstheorie of the Deutsche Forschungsgemeinschaft (DFG). J. Bastian and T. Holm gratefully acknowledge financial support through the grant HO 1880/4-1.

S. Ladkani was supported by a European Postdoctoral Institute (EPDI) fellowship. He also acknowledges support from DFG grant LA 2732/1-1 in the framework of the priority program SPP 1388 “Representation theory”.

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Bastian, J., Holm, T. & Ladkani, S. Derived Equivalence Classification of the Cluster-Tilted Algebras of Dynkin Type E . Algebr Represent Theor 16, 527–551 (2013). https://doi.org/10.1007/s10468-011-9318-y

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  • DOI: https://doi.org/10.1007/s10468-011-9318-y

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