Mathematische Zeitschrift

, Volume 277, Issue 3–4, pp 665–690 | Cite as

Classifying \(\tau \)-tilting modules over preprojective algebras of Dynkin type

  • Yuya Mizuno


We study support \(\tau \)-tilting modules over preprojective algebras of Dynkin type. We classify basic support \(\tau \)-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and many other important objects in representation theory. We also study \(g\)-matrices of support \(\tau \)-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of \(g\)-matrices and prove that cones given by \(g\)-matrices coincide with chambers of the associated root systems.



First and foremost, the author would like to thank Osamu Iyama for his support and patient guidance. He would like to express his gratitude to Steffen Oppermann, who kindly explain results of his paper. He is grateful to Joseph Grant, Laurent Demonet and Dong Yang for answering questions and helpful comments. He thanks Kota Yamaura, Takahide Adachi and Gustavo Jasso for their help and stimulating discussions. He is very grateful to the anonymous referee for valuable comments, especially for suggesting the better terms and sentences.


  1. 1.
    Adachi, T., Iyama, O., Reiten, I.: \(\tau \)-tilting theory, arXiv: 1210.1036 (to appear in Compos. Math., 2012)Google Scholar
  2. 2.
    Amiot, C., Iyama, O., Reiten, I., Todorov, G.: Preprojective algebras and c-sortable words. Proc. Lond. Math. Soc. (3) 104(3), 513–539 (2012)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1, London Mathematical Society Student Texts 65. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  4. 4.
    Auslander, M., Reiten, I.: Modules determined by their composition factors, Ill. J. Math. 29, 280–301 (1985)MATHMathSciNetGoogle Scholar
  5. 5.
    Auslander, M., Reiten, I.: \(DTr\)-periodic modules and functors. Representation theory of algebras (Cocoyoc, 1994), pp. 39–50. In: CMS Conference Proceedings, 18, American Mathematical Society, Providence, RI (1996)Google Scholar
  6. 6.
    Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  7. 7.
    Baer, D., Geigle, W., Lenzing, H.: The preprojective algebra of a tame hereditary Artin algebra. Commun. Algebra 15(1–2), 425–457 (1987)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Baumann, P., Kamnitzer, J., Tingley, P.: Affine Mirković–Vilonen polytopes, arXiv:1110.3661 (to appear in Publ. IHES)Google Scholar
  9. 9.
    Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)Google Scholar
  10. 10.
    Bocklandt, R.: Graded Calabi Yau algebras of dimension 3. J. Pure Appl. Algebra 212(1), 14–32 (2008)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Bourbaki, N.: Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), transl. from the 1968 French original by Andrew Pressley. Springer, Berlin (2002)Google Scholar
  12. 12.
    Brenner, S., Butler, M.C.R., King, A.D.: Periodic algebras which are Almost Koszul. Algebr. Represent. Theory 5(4), 331–367 (2002)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Brüstle, T., Yang, D.: Ordered exchange graphs, arXiv:1302.6045 (to appear in Adv. Represent. Theory Algebras (ICRA Bielefeld 2012))Google Scholar
  14. 14.
    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)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Crawley-Boevey, W.: On the exceptional fibres of Kleinian singularities. Am. J. Math. 122(5), 1027–1037 (2000)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Dehy, R., Keller, B.: On the combinatorics of rigid objects in 2-Calabi–Yau categories. Int. Math. Res. Not. IMRN 2008, no. 11, Art. ID rnn029Google Scholar
  17. 17.
    Dlab, V., Ringel, C.M.: The preprojective algebra of a modulated graph, Representation theory, II. In: Proceedings of Second International Conference on Carleton University, Ottawa, Ontario, 1979, pp. 216–231, Lecture Notes in Mathematics, vol. 832. Springer, Berlin, New York (1980)Google Scholar
  18. 18.
    Dlab, V., Ringel, C.M.: The module theoretical approach to quasi-hereditary algebras. Representations of algebras and related topics (Kyoto, 1990), pp. 200–224, London Mathematical Society Lecture Note Series, 168, Cambridge University Press, Cambridge (1992)Google Scholar
  19. 19.
    Geiss, C., Leclerc, B., Schröer, J.: Rigid modules over preprojective algebras. Invent. Math. 165(3), 589–632 (2006)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Geiss, C., Leclerc, B., Schröer, J.: Semicanonical bases and preprojective algebras II: a multiplication formula. Compos. Math. 143(5), 1313–1334 (2007)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Geiss, C., Leclerc, B., Schröer, J.: Kac–Moody groups and cluster algebras. Adv. Math. 228(1), 329–433 (2011)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Gelfand, I.M., Ponomarev, V.A.: Model algebras and representations of graphs. Funktsional. Anal. i Prilozhen. 13(3), 1–12 (1979)MathSciNetGoogle Scholar
  23. 23.
    Humphreys, J.E.: Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  24. 24.
    Ingalls, C., Thomas, H.: Noncrossing partitions and representations of quivers. Compos. Math. 145(6), 1533–1562 (2009)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Iyama, O., Jasso, G.: On \(g\)-vectors of \(\tau \) -tilting modules and \(\tau \)-rigid-finite algebras (in preparation)Google Scholar
  26. 26.
    Iyama, O., Reiten, I.: Fomin–Zelevinsky mutation and tilting modules over Calabi–Yau algebras. Am. J. Math. 130(4), 1087–1149 (2008)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Koenig, S., Yang, D.: Silting objects, simple-minded collections, \(t\)-structures and co-\(t\)-structures for finite-dimensional algebras. arXiv:1203.5657Google Scholar
  28. 28.
    Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Am. Math. Soc. 4(2), 365–421 (1991)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Lusztig, G.: Semicanonical bases arising from enveloping algebras. Adv. Math. 151(2), 129–139 (2000)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Mizuno, Y.: \(\nu \)-stable \(\tau \)-tilting modules, arXiv:1210.8322 (to appear in Commun. Algebra, 2012)Google Scholar
  31. 31.
    Oppermann, S., Reiten, I., Thomas, H.: Quotient closed subcategories of quiver representations, arXiv:1205.3268 (2012)Google Scholar
  32. 32.
    Palu, Y.: Cluster characters for 2-Calabi–Yau triangulated categories. Ann. Inst. Fourier (Grenoble) 58(6), 2221–2248 (2008)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Reading, N.: Clusters, coxeter-sortable elements and noncrossing partitions. Trans. Am. Math. Soc. 359(12), 5931–5958 (2007)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Ringel, C.M.: The preprojective algebra of a quiver, Algebras and modules, II (Geiranger, 1996), pp. 467–480. In: CMS Conference Proceedings, 24, American Mathematical Society, Providence, RI (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityFrocho, Chikusaku, NagoyaJapan

Personalised recommendations