Algebras and Representation Theory

, Volume 21, Issue 4, pp 817–832 | Cite as

Mutation of Torsion Pairs in Triangulated Categories and its Geometric Realization

  • Yu ZhouEmail author
  • Bin Zhu


We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type A n or A is given via rotation of Ptolemy diagrams.


Torsion pairs Triangulated categories Mutations Ptolemy diagrams 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors thank the anonymous referee for his/her useful suggestions to improve this article.


  1. 1.
    Amiot, C.: On the structure of triangulated categories with finitely many indecomposables. Bull. Soc. Math. France 135(3), 435–474 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Assem, I., Simson, D., Skowronski, A.: Elements of the Representation Theory of Associative Algebras. vol. 1. Techniques of Representation Theory London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  3. 3.
    Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux Pervers Astérisque, 100, Soc. Math. France, Paris (1982)Google Scholar
  4. 4.
    Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc. 188(883), viii+207 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Buan, A.B., Iyama, O., Reiten, I., Scott, J.: Cluster structures for 2-Calabi-Yau categories and unipotent groups. Compos. Math. 145(4), 1035–1079 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Buan, A.B., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buan, A.B., Marsh, R., Vatne, D.: Cluster structures from 2-Calabi-Yau categories with loops. Math. Z. 265(4), 951–970 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Burban, I., Iyama, O., Keller, B., Reiten, I.: Cluster tilting for one-dimensional hypersurface singularities. Adv. Math. 217(6), 2443–2484 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bondal, A., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions. Math. USSR-Izv. 35(3), 519–541 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (A n case). Trans. Amer. Math. Soc. 358(3), 1347–1364 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dickson, S.E.: A torsion theory for abelian categories. Trans. Amer. Math. Soc. 121, 223–235 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15(2), 497–529 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gratz, S.: Mutation of torsion pairs in cluster categories of dynkin type D. Appl. Categ. Struct.
  14. 14.
    Holm, T., Jørgensen, P.: On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon. Math. Z. 270(1–2), 277–295 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Holm, T., Jørgensen, P., Rubey, M.: Ptolemy diagrams and torsion pairs in the cluster category of Dynkin tpye A n. J. Algebraic Combin. 34(3), 507–523 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Iyama, O., Yoshino, Y.: Mutations in triangulated categories and rigid Cohen-Macaulay modules. Invent. Math. 172(1), 117–168 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Keller, B.: Cluster Algebras, Quiver Representations and Triangulated Categories. Triangulated Categories, 76–160, London Math. Soc Lecture Note Ser., 375. Cambridge University Press, Cambridge (2010)Google Scholar
  18. 18.
    Keller, B.: Calabi-yau triangulated categories. Trends in representation theory of algebras and related topics, 467–489, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich (2008)Google Scholar
  19. 19.
    Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. Adv. Math. 211, 123–151 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Koehler, C.: Thick subcategories of finite algebraic triangulated categories. arXiv:1010.0146
  21. 21.
    Koenig, S., Zhu, B.: From triangulated categories to abelian categories: cluster tilting in a general framework. Math. Z. 258(1), 143–160 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Krause, H.: Report on locally finite triangulated categories. J. K-Theory 9(3), 421–458 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Marsh, R., Palu, Y.: Coloured quivers for rigid objects and partial triangulations: the unpunctured case. Proc. Lond. Math. Soc. 108(2), 411–440 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nakaoka, H.: General heart construction on a triangulated category (i): unifying t-structures and cluster tilting subcategories. Appl. Categ. Structures 19(6), 879–899 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ng, P.: A characterization of torsion theories in the cluster category of dynkin type A . arXiv:1005.4364
  26. 26.
    Palu, Y.: Grothendieck group and generalized mutation rule for 2-Calabi-Yau triangulated categories. J. Pure Appl. Algebra 213(7), 1438–1449 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Reiten, I.: Cluster categories. In: Proceedings of the International Congress of Mathematicians, vol. I, pp. 558–594. Hindustan Book Agency, New Delhi (2010)Google Scholar
  28. 28.
    Xiao, J., Zhu, B.: Relations for the Grothendieck groups of triangulated categories. J. Algebra 257(1), 37–50 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Xiao, J., Zhu, B.: Locally finite triangulated categories. J. Algebra 290(2), 473–490 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zhang, J., Zhou, Y., Zhu, B.: Cotorsion pairs in the cluster category of a marked surface. J. Algebra 391, 209–226 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zhou, Y., Zhu, B.: Maximal rigid subcategories in 2-Calabi-Yau triangulated categories. J. Algebra 348, 49–60 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zhou, Y., Zhu, B.: T-structures and torsion pairs in a 2-Calabi-Yau triangulated category. J. Lond. Math. Soc. 89(1), 213–234 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations