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Mathematische Annalen

, Volume 351, Issue 4, pp 1005–1017 | Cite as

Braid groups and Kleinian singularities

  • Christopher Brav
  • Hugh Thomas
Article

Abstract

We establish faithfulness of braid group actions generated by twists along an ADE configuration of 2-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland’s description of a space of stability conditions associated to a Kleinian singularity.

Keywords

Braid Group Dynkin Diagram Exceptional Divisor Coherent Sheave Triangulate Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Anno, R.: Spherical functors (2007). arXiv:0711.4409Google Scholar
  2. 2.
    Bondal A.I., Kapranov M.M.: Framed triangulated categories. Math. Sb. 181(5), 669–683 (1990)zbMATHGoogle Scholar
  3. 3.
    Bridgeland T.: Stability conditions on triangulated categories. Ann. Math. (2) 166(2), 317–345 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bridgeland, T.: Stability conditions and Kleinian singularities. Int. Math. Res. Notices 2009, rnp081 (2009)Google Scholar
  5. 5.
    du Val P.: On isolated singularities of surfaces which do not affect the conditions of adjunction. Proc. Camb. Phil. Soc. 30, 453–459 (1934)CrossRefGoogle Scholar
  6. 6.
    Ishii A., Ueda K., Uehara H.: Stability conditions on A n-singularities. J. Differ. Geom. 84(1), 87–126 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kassel, C., Turaev, V.: Braid Groups, vol. 247 of Graduate Texts in Mathematics. Springer, New York (2008). (With the graphical assistance of Olivier Dodane)Google Scholar
  8. 8.
    Keller B.: Derived categories and tilting. In: Angeleri Hügel, L., Happel, D., Krause, H. (eds) Handbook of Tilting Theory, vol. 332 of London Mathematical Society Lecture Note Series., pp. 49–104. Cambridge University Press, Cambridge (2007)Google Scholar
  9. 9.
    Khovanov M., Seidel P.: Quivers, Floer cohomology, and braid group actions. J. Am. Math. Soc. 15(1), 203–271 (2002) (electronic)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Klein, F.: Vorlesungenüber das Ikosaeder und die Auflösung der Gleichungen vom 5ten Grade. B. G. Teubner, Leipzig, Germany (1884)Google Scholar
  11. 11.
    Kontsevich, M.: Homological algebra of mirror symmetry. In: Proc. Int. Congr. Math. 1, 2(Zürich, 1994), 120–139. Basel, Birkhäuser (1995)Google Scholar
  12. 12.
    Lurie, J.: Stable infinity categories (2006). arXiv:math/0608228Google Scholar
  13. 13.
    Rouquier, R.: Categorification of the braid groups (2004). arXiv:math/0409593Google Scholar
  14. 14.
    Seidel P., Thomas R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1), 37–108 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Thomas R.P.: Stability conditions and the braid group. Comm. Anal. Geom. 14(1), 135–161 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Leibniz Universität HannoverHannoverGermany
  2. 2.University of New BrunswickNew BrunswickCanada

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