Skip to main content
Log in

Braid groups and Kleinian singularities

Mathematische Annalen Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Anno, R.: Spherical functors (2007). arXiv:0711.4409

  2. Bondal A.I., Kapranov M.M.: Framed triangulated categories. Math. Sb. 181(5), 669–683 (1990)

    MATH  Google Scholar 

  3. Bridgeland T.: Stability conditions on triangulated categories. Ann. Math. (2) 166(2), 317–345 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bridgeland, T.: Stability conditions and Kleinian singularities. Int. Math. Res. Notices 2009, rnp081 (2009)

  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)

    Article  Google Scholar 

  6. Ishii A., Ueda K., Uehara H.: Stability conditions on A n -singularities. J. Differ. Geom. 84(1), 87–126 (2010)

    MathSciNet  MATH  Google Scholar 

  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)

  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. Khovanov M., Seidel P.: Quivers, Floer cohomology, and braid group actions. J. Am. Math. Soc. 15(1), 203–271 (2002) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  10. Klein, F.: Vorlesungenüber das Ikosaeder und die Auflösung der Gleichungen vom 5ten Grade. B. G. Teubner, Leipzig, Germany (1884)

  11. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proc. Int. Congr. Math. 1, 2(Zürich, 1994), 120–139. Basel, Birkhäuser (1995)

  12. Lurie, J.: Stable infinity categories (2006). arXiv:math/0608228

  13. Rouquier, R.: Categorification of the braid groups (2004). arXiv:math/0409593

  14. Seidel P., Thomas R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1), 37–108 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Thomas R.P.: Stability conditions and the braid group. Comm. Anal. Geom. 14(1), 135–161 (2006)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Christopher Brav.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brav, C., Thomas, H. Braid groups and Kleinian singularities. Math. Ann. 351, 1005–1017 (2011). https://doi.org/10.1007/s00208-010-0627-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-010-0627-y

Keywords

Navigation