Inventiones mathematicae

, Volume 152, Issue 2, pp 349–368

The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli

  • Markus Reineke

DOI: 10.1007/s00222-002-0273-4

Cite this article as:
Reineke, M. Invent. math. (2003) 152: 349. doi:10.1007/s00222-002-0273-4


Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a Kac-Moody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution of the recursion, an explicit formula for the HN system is given. As an application, explicit formulas for Betti numbers of the cohomology of quiver moduli are derived, generalizing several results on the cohomology of quotients in ‘linear algebra type’ situations.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Markus Reineke
    • 1
  1. 1.BUGH Wuppertal, Gaussstr. 20, D-42097 Wuppertal, Germany (e-mail:

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