Letters in Mathematical Physics

, Volume 100, Issue 3, pp 261–278 | Cite as

On Fundamental Domains and Volumes of Hyperbolic Coxeter–Weyl Groups

  • Philipp Fleig
  • Michael Koehn
  • Hermann NicolaiEmail author


We present a simple method for determining the shape of fundamental domains of generalized modular groups related to Weyl groups of hyperbolic Kac–Moody algebras. These domains are given as subsets of certain generalized upper half planes, on which the Weyl groups act via generalized modular transformations. Our construction only requires the Cartan matrix of the underlying finite-dimensional Lie algebra and the associated Coxeter labels as input information. We present a simple formula for determining the volume of these fundamental domains. This allows us to re-produce in a simple manner the known values for these volumes previously obtained by other methods.

Mathematics Subject Classification (2000)

11F99 11R52 17B67 20F55 20H10 22E40 33B30 51F15 


Weyl groups Kac–Moody algebras hyperbolic volume polylogarithm 


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Copyright information

© Springer 2011

Authors and Affiliations

  • Philipp Fleig
    • 1
    • 2
  • Michael Koehn
    • 1
  • Hermann Nicolai
    • 1
    Email author
  1. 1.Max-Planck-Institut für GravitationsphysikAlbert-Einstein-InstitutPotsdamGermany
  2. 2.Université de Nice-Sophia AntipolisNice Cedex 2France

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