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Communications in Mathematical Physics

, Volume 288, Issue 2, pp 403–429 | Cite as

Noncommutative Riemann Surfaces by Embeddings in \({\mathbb{R}^{3}}\)

  • Joakim Arnlind
  • Martin Bordemann
  • Laurent Hofer
  • Jens HoppeEmail author
  • Hidehiko Shimada
Open Access
Article

Abstract

We introduce C-Algebras of compact Riemann surfaces \({\Sigma}\) as non-commutative analogues of the Poisson algebra of smooth functions on \({\Sigma}\) . Representations of these algebras give rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.

Keywords

Riemann Surface Poisson Bracket Morse Function Eigenvalue Distribution Poisson Algebra 
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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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Joakim Arnlind
    • 1
    • 2
  • Martin Bordemann
    • 3
  • Laurent Hofer
    • 4
  • Jens Hoppe
    • 5
    Email author
  • Hidehiko Shimada
    • 2
  1. 1.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  2. 2.Max Planck Institute for Gravitational PhysicsGolmGermany
  3. 3.Laboratoire de MIA, 4, rue des Frères LumièreUniversité de Haute-AlsaceMulhouseFrance
  4. 4.Université du LuxembourgLuxembourg CityLuxembourg
  5. 5.Department of MathematicsKTHStockholmSweden

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