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


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.


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.


  1. ASS06.
    Assem, I., Simson, D., Skowronski, A.: Elements of the Representation Theory of Associative Algebras. LMS Student Texts 65, Cambridge: Cambridge University Press, 2006Google Scholar
  2. Ber78.
    Bergman G.M.: The Diamond Lemma for Ring Theory. Adv. Math. 29, 178–218 (1978)CrossRefMathSciNetGoogle Scholar
  3. BHSS91.
    Bordemann M., Hoppe J., Schaller P., Schlichenmaier M.: \({\mathfrak{gl}(\infty)}\) and geometric quantization. Commun. Math. Phys. 138, 209–244 (1991)zbMATHCrossRefADSMathSciNetGoogle Scholar
  4. BKL05.
    Bak D., Kim S., Lee K.: All higher genus BPS membranes in the plane wave background. JHEP 0506, 035 (2005)CrossRefADSMathSciNetGoogle Scholar
  5. BMS94.
    Bordemann M., Meinrenken E., Schlichenmaier M.: Toeplitz Quant. of Kähler Manifolds and \({\mathfrak{gl}(N), N \rightarrow \infty}\) Limits. Commun. Math. Phys. 165, 281–296 (1994)zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. FFZ89.
    Fairlie D., Fletcher P., Zachos C.: Trigonometric structure constants for new infinite algebras. Phys. Lett. B 218, 203 (1989)zbMATHCrossRefADSMathSciNetGoogle Scholar
  7. GH82.
    Hoppe, J.: Quantum theory of a massless relativistic surface. Ph.D. Thesis (Advisor: J. Goldstone), MIT., 1982
  8. FH94.
    Harary F.: Graph Theory. Addison-Wesley, Reading MA (1969)Google Scholar
  9. Hir76.
    Hirsch M.W.: Differential topology. Springer, New-York (1976)zbMATHGoogle Scholar
  10. Hof02.
    Hofer, L.: Surfaces de Riemann compactes. Master’s thesis, Université de Haute-Alsace Mulhouse,, 2002
  11. Hof07.
    Hofer, L.: Aspects algébriques et quantification des surfaces minimales. Ph.D. thesis, Université de Haute-Alsace de Mulhouse,, June 2007
  12. Hop89/88.
    Hoppe, J.: diffeomorphism groups, quantization, and SU(∞). Int. J. of Mod. Phys. A, 4(19), 5235–5248 (1989); DiffA T 2, and the curvature of some infinite dimensional manifolds. Phys. Lett. B 215, 706–710 (1988)Google Scholar
  13. KL92.
    Klimek, S., Lesniewski, A.: Quantum Riemann surfaces I. The unit disc Commun. Math. Phys. 146, 103–122 (1992); Quantum Riemann surfaces II. The discrete series. Lett. Math. Phys. 24, 125–139 (1992)Google Scholar
  14. Mad92.
    Madore J.: The Fuzzy Sphere. Class. Quant. Grav. 9, 69–88 (1992)zbMATHCrossRefADSMathSciNetGoogle Scholar
  15. NN99.
    Natsume T., Nest R.: Topological approach to quantum surfaces. Commun. Math. Phys. 202, 65–87 (1999)zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. Now97.
    Nowak, C.: \"Uber Sternprodukte auf nichtregulren Poissonmannigfaltigkeiten (Ph.D Thesis, Freiburg University); Star Products for integrable Poisson Structures on \({\mathbb{R}^3}\) ., 1997
  17. Shi04.
    Shimada H.: Membrane topology and matrix regularization. Nucl. Phys. B 685, 297–320 (2004)zbMATHCrossRefADSMathSciNetGoogle Scholar

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

Personalised recommendations