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The quantum 2-sphere as a complex quantum manifold

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Zeitschrift für Physik C Particles and Fields

Abstract

We describe the quantum sphere of Podles for c = 0 by means of a stereographic projection which is analogous to that which exibits the classical sphere as a complex manifold. We show that the algebra of functions and the differential calculus on the sphere are covariant under the coaction of fractional transformations with SU q(2) coefficients as well as under the action of SU q(2) vector fields. Going to the classical limit we obtain the Poisson sphere. Finally, we study the invariant integration of functions on the sphere and find its relation with the translationally invariant integration on the complex quantum plane.

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References

  1. N.Yu. Reshetikhin, L. A. Takhtajan, L. D. Faddeev: Quantization of Lie Groups and Lie Algebras, Alg. i Anal. 1, 178 (1989)

    MathSciNet  Google Scholar 

  2. O. Ogievetsky, B. Zumino: Reality in The Differential Calculus on q-Euclidean Spaces, Lett. Math. Phys. 25, 121 (1992)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. P. Podleś: Quantum Spheres, Lett.Math.Phys. 14, 193 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. P. Podleś: Differential Calculus on Quantum Spheres, Lett. Math. Phys. 18, 107 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. P. Podleś: Differential Calculus on Quantum Spheres, RIMS Kokyuroku Series, No. 751, May (1991)

  6. P. Podleś: The Classification of Differential Structures on Quantum 2-Spheres, Comm. Math. Phys. 150, 167 (1992)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. A.J.-L. Sheu with J.-H. Lu, A. Weinstein: Quantization of Poisson SU(2) and the Two-Sphere, Comm. Math. Phys. 135, 217 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. P. Šť⩈íč≛, Quantum Line Bundles on S2 and Method of Orbits for SUq(2), J. Math. Phys., 34, 1606 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  9. T. Brzezinski, S. Majid: Quantum Group Gauge Theory on Quantum Spaces, Comm. Math. Phys. 157, 591 (1993).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. S.L. Woronowicz: Compact Matrix Pseudogroups, Comm. Math. Phys. 111, 613 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. S.L. Woronowicz: Twisted SU(2) Group. An Example of a Non-Commutative Differential Calculus, Publ.RIMS, Kyoto Univ. 23, 117 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. S.L. Woronowicz: Differential Calculus on Compact Matrix Pseudogroups (Quantum Groups), Comm. Math. Phys. 122, 125 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. B. Zumino: Introduction to the Differential Geometry of Quantum Groups, In: K. Schmüdgen (Ed.), Math. Phys. X, Proc. X-th IAMP Conf. Leipzig (1991), Springer-Verlag (1992)

  14. A. Connes: Non-Commutative Geometry, Academic Press (1994)

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Authors and Affiliations

  1. Department of Physics, University of California, USA

    Chong-Sun Chu & Pei-Ming Ho

  2. Theoretical Physics Group, Lawrence Berkeley Laboratory, University of California, Berkeley, CA, 94720, USA

    Bruno Zumino

Authors
  1. Chong-Sun Chu
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  2. Pei-Ming Ho
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  3. Bruno Zumino
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Corresponding author

Correspondence to Chong-Sun Chu.

Additional information

This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-90-21139

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Chu, CS., Ho, PM. & Zumino, B. The quantum 2-sphere as a complex quantum manifold. Z Phys C - Particles and Fields 70, 339–344 (1996). https://doi.org/10.1007/s002880050111

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  • DOI: https://doi.org/10.1007/s002880050111

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