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Laplacians and gauged Laplacians on a quantum Hopf bundle

  • Alessandro Zampini

Abstract

This paper presents an analysis of the set of connections and covariant derivatives on a U(1) quantum Hopf bundle on the standard quantum sphere \( S_q^2 \), whose total space algebra SUq(2) is equipped with the 3d left covariant differential calculus by Woronowicz. The introduction of a Hodge duality on both \( \Omega \left( {S{U_q}\left( 2 \right)} \right) \) and on \( \Omega \left( {S_q^2} \right) \) allows for the study of Laplacians and of gauged Laplacians.

Keywords

Hopf Algebra Quantum Group Principal Bundle Hodge Structure High Weight Vector 
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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References

  1. [1]
    [1] R. Abraham, J.E. Marsden, T. Ratiu, Manifolds, tensor analysis and applications, App. Math. Scie. 75, Springer 1988.Google Scholar
  2. [2]
    [2] P. Aschieri, L. Castellani, An introduction to non-commutative differential geometry on quantum groups, Int.J.Mod.Phys. A8 (1993) 1667–1706.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    [3] P. Baum, P.M. Hajac, R. Matthes, W. Szymanski, Noncommutative geometry approach to principal and associated bundles, arXiv:math/0701033.Google Scholar
  4. [4]
    [4] N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer 1991.Google Scholar
  5. [5]
    [5] T. Brzezinski, Quantum fibre bundles. An introduction, Banach Center Publications, Warsaw (1995), arXiv:q-alg/9508008.Google Scholar
  6. [6]
    [6] T. Brzezinski, S. Majid, Quantum group gauge theory on quantum spaces, Comm. Math. Phys. 157 (1993) 591–638; Erratum 167 (1995) 235.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    [7] T. Brzezinski, S. Majid, Quantum differential and the q-monopole revisited, Acta Appl. Math. 54 (1998) 185–233.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    [8] T. Brzezinski, S. Majid, Line bundles on quantum spheres, AIP Conf. Proc. 345 (1998) 3–8.MathSciNetGoogle Scholar
  9. [9]
    [9] C.-S. Chu, P.-M. Ho, H. Steinacker, q-deformed Dirac monopole with arbitrary charge, Z. Phys. C71 (1996) 171–177.MathSciNetGoogle Scholar
  10. [10]
    [10] P.A.M. Dirac, Proc.R.Soc A 133 (1931) 60.Google Scholar
  11. [11]
    [11] F. D’Andrea, G. Landi, Anti-self dual connections on the quantum projective plane: monopole, arXiv:0903.3551.Google Scholar
  12. [12]
    [12] M. Durdevic, Geometry of quantum principal bundles I, Comm.Math.Phys. 175 (1996), 457–520.CrossRefMathSciNetGoogle Scholar
  13. [13]
    [13] M. Durdevic, Geometry of quantum principal bundles II, Rev.Math.Phys 9 (1997) 531–607.CrossRefMathSciNetGoogle Scholar
  14. [14]
    [14] M. Durdevic, Quantum principal bundles as Hopf-Galois extensions, arXiv:q-alg/9507022.Google Scholar
  15. [15]
    [15] M. Göckeler, T. Schücker, Differential geometry, gauge theories and gravity, Cambridge University Press 1987.Google Scholar
  16. [16]
    [16] P.M.Hajac, Strong connections on quantum principal bundles, Comm.Math.Phys. 182 (1996) 579–617.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    [17] P.M. Hajac, S. Majid, Projective module description of the q-monopole, Comm. Math. Phys. 206 (1999) 247–264.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    [18] H. Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Mathematische Annalen 104 (1931) 637–635.CrossRefMathSciNetGoogle Scholar
  19. [19]
    [19] H. Hopf, Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension, Fundamenta Mathematicae 25 (1935) 427–440Google Scholar
  20. [20]
    [20] D. Husemoller, Principal Bundles, Springer-Verlag - New York 1998.Google Scholar
  21. [21]
    [21] A. Klimyk, K. Schmüdgen, Quantum Groups and Their Representations, Springer 1997.Google Scholar
  22. [22]
    [22] J. Kustermans, G.J. Murphy, L. Tuset, Quantum groups, differential calculi and the eigenvalues of the Laplacian, Trans.Amer.Math.Soc. 357(2005) 4681–4717.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    [23] G. Landi, Projective modules of finite type and monopoles over S2, J.Geom.Phys. 37 (2001) 47–62.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    [24] G. Landi, C. Reina, A. Zampini, Gauged Laplacians on quantum Hopf bundles, Comm.Math.Phys. 287 (2009) 179–209.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    [25] S. Majid, Foundations of Quantum Group Theory, Cambridge Univ. Press, 1995.Google Scholar
  26. [26]
    [26] S. Majid, Noncommutative Riemannian and spin geometry of the standard q-sphere, Comm. Math. Phys. 256 (2005) 255–285.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    [27] T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, K. Ueno, Representations of the Quantum Group SUq(2) and the Little q-Jacobi Polynomials, J. Funct. Anal. 99 (1991) 357–387.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    [28] P.W. Michor, Topics in differential geometry, Graduate Studies in Mathematics vol 93, AMS 2008.Google Scholar
  29. [29]
    [29] J.A. Mignaco, C. Sigaud, A.R. da Silva, F.J. Vanhecke, The Connes-Lott program on the sphere, Rev.Math.Phys. 9 (1997) 689–718.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    [30] P. Podleś, Quantum spheres, Lett. Math. Phys. 14 (1987) 193–202.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    [31] K. Schmüdgen, E. Wagner, Representations of cross product algebras of Podleś quantum spheres, J. Lie Theory 17 (2007) 751–790.MATHMathSciNetGoogle Scholar
  32. [32]
    [32] K. Schmüdgen, E.Wagner, Dirac operator and a twisted cyclic cocycle on the standard Podleś quantum sphere, J. reine angew. Math. 574 (2004) 219–235.MATHMathSciNetGoogle Scholar
  33. [33]
    [33] H. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990) 167–195.MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    [34] R.G. Swan, Vector bundles and projective modules, Trans.Am.Math.Soc. 105 (1962) 264–277.MATHMathSciNetGoogle Scholar
  35. [35]
    [35] A. Trautman, Solutions of the Maxwell and Yang-Mills equations associated with Hopf fibrings, Int.J.Theor.Phys. 16, 8 (1977) 561–565.CrossRefGoogle Scholar
  36. [36]
    [36] D.A. Varshalovic, A.N. Moskalev, V.K. Khersonskii, Quantum theory of angular momentum, World Scientific 1988.Google Scholar
  37. [37]
    [37] S.L. Woronowicz, Twisted SUq(2) group. An example of a noncommutative differential calculus, Publ. Rest. Inst. Math.Sci., Kyoto Univ. 23 (1987) 117–181.MATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    [38] S.L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989) 125–170.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011

Authors and Affiliations

  • Alessandro Zampini
    • 1
  1. 1.Hausdorff Zentrum für MathematikUniversität BonnBonnGermany

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