Advertisement

Communications in Mathematical Physics

, Volume 263, Issue 1, pp 65–88 | Cite as

A Hopf Bundle Over a Quantum Four-Sphere from the Symplectic Group

  • Giovanni LandiEmail author
  • Chiara Pagani
  • Cesare Reina
Article

Abstract

We construct a quantum version of the SU(2) Hopf bundle S 7S 4. The quantum sphere S 7 q arises from the symplectic group Sp q (2) and a quantum 4-sphere S 4 q is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(S 4 q ) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S 4. We compute the fundamental K-homology class of S 4 q and pair it with the class of p in the K-theory getting the value −1 for the topological charge. There is a right coaction of SU q (2) on S 7 q such that the algebra A(S 7 q ) is a non-trivial quantum principal bundle over A(S 4 q ) with structure quantum group A(SU q (2)).

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Polynomial Function 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah, M.: The geometry of Yang-Mills fields. Lezioni Fermiane. Accademia Nazionale dei Lincei e Scuola Normale Superiore, Pisa 1979Google Scholar
  2. 2.
    Belavin, A., Polyakov, A., Schwartz, A., Tyupkin, Y.: Pseudoparticles solutions of the Yang-Mills equations. Phys. Lett. 58 B, 85–87 (1975)MathSciNetGoogle Scholar
  3. 3.
    Bonechi, F., Ciccoli, N., Tarlini, M.: Noncommutative instantons on the 4-sphere from quantum groups. Commun. Math. Phys. 226, 419–432 (2002)CrossRefADSzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bonechi, F., Ciccoli, N., Dabrowski, L., Tarlini, L.M.: Bijectivity of the canonical map for the non-commutative instanton bundle. J. Geom. Phys. 51, 71–81 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Brzeziński, T., Dabrowski, L., Zielinski, B.: Hopf fibration and monopole connection over the contact quantum spheres. J. Geom. Phys. 50, 345–359 (2004)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Brzeziński, T., Hajac, P.M.: Coalgebra extensions and algebra coextensions of Galois type. Commun. Algebra 27, 1347–1368 (1999)Google Scholar
  7. 7.
    Brzeziński, T., Hajac, P.M.: The Chern-Galois character. C. R. Acad. Sci. Paris, Ser. I 333, 113–116 (2004)Google Scholar
  8. 8.
    Brzeziński, T., Majid, S.: Quantum group gauge theory on quantum spaces. Commun. Math. Phys. 157, 591–638 (1993) Erratum 167, 235 (1995)CrossRefADSGoogle Scholar
  9. 9.
    Brzeziński, T., Majid, S.: Coalgebra Bundles. Commun. Math. Phys. 191, 467–492 (1998)CrossRefADSGoogle Scholar
  10. 10.
    Connes, A.: Noncommutative geometry. London-New York: Academic Press, 1994Google Scholar
  11. 11.
    Durdevich, M.: Geometry of quantum principal bundles I. Commun. Math. Phys. 175, 427–521 (1996); Geometry of quantum principal bundles II. Rev. Math. Phys. 9, 531–607 (1997)ADSGoogle Scholar
  12. 12.
    Dabrowski, L., Grosse, H., Hajac, P.M.: Strong connections and Chern-Connes pairing in the Hopf-Galois theory. Commun. Math. Phys. 206, 247–264 (1999)CrossRefGoogle Scholar
  13. 13.
    Hajac, P.M.: Strong connections on quantum principal bundles. Commun. Math. Phys. 182, 579–617 (1996)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Hajac, P.M., Majid, S.: Projective module description of the q-monopole. Commun. Math. Phys. 206, 247–264 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hajac, P.M., Matthes, R., Szymański, W.: A locally trivial quantum Hopf fibration. http://arXiv.org/list/math.QA/0112317, 2001; to appear in Algebra and Representation Theory
  16. 16.
    Landi, G.: Deconstructing monopoles and instantons. Rev. Math. Phys. 12, 1367–1390 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Majid, S.: Quantum and braided group Riemannian geometry. J. Geom. Phys. 30, 113–146 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Kassel, C.: Quantum groups. Berlin-Heidelberg-New York: Springer 1995Google Scholar
  19. 19.
    Klimyk, A., Schmüdgen, K.: Quantum groups and their representations. Berlin-Heidelberg: Springer Verlag, 1997Google Scholar
  20. 20.
    Kreimer, H.F., Takeuchi, M.: Hopf algebras and Galois extensions of an algebra. Indiana Univ. Math. J. 30, 675–692 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum SU(2). I:An algebraic viewpoint. K-Theory 4, 157–180 (1990); Noncommutative differential geometry on the quantum two sphere of P.Podleś. I: An algebraic viewpoint. K-Theory 5, 151–175 (1991)Google Scholar
  22. 22.
    Montgomery, S.: Hopf algebras and their actions on rings. Providence, RI: AMS 1993Google Scholar
  23. 23.
    Pagani, C.: In preparationGoogle Scholar
  24. 24.
    Podleś, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Reshetikhin, N.Yu., Takhtadzhyan, L.A., Faddeev, L.D.: Quantization of Lie groups and Lie algebras. Leningrad Math. J. 1, 193–225 (1990)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Schauenburg, P.: Bi-Galois objects over Taft algebras. Israel J. Math. 115, 101–123 (2000)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Schauenburg, P., Schneider, H.: Galois type extensions of noncommutative algebras. In preparationGoogle Scholar
  28. 28.
    Schneider, H.: Principal homogeneous spaces for arbitrary Hopf algebras. Israel J. Math. 72, 167–195 (1990)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Simon, B.: Trace ideals and their applications. Cambridge: Cambridge Univ. Press, 1979Google Scholar
  30. 30.
    Woronowicz, S.L.: Twisted SU(2) group. An example of a noncommutative differential calculus. Publ. Res. Inst. Math. Sci. 23, 117–181 (1987)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di TriesteTriesteItaly
  2. 2.S.I.S.S.A. International School for Advanced StudiesTriesteItaly
  3. 3.I.N.F.NNapoliItaly

Personalised recommendations