Communications in Mathematical Physics

, Volume 166, Issue 2, pp 279–315 | Cite as

Quantum groups on fibre bundles

  • Markus J. Pflaum


It is shown that the principle of locality and noncommutative geometry can be connected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. Within the language of quantum spaces noncommutative principal and vector bundles are defined and their properties are studied. Important constructions in the classical theory of principal fibre bundles like associated bundles and differential calculi are carried over to the quantum case. At the endq-deformed instanton models are introduced for every integral index.


Neural Network Mathematical Physic Nonlinear Dynamics Vector Bundle Classical Theory 
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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  1. 1.
    Abe, E.: Hopf Algebras. Cambridge Tracts in Math. N74. Cambridge, New York: Cambridge University Press, 1980Google Scholar
  2. 2.
    Blattner, R.J., Cohen, M., Montgomery, S.: Crossed products and inner actions of Hopf algebras. Trans. Am. Math. Soc.298, 671–711 (1986)Google Scholar
  3. 3.
    Blattner, R.J., Montgomery, S.: Crossed Product and Galois Extensions of Hopf Algebras. Pac. J. Math. Vol.137, No. 1. 37–54 (1989)Google Scholar
  4. 4.
    Brzezinski, T., Majid, S.: Quantum group gauge theory on quantum spaces. Commun. Math. Phys.157, No. 3 (1993)Google Scholar
  5. 5.
    Connes, A.: Non-Commutative Differential Geometry. Publ. Math. IHES62 (1985)Google Scholar
  6. 6.
    Daniel, M., Viallet, C.M.: The geometrical setting of gauge theories of the Yang-Mills type. Rev. Mod. Phys.52, No. 1 (1980)Google Scholar
  7. 7.
    Drinfeld, M.: Quantum Groups. Proc. of the ICM 1986, Berkeley (1986), pp. 798–820Google Scholar
  8. 8.
    Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature and Cohomology, Volume I. New York. London: Academic Press, 1972Google Scholar
  9. 9.
    Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature and Cohomology, Volume II. New York, London: Academic Press, 1973Google Scholar
  10. 10.
    Grothendieck, A.: A General Theory of Fibre Spaces with Structure Sheaf. University of Kansas, Department of Mathematics, Report No.4 (Lawrence, Kansas 19652)Google Scholar
  11. 11.
    Haag, R.: Local Quantum Physics. Texts and Monographs in Physics, Berlin, Heidelberg, New York: Springer, 1992Google Scholar
  12. 12.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, Berlin, Heidelberg, New York: Springer, 1977Google Scholar
  13. 13.
    Itzykson, C., Zuber, J.B.: Quantum Field Theory. New York: McGraw-Hill, 1980Google Scholar
  14. 14.
    Kasper, U.: Fibre Bundles: An Introduction to Concepts of Modern Differential Geometry. In: Geometry and Theoretical Physics, eds. Debrus, J., Hirshfeld, A.C., Berlin, Heidelberg, New York: Springer, 1991Google Scholar
  15. 15.
    Kaup, L. Kaup, R.: Holomorphic Functions of Several Variable. de Gruyter Studies in Mathematics, Berlin, New York, 1983Google Scholar
  16. 16.
    Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic. Berlin, Heidelberg, New York: Springer-Verlag 1992Google Scholar
  17. 17.
    Manin, Y.I.: Quantum Group and Non-Commutative GEometry. Montreal Lecture Notes (1988)Google Scholar
  18. 18.
    Manin, Y.I.: Gauge Field Theory and Complex Geometry. Grundlehren der mathematischen Wissennschaften289, Berlin, Heidelberg, New York: Springer, 1988Google Scholar
  19. 19.
    Pflaum, M.J.: Quantengruppen auf Faserbündeln. Diplomarbeit and der Sektion Physik der Universität München, München, 1. November 1992Google Scholar
  20. 20.
    Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics and all that. New York, Amsterdam: Academic Press, 1964Google Scholar
  21. 21.
    Tennison, B.R.: Sheaf Theory. Cambridge University Press, London Math. So. Lecture Note Series20 Cambridge 1975Google Scholar
  22. 22.
    Wess, J., Bagger, J.: Supersymmetry and supergravity. Princeton, N.J.: University Press, 1983Google Scholar
  23. 23.
    Wess, J., Zumino, B.: Covariant Differential Calculus on the Quantum Hyperphlane. preprint CERN-TH-5697/90 (April 1990)Google Scholar
  24. 24.
    Woronowicz, S.L.: Differential Calculus on Compact Matrix Pseudogroups. Commun. Math. Phys.122, 125–170 (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Markus J. Pflaum
    • 1
  1. 1.Mathematisches Institut der Universität MünchenMünchenGermany

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