Letters in Mathematical Physics

, Volume 27, Issue 4, pp 287–300 | Cite as

Remarks on bicovariant differential calculi and exterior Hopf algebras

  • Tomasz Brzeziński


We show that every bicovariant differential calculus over the quantum groupA defines a bialgebra structure on its exterior algebra. Conversely, every exterior bialgebra ofA defines bicovariant bimodule overA. We also study a quasitriangular structure on exterior Hopf algebras in some detail.

Mathematics Subject Classifications (1991)

81R50 16W30 58B30 46L87 05A30 


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  1. 1.
    Abe, E.,Hopf Algebras, Cambridge Univ. Press, 1980.Google Scholar
  2. 2.
    Bourbaki, N.,Elements of Mathematics. Algebra I, Addison-Wesley, New York, 1973.Google Scholar
  3. 3.
    Brzeziński, T., Exterior bialgebras, Cambridge University preprint, 1991.Google Scholar
  4. 4.
    Brzeziński, T., Dabrowski, H., and Rembieliński, J., On the quantum differential calculus and the quantum holomorphicity,J. Math. Phys. 33, 19 (1992).Google Scholar
  5. 5.
    Brzeziński, T. and Majid, S., A class of bicovariant differential calculi on Hopf algebras,Lett. Math. Phys. 26, 67–78 (1992).Google Scholar
  6. 6.
    Carow-Watamura, U., Schlieker, M., Watamura, S., and Weich, W.,Comm. Math. Phys. 142, 605 (1991).Google Scholar
  7. 7.
    Castellani, L., Bicovariant differential calculus on the quantumD = 2 Poincaré group,Phys. Lett. B279, 291 (1992).Google Scholar
  8. 8.
    Connes, A., Non-commutative differential geometry,IHES 62 (1986).Google Scholar
  9. 9.
    Coqueraux, R. and Kastler, D., Remarks on the differential envelopes of associative algebras,Pacific J. Math. 137, 245 (1989).Google Scholar
  10. 10.
    Drinfeld, V. G., Quantum groups, inProc. Internat. Congr. Mathematicians, Berkeley, Calif., Vol. 1, Academic Press, New York, 1986, p. 798.Google Scholar
  11. 11.
    Fadeev, L. D., Reshetikhin, N. Yu., and Takhtajan, L. A., Quantization of Lie groups and Lie algebras,Algebra i Analiz 1 (1989).Google Scholar
  12. 12.
    Feng, P. and Tsygan, B., Hochschild and cyclic homology of quantum groups,Comm. Math. Phys. 142, 481 (1991).Google Scholar
  13. 13.
    Helgason, S.,Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.Google Scholar
  14. 14.
    Jurco, B., Differential calculus on quantized simple Lie groups,Lett. Math. Phys. 22, 177 (1991).Google Scholar
  15. 15.
    Kane, R. M.,The Homology of Hopf Spaces, North-Holland, Amsterdam, 1988.Google Scholar
  16. 16.
    Kastler, D.,Cyclic Cohomology within Differential Envelope, Hermann, Paris, 1988.Google Scholar
  17. 17.
    Majid, S.,C-statistical quantum groups and Weyl algebras,J. Math. Phys. 33, 3431–3344 (1992).Google Scholar
  18. 18.
    Maltsiniotis, G., Groupes quantiques et structures différentielles,C.R. Acad. Sci. Paris, Serie I 311, 831 (1990).Google Scholar
  19. 19.
    Manin, Yu. I.,Quantum Groups and Non-commutative Geometry, Montreal Notes, 1989.Google Scholar
  20. 20.
    Manin, Yu. I., Notes on quantum groups and quantum de Rham complexes, Preprint MPI, 1991.Google Scholar
  21. 21.
    Masuda, T., Nakagami, Y., and Watanabe, J., Noncommutative differential geometry and the quantum spheres of Podleś I: An algebraic viewpoint,K-Theory 5, 151 (1991).Google Scholar
  22. 22.
    Podleś, P., Quantum spheres,Lett. Math. Phys. 14, 117 (1987).Google Scholar
  23. 23.
    Savo, A., Private communication, 1992.Google Scholar
  24. 24.
    Sudbery, A., The algebra of differential forms on a full matric bialgebra, York University preprint, 1991.Google Scholar
  25. 25.
    Sweedler, M. E.,Hopf Algebras, Benjamin, New York, 1969.Google Scholar
  26. 26.
    Wess, J. and Zumino, B., Covariant differential calculus on the quantum hyperplane,Nuclear Phys. B (Proc. Supl.) 18, 302 (1990).Google Scholar
  27. 27.
    Woronowicz, S. L., Twisted SU2 group. An example of a non-commutative differential calculus,Publ. RIMS Kyoto University 23, 117 (1987).Google Scholar
  28. 28.
    Woronowicz, S. L., Differential calculus on compact matric pseudogroups (quantum groups),Comm. Math. Phys. 122, 125 (1989).Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Tomasz Brzeziński
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK

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