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Remarks on the Use of R-matrices

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Groups and Related Topics

Part of the book series: Mathematical Physics Studies ((MPST,volume 13))

Abstract

Examples for the application of the R-matrix formalism on problems related to GL(n) quantum groups are given. (1) The Yang-Baxter property of the R-matrix provides a simple means to replace the explicit use of the diamond lemma and thus eliminates lengthy calculations. (2) The Rmatrix description induces a pair of bicovariant differential calculi on the group. Reasons are given that these may be the only suitable ones.

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References

  1. Yu. I. Manin: Quantum groups and non-commutative geometry, CRM University of Montreal (1988).

    Google Scholar 

  2. J. Wess, B. Zumino: Covariant differential calculus on the quantum hyperplane, Nucl. Phys. B (Proc. Suppl.)18B, 302–312, (1990).

    MathSciNet  ADS  MATH  Google Scholar 

  3. A. Schirrmacher, J. Wess, B. Zumino: The two-parameter deformation of GL(2) its differential calculus, and Lie algebra, Z. Phys. C 49, 317, (1991).

    Article  MathSciNet  Google Scholar 

  4. N. Yu. Reshetikhin, L. A. Takhtadzhyan, L. D. Faddeev,: Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1, 193–225, (1990).

    MathSciNet  MATH  Google Scholar 

  5. G. M. Bergman: The diamond lemma for ring theory, Adv. Math., 29, (1978) 178.

    Article  MathSciNet  Google Scholar 

  6. Yu. I. Manin: Quantum groups and non-commutative de Rahm complexes, Bonn preprint MPI/91–47 (1991).

    Google Scholar 

  7. Yu. I. Manin: Notes on quantum groups and quantum de Rahm complexes, Bonn preprint MPI/91–60 (1991).

    Google Scholar 

  8. G. Maltsiniotis: Groupes quantiques et structures differentilles, C.R. Acad. Sci. Paris, 331, (1990), 831.

    MathSciNet  Google Scholar 

  9. G. Maltsiniotis: Calcul differentiell sur le groupe lineaire quantique, exposé, ENS (1990).

    Google Scholar 

  10. E. Corrigan, D. B. Fairlie, P. Fletcher, R. Sasaki: Some aspects of quantum groups and super groups J. Math. Phys., 36, (1990), 776.

    Article  MathSciNet  ADS  Google Scholar 

  11. A. Alekseev, L. Faddeev: (T*;G)t; A toy model for conformal field theory, Commun. Math. Phys. 141, (1991), 413.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. L. Faddeev: Cargése Lectures 1991, to appear at Plenum press.

    Google Scholar 

  13. S. L. Woronowicz: Differential calculus on compact matrix pseudogroups, Commun. Math. Phys., 122, (1989), 125.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. A. Schirrmacher: Multiparameter R-matrices and their quantum groups, J. Phys. A. 50, 321, (1991).

    MathSciNet  Google Scholar 

  15. U. Carow-Watamura, M. Schlieker, S. Watamura, W. Weich: Bicovariant differential calculus on Quantum groups SUq(N) and SOq(N), Commun. Math. Phys. 142, 605 (1991).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. D. Bernard: Quantum Lie algebras and differential calculus on quantum groups, Saclay preprint SPhT-90–124 (1990).

    Google Scholar 

  17. F. Müller-Hoissen Differential calculi on the quantum group GLp,q(2), Göttingen preprint GOET-TP 55/1991.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Max-Planck-Institut für Physik, München, Germany

    Arne Schirrmacher

  2. Föhringer Ring 6, München 40, Germany, W-8000

    Arne Schirrmacher

Authors
  1. Arne Schirrmacher
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Editor information

Editors and Affiliations

  1. Institute of Theoretical Physics, University of Wrocław, Wrocław, Poland

    R. Gielerak , J. Lukierski  & Z. Popowicz ,  & 

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© 1992 Springer Science+Business Media Dordrecht

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Schirrmacher, A. (1992). Remarks on the Use of R-matrices. In: Gielerak, R., Lukierski, J., Popowicz, Z. (eds) Groups and Related Topics. Mathematical Physics Studies, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2801-8_6

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  • DOI: https://doi.org/10.1007/978-94-011-2801-8_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5244-3

  • Online ISBN: 978-94-011-2801-8

  • eBook Packages: Springer Book Archive

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