Skip to main content
SpringerLink
Log in

Representations of Multiparameter Quantum Groups

  • Elementary Particles and Fields
  • Theory
  • Published:
Physics of Atomic Nuclei Aims and scope Submit manuscript

Abstract

We construct representations of the quantum algebras Uq,q(gl(n)) and Uq,q(sl(n)) which are in duality with the multiparameter quantum groups GLqq(n), SLqq(n), respectively. These objects depend on n(n − 1)/2+ 1 deformation parameters q, qij (1 ≤ i< j ≤ n) which is the maximal possible number in the case of GL(n). The representations are labelled by n − 1 complex numbers ri and are acting in the space of formal power series of n(n − 1)/2 non-commuting variables. These variables generate quantum flag manifolds of GLqq(n), SLqq(n). The case n = 3 is treated in more detail.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. G. Drinfeld, Soviet. Math. Dokl. 32, 254 (1985).

    Google Scholar 

  2. M. Jimbo, Lett. Math. Phys. 10, 63 (1985); 11, 247 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  3. G. Lusztig, Adv. Math. 70, 237 (1988).

    Article  Google Scholar 

  4. Yu. I. Manin, Commun. Math. Phys. 123, 163 (1989).

    Article  ADS  Google Scholar 

  5. S. L. Woronowicz, Commun. Math. Phys. 122, 125 (1989).

    Article  ADS  Google Scholar 

  6. L. C. Biedenharn, J. Phys. A 22, L873 (1989).

    Article  ADS  Google Scholar 

  7. N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, Alg. Anal. 1, 178 (1989) [Leningrad Math. J. 1, 193 (1990)].

    Google Scholar 

  8. C. De Concini and V. G. Kac, in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Prog. Math. 92, Ed. by A. Connes, M. Duflo, A. Joseph, and R. Rentschler (Birkhäuser, 1990), p. 471.

  9. S. Majid, Int. J. Mod. Phys. A 5, 1 (1990).

    Article  ADS  Google Scholar 

  10. P. P. Kulish and E. K. Sklyanin, J. Phys. A 24, L435 (1991).

    Article  ADS  Google Scholar 

  11. V. K. Dobrev, Invariant Differential Operators, Vol. 2: Quantum Groups, De Gruyter Studies in Mathematical Physics, vol. 39 (De Gruyter, Berlin, Boston, 2017).

    Google Scholar 

  12. V. K. Dobrev, Rep. Math. Phys. 25, 159 (1988).

    Article  ADS  Google Scholar 

  13. V. K. Dobrev, J. Math. Phys. 33, 3419 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  14. V. K. Dobrev, J. Phys. A 27, 4841, 6633 (Erratum) (1994).

    Article  ADS  MathSciNet  Google Scholar 

  15. A. Sudbery, J. Phys. A 23, L697 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  16. V. K. Dobrev and P. Parashar, J. Phys. A 26, 6991 (1993); 32, 443 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  17. V. K. Dobrev, G. Mack, V. B. Petkova, S. G. Petrova, and I. T. Todorov, Rep. Math. Phys. 9, 219 (1976).

    Article  ADS  Google Scholar 

  18. V. K. Dobrev and P. J. Moylan, Phys. Lett. B 315, 292 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  19. V. K. Dobrev, J. High Energy Phys. 1302, 015 (2013).

    Article  ADS  Google Scholar 

  20. V. K. Dobrev, in preparation.

Download references

Author information

Authors and Affiliations

  1. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, 1784, Bulgaria

    V. K. Dobrev

Authors
  1. V. K. Dobrev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. K. Dobrev.

Additional information

The text was submitted by the author in English.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dobrev, V.K. Representations of Multiparameter Quantum Groups. Phys. Atom. Nuclei 81, 826–831 (2018). https://doi.org/10.1134/S1063778818060121

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063778818060121

Search

Navigation