Skip to main content
SpringerLink
Log in

Reflection equation, twist, and equivariant quantization

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We prove that the reflection equation (RE) algebraL R associated with a finite dimensional representation of a quasitriangular Hopf algebraH is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show thatL R is a module algebra over the twisted tensor square\({\mathcal{H}}\mathop \otimes \limits^{\mathcal{R}} {\mathcal{H}}\) and the double D(\({\mathcal{H}}\)). We define FRT- and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces.

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.

Similar content being viewed by others

References

  1. A. Alekseev, L. Faddeev and M. Semenov-Tian-Shansky,Hidden quantum group inside Kac-Moody algebra, Proceedings of the Euler International Mathematical Institute on Quantum Groups, Lecture Notes in Mathematics1510, Springer, Berlin, 1992, p. 148.

    Google Scholar 

  2. I. Cherednik,Factorizable particles on a half-line and root systems, Theoretical Mathematical Physics64 (1984), 35.

    Google Scholar 

  3. J. Donin and S. Shnider,Deformations of certain quadratic algebras and the corresponding quantum semigroups, Israel Journal of Mathematics104 (1998), 285–300.

    Article  MATH  MathSciNet  Google Scholar 

  4. V. G. Drinfeld,Quantum Groups, inProceedings of the International Congress of Mathematicians, Berkeley, 1986 (A. V. Gleason, ed.), American Mathematical Society, Providence, RI, 1987, p. 798.

    Google Scholar 

  5. V. G. Drinfeld,Quasi-Hopf algebras, Leningrad Mathematical Journal1 (1990), 1419–1457.

    MathSciNet  Google Scholar 

  6. L. Faddeev, N. Reshetikhin and L. Takhtajan,Quantization of Lie groups and Lie algebras, Leningrad Mathematical Journal1 (1990), 193.

    MATH  MathSciNet  Google Scholar 

  7. P. P. Kulish,Quantum groups, q-oscillators, and covariant algebras, Theoretical Mathematical Physics94 (1993), 193.

    MathSciNet  Google Scholar 

  8. P. P. Kulish and E. K. Sklyanin,Algebraic structure related to the reflection equation, Journal of Physics A25 (1992), 5963.

    Article  MATH  MathSciNet  Google Scholar 

  9. P. P. Kulish and R. Sasaki,Covariance properties of reflection equation algebras, Progress in Theoretical Physics89 (1993), 741.

    Article  MathSciNet  Google Scholar 

  10. S. Majid,Foundations of Quantum Group Theory, Cambridge University Press, 1995.

  11. N. Yu. Reshetikhin and M. A. Semenov-Tian-Shansky,Quantum R-matrices and factorization problem, Journal of Geometry and Physics5 (1988), 533.

    Article  MATH  MathSciNet  Google Scholar 

  12. E. K. Sklyanin,Boundary conditions for integrable quantum systems, Journal of Physics A21 (1988), 2375–2389.

    Article  MathSciNet  MATH  Google Scholar 

  13. L. A. Takhtajan,Introduction to Quantum Groups, International Press Inc., Boston, 1993; Lecture Notes in Physics370 (1989), 3–28.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    J. Donin & A. Mudrov

Authors
  1. J. Donin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Mudrov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Donin.

Additional information

This research is partially supported by the Israel Academy of Sciences grant no. 8007/99-01.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Donin, J., Mudrov, A. Reflection equation, twist, and equivariant quantization. Isr. J. Math. 136, 11–28 (2003). https://doi.org/10.1007/BF02807191

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02807191

Keywords

Search

Navigation