Skip to main content
SpringerLink
Log in

Quantized affine Lie algebras and diagonalization of braid generators

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

Let U q \((\hat{\mathcal{G}})\) be a quantized affine Lie algebra. It is proven that the universal R-matrix R of U q \((\hat{\mathcal{G}})\) satisfies the celebrated conjugation relationR + =TR withT the usual twist map. As applications, the braid generator is shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight U q \((\hat{\mathcal{G}})\)-module and a spectral decomposition formula for the braid generator is obtained which is the generalization of Reshetikhin and Gould forms to the present affine case. Casimir invariants are constructed and their eigenvalues computed by means of the spectral decomposition formula. As a by-product, an interesting identity is found.

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. Reshetikhin, N., Quantized universal enveloping algebras, the Yang-Baxter equation and inveriants of links: I, II, Preprints LOMI E-4-87, E-17-87.

  2. Gould, M. D.,Lett. Math. Phys. 24, 183 (1992); Links, J. R., Gould, M. D., and Zhang, R. B.,Rev. Math. Phys. 5, 345 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  3. Kac, V. G.,Infinite- Dimensional Lie Algebras, Prog. Math. 44, Birkhäuser, Boston, Basel, Stuttgart, 1983.

    Google Scholar 

  4. Drinfeld, V. G.,Proc. ICM, Berkeley 1, 798 (1986).

    Google Scholar 

  5. Jimbo, M.,Lett. Math. Phys. 10, 63 (1985),11, 247 (1986); Topics from representations of U q (277-1) - an introductory guide for physicists, Nankai Lectures, 1991, in M.-L. Ge (ed),Quantum Groups and Quantum Integrable Systems, World Scientific, Singapore (1992).

    Article  ADS  MathSciNet  Google Scholar 

  6. Rosso, M.,Comm. Math. Phys. 117, 581 (1989); Lusztig, G.,Adv. Math. 70, 237 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  7. Zhang, Y.-Z. and Gould, M. D., Unitarity and complete reducibility of certain modules over quantized affine Lie algebras, Univ. Queensland preprint UQMATH-93-02, hepth/9303096;Lett. Math. Phys. 29, 19 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  8. Goddard, P. and Olive, D.,Internat. J. Modern Phys. A 1, 303 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  9. Zhang, R. B., Gould, M. D., and Bracken, A. J.,Comm. Math. Phys. 137, 13 (1991); Gould, M. D., Zhang, R. B., and Bracken, A. J.,J. Math. Phys. 32, 2298 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  10. Frenkel, I. B. and Reshetikhin, N.,Comm. Math. Phys. 146, 1 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  11. Khoroshkin, S. M. and Tolstoy, U. N., The universal R-matrix for quantized nontwisted affine Lie algebras, inProc. 4th Workshop, Obniusk, 1990, to appear inFunktsional Anal. i Prilozhen.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Queensland, Brisbane, 4072, Qld, Australia

    M. D. Gould & Y. -Z. Zhang

Authors
  1. M. D. Gould
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. -Z. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gould, M.D., Zhang, Y.Z. Quantized affine Lie algebras and diagonalization of braid generators. Lett Math Phys 30, 267–277 (1994). https://doi.org/10.1007/BF00751063

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00751063

Mathematics Subject Classifications (1991)

Search

Navigation