Skip to main content
SpringerLink
Log in

On the FRTS Approach to Quantized Current Algebras

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

Abstract

We study the possibility to establish L-operator's formalism by Faddeev–Reshetikhin–Takhtajan–Semenov-Tian-Shansky (FRST) for quantized current algebras, that is, for quantum affine algebras in the ‘new realization’ by V. Drinfeld with the corresponding Hopf algebra structure and for their Yangian counterpart. We establish this formalism using the twisting procedure by Tolstoy and the second author and explain the problems which on FRST approach encounters for quantized current algebras. We also show that, for the case of Uq(ŝln), entries of the L-operators of the FRTS type give the Drinfeld current operators for the nonsimple roots, which we discovered recently. As an application, we deduce the commutation relations between these current operators for Uq(ŝl3).

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. Ding, J. and Feigin, B.: Quantum current operators (III): Commutative quantum current operators, semi-infinite construction and functional models, q-alg/9612009.

  2. Ding, J. and Feigin, B.: Drinfeld realization and it KZ-equation, in preparation.

  3. Ding, J. and Frenkel, B.: Isomorphism of two realizations of quantum affine algebra Uq (\({\mathfrak{g}}^ \wedge {\mathfrak{l}}\)((n)), Comm. Math. Phys. 156, (1993), 277-300.

    Google Scholar 

  4. Ding, J. and Iohara, K.: Generalization and deformation of Drinfeld quantum affine algebras, q-alg/9608002, RIMS-1091

  5. Ding, J. and Iohara, K.: Drinfeld comultiplication and vertex operators, q-alg/9608003.

  6. Ding, J. and Khoroshkin, S.: Weyl group extension of quantized current algebras, qalg/ 9804xxx.

  7. Ding, J. and Miwa, T.: Zeros and poles of quantum current operators and the condition of quantum integrability, q-alg/9608001, RIMS-1092.

  8. Drinfeld, V. G.: New realization of Yangian and quantum affine algebra, Soviet Math. Dokl. 36(1988), 212-216.

    Google Scholar 

  9. Drinfeld, V. G.: Hopf algebra and the quantum Yang-Baxter equation, Dokl. Akad. Nauk. SSS 283(1985).

  10. Drinfeld, V. G.: Quasi-Hopf algebra, Algebra Anal. (Petersburg Math. J.)(1990), 1419-1457.

  11. Drinfeld, V. G.: Quantum groups, Proc. ICM-86 (Berkeley USA) vol. 1, Amer. Math. Soc. Providence, 1987, pp. 798-820.

    Google Scholar 

  12. Enriquez, B. and Felder, G.: Elliptic quantum groups E τ,η (sl 2 ) and quasi-Hopf algebras, qalg/ 9703018.

  13. Enriquez, B. and Felder, G.: A construction of Hopf algebra cocycles for the double Yangian DY(SL2), q-alg/9703012.

  14. Frenkel, I. B. and Reshetikhin, N. Yu: Quantum affine algebras and holomorphic difference equation, Comm. Math. Phys. 146(1992), 1-60.

    Google Scholar 

  15. Faddeev, L. D., Reshetikhin, N. Yu and Takhtajan, L. A.: Quantization of Lie groups and Lie algebras, in: Yang-Baxter Equation in Integrable Systems, Adv. Series Math. Phys. 10, World Scientific, Singapore, 1989, 299-309.

    Google Scholar 

  16. Kedem, R.: Singular R-matrices and Drinfeld’s comultiplication, q-alg/9611001.

  17. Khoroshkin, S., Lebedev, D. and Pakuliak, S.: Elliptic algebra A q,p (s \(\hat l\) 2 )in the scaling limit, q-alg/970200.

  18. Khoroshkin, S., Stolin, A. and Tolstoy, V.: Generalized Gauss decomposition of trigonometric R-matrices, Modern Phys. Lett. A, 19(1995), 1375-1392.

    Google Scholar 

  19. Khoroshkin, S. M. and Tolstoy, V. N.: Twisting of quantum (super)algebras. Connection of Drinfeld’s and Cartan-Weyl realizations for quantum affine algebras, hep-th/9404036.

  20. Khoroshkin, S. M. and Tolstoy, V. N.: Universal R-matrix for quantized (super)algebras, Comm. Math. Phys. 141(1991), 599-617.

    Google Scholar 

  21. Khoroshkin, S. M. and Tolstoy, V. N.: Yangian double, Lett. Math. Phys. 36(1996), 373-402.

    Google Scholar 

  22. Levendorskii, S. Z. and Soibelman, Ya. S.: Some application of quantum Weyl groups. The multiplicative formula for universal R-matrix for simple Lie algebras, Geom. Phys 7(4) (1990), 1-14.

    Google Scholar 

  23. Lusztig, G.: Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3(1990), 447-498.

    Google Scholar 

  24. Reshetikhin, N. Yu.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys. 20(1990), 331-335.

    Google Scholar 

  25. Reshetikhin, N. Yu and Semenov-Tian-Shansky, M. A.: Central extensions of quantum current groups, Lett. Math. Phys. 19(1990).

Download references

Author information

Authors and Affiliations

  1. RIMS, Kyoto University, Kyoto, 606, Japan

    Jintai Ding

  2. ITEP, 117259, Moscow, Russia

    Sergei Khoroshkin

Authors
  1. Jintai Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergei Khoroshkin
    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

Ding, J., Khoroshkin, S. On the FRTS Approach to Quantized Current Algebras. Letters in Mathematical Physics 45, 331–352 (1998). https://doi.org/10.1023/A:1007479807541

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1007479807541

Search

Navigation