Communications in Mathematical Physics

, Volume 156, Issue 2, pp 277–300 | Cite as

Isomorphism of two realizations of quantum affine algebra\(U_q (\widehat{\mathfrak{g}\mathfrak{l}{\text{(}}n{\text{)}}})\)

  • Jintai Ding
  • Igor B. Frenkel


We establish an explicit isomorphism between two realizations of the quantum affine algebra\(U_q (\widehat{\mathfrak{g}\mathfrak{l}{\text{(}}n{\text{)}}})\) given previously by Drinfeld and Reshetikhin-Semenov-Tian-Shansky. Our result can be considered as an affine version of the isomorphism between the Drinfield/Jimbo and the Faddeev-Reshetikhin-Takhtajan constructions of the quantum algebra\(U_q (\mathfrak{g}\mathfrak{l}(n))\).


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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  1. [ABE] Abada, A., Bougourzi, H., El Gradechi, M.A.: Deformation of the Wakimoto construction. CRM-1829, 1992Google Scholar
  2. [D1] Drinfeld, V.G.: Hopf algebras and the Quantum Yang-Baxter equation. Sov. Math. Doklady32, 254–258 (1985)Google Scholar
  3. [D2] Drinfeld, V.G.: Quantum groups. Proc. ICM-86 (Berkeley), Vol. 1, New York: Academic Press 1986, pp. 789–820Google Scholar
  4. [D3] Drinfeld, V.G.: New realization of Yangian and quantum affine algebra. Sov. Math. Doklady36, 212–216 (1988)Google Scholar
  5. [D4] Drinfeld, V.G.: Quasi-Hopf algebras. Leningrad Math. J.1, 1419–1457 (1990)Google Scholar
  6. [D5] Drinfeld, V.G.: Private communicationGoogle Scholar
  7. [DF] Ding, J. Frenkel, I.B.: Spinor and oscillator representations of quantum algebras in invariant form. Preprint, Yale UniversityGoogle Scholar
  8. [DF2] Ding, J., Frenkel, I.B.: In preparationGoogle Scholar
  9. [Di] Dixmier, J.: Enveloping algebras. Amsterdam: North-Holland Publishing Company, 1977Google Scholar
  10. [FRT1] Faddeev, L.D., Reshetikhin, N.Yu., Takhtajan, L.A.: Quantization of Lie groups and Lie algebras. Algebra and Analysis (Russian)1.1, 118–206 (1989)Google Scholar
  11. [FRT2] Faddeev, L.D., Reshetikhin, N.Yu., Takhtajan, L.A.: Quantization of Lie groups and Lie algebras, Yang-Baxter equation in Integrable Systems. Advanced Series in Mathematical Physics, Vol.10, Singapore: World Scientific 1989, pp. 299–309Google Scholar
  12. [FeF] Feigin, B.L., Frenkel, E.V.: The family of representations of affine Lie algebras. Ups. Mat. Nauk43, 227–228 (1988)Google Scholar
  13. [FF] Feingold, A., Frenkel, I.B.: Classical affine algebras. Adv. Math.56, 117–172 (1985)Google Scholar
  14. [F] Frenkel, I.B.: Spinor representation of affine Lie algebras. Proc. Natl. Acad. Sci. USA77, 6303–6306 (1980)Google Scholar
  15. [FK] Frenkel, I.B., Kac, V.G.: Baqsic representations of affine Lie algebras and dual resonance model. Invent. Math.62, 23–66 (1980)Google Scholar
  16. [FJ] Frenkel, I.B., Jing, N.: Vertex representation of quantum affine algebras. Proc. Natl. Acad. Sci. USA85, 9373–9377 (1988)Google Scholar
  17. [G] Garland, H.: The arithmetic theory of loop groups. Publ. Math. IHES52, 5–136 (1980)Google Scholar
  18. [GK] Gabber, O., Kac, V.G.: On defining relations of certain infinite-dimensional Lie algebras. Bull. Am. Math. Soc.5, 185–189 (1981)Google Scholar
  19. [H] Hayashi, T.:Q-analogues of Clifford and Weyl algebras — Spinor and Oscillator representations of quantum enveloping algebra. Commun. Math. Phys.127, 129–144 (1990)Google Scholar
  20. [J1] Jimbo, M.: Aq-difference analogue ofU(g) and the Yang-Baxter equation. Lett. Math. Phys.10, 63–69 (1985)Google Scholar
  21. [J2] Jimbo, M.: QuantumR-matrix for the generalized Toda system. Commun. Math. Phys.102, 537–548 (1986)Google Scholar
  22. [J3] Jimbo, M.: Aq-analog of\(U(\mathfrak{g}\mathfrak{l}(N + 1))\), Hecke algebras, and the Yang-Baxter equation. Lett. Math. Phys.11, 247–252 (1986)Google Scholar
  23. [KP] Kac, V.G., Peterson, D.H.: Spinor and wedge representations of infinite-dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA78, 3308–3312 (1981)Google Scholar
  24. [L] Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math.70, 237–249 (1988)Google Scholar
  25. [M] Matsuo, A.: Free field representation of quantum affine algebra\(U_q \widehat{(\mathfrak{s}\mathfrak{l}{\text{)}}}\). Preprint, 1992Google Scholar
  26. [Re] Reshetikhin, N.Yu.: Private communicationGoogle Scholar
  27. [R1] Rosso, M.: Finite dimensional representations of the quantum analogue of the universal enveloping algebgra of a complex simple Lie algebra. Commun. Math. Phys.117, 558–593 (1988)Google Scholar
  28. [R2] Rosso, M.: An analogue of P.B.W. theorem and the universal R-matrix for\(U_h \mathfrak{s}\mathfrak{l}{\text{(}}N + 1{\text{)}}\). Commun. Math. Phys.124, 307–318 (1989)Google Scholar
  29. [RS] Reshetikhin, N.Yu., Semenov-Tian-Shansky, M.A.: Central extensions of quantum current groups. Lett. Math. Phys.19, 133–142 (1990)Google Scholar
  30. [S] Segal, G.: Unitary representation of some infinite dimensional groups. Commun. Math. Phys.80, 301–342 (1981)Google Scholar
  31. [TK] Tsuchiya, A., Kanie, Y.: Vertex operators in conformal, field theory onP 1 and monodromy representation of braid group. Adv. Stud. Pure Math.16, 297–372 (1988)Google Scholar
  32. [W] Wakimoto, M.: Fock representations of affine Lie algebraA 1(1). Commun. Math. Phys.104, 605–609 (1986)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Jintai Ding
    • 1
  • Igor B. Frenkel
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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