Isomorphism of two realizations of quantum affine algebra \(U_ q(\widehat{\mathfrak{gl}}(n))\). (English) Zbl 0786.17008
There are known at least four different descriptions of the quantum affine algebra \(U_q(\widehat {\mathfrak g})\): (1) in terms of Chevalley generators by Drinfeld-Jimbo; (2) the so called “new” realization by Drinfeld where the whole currents are quantized; (3) in terms of \({\mathcal L}\)-operator by Semenov-Tian-Shansky, Reshetikhin and Frenkel; and (4) in terms of Cartan-Weyl generators by Khoroshkin-Tolstoĭ.
Here an explicit isomorphism of the forms (2) and (3) is established for the case of \(U_ q(\widehat {\mathfrak{gl}}_ n)\). More precisely, it is proved that Drinfeld generators are the entries of the Gauss decomposition of the corresponding \(\mathcal L\)-operators.
Here an explicit isomorphism of the forms (2) and (3) is established for the case of \(U_ q(\widehat {\mathfrak{gl}}_ n)\). More precisely, it is proved that Drinfeld generators are the entries of the Gauss decomposition of the corresponding \(\mathcal L\)-operators.
Reviewer: Sergeĭ Khoroshkin (Moskva)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 16T20 | Ring-theoretic aspects of quantum groups |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
References:
| [1] | [ABE] Abada, A., Bougourzi, H., El Gradechi, M.A.: Deformation of the Wakimoto construction. CRM-1829, 1992 · Zbl 0929.17015 |
| [2] | [D1] Drinfeld, V.G.: Hopf algebras and the Quantum Yang-Baxter equation. Sov. Math. Doklady32, 254–258 (1985) |
| [3] | [D2] Drinfeld, V.G.: Quantum groups. Proc. ICM-86 (Berkeley), Vol. 1, New York: Academic Press 1986, pp. 789–820 |
| [4] | [D3] Drinfeld, V.G.: New realization of Yangian and quantum affine algebra. Sov. Math. Doklady36, 212–216 (1988) |
| [5] | [D4] Drinfeld, V.G.: Quasi-Hopf algebras. Leningrad Math. J.1, 1419–1457 (1990) |
| [6] | [D5] Drinfeld, V.G.: Private communication |
| [7] | [DF] Ding, J. Frenkel, I.B.: Spinor and oscillator representations of quantum algebras in invariant form. Preprint, Yale University |
| [8] | [DF2] Ding, J., Frenkel, I.B.: In preparation |
| [9] | [Di] Dixmier, J.: Enveloping algebras. Amsterdam: North-Holland Publishing Company, 1977 · Zbl 0339.17007 |
| [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) |
| [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. 299–309 |
| [12] | [FeF] Feigin, B.L., Frenkel, E.V.: The family of representations of affine Lie algebras. Ups. Mat. Nauk43, 227–228 (1988) |
| [13] | [FF] Feingold, A., Frenkel, I.B.: Classical affine algebras. Adv. Math.56, 117–172 (1985) · Zbl 0601.17012 · doi:10.1016/0001-8708(85)90027-1 |
| [14] | [F] Frenkel, I.B.: Spinor representation of affine Lie algebras. Proc. Natl. Acad. Sci. USA77, 6303–6306 (1980) · Zbl 0451.17004 · doi:10.1073/pnas.77.11.6303 |
| [15] | [FK] Frenkel, I.B., Kac, V.G.: Baqsic representations of affine Lie algebras and dual resonance model. Invent. Math.62, 23–66 (1980) · Zbl 0493.17010 · doi:10.1007/BF01391662 |
| [16] | [FJ] Frenkel, I.B., Jing, N.: Vertex representation of quantum affine algebras. Proc. Natl. Acad. Sci. USA85, 9373–9377 (1988) · Zbl 0662.17006 · doi:10.1073/pnas.85.24.9373 |
| [17] | [G] Garland, H.: The arithmetic theory of loop groups. Publ. Math. IHES52, 5–136 (1980) · Zbl 0475.17004 |
| [18] | [GK] Gabber, O., Kac, V.G.: On defining relations of certain infinite-dimensional Lie algebras. Bull. Am. Math. Soc.5, 185–189 (1981) · Zbl 0474.17007 · doi:10.1090/S0273-0979-1981-14940-5 |
| [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) · Zbl 0701.17008 · doi:10.1007/BF02096497 |
| [20] | [J1] Jimbo, M.: Aq-difference analogue ofU(g) and the Yang-Baxter equation. Lett. Math. Phys.10, 63–69 (1985) · Zbl 0587.17004 · doi:10.1007/BF00704588 |
| [21] | [J2] Jimbo, M.: QuantumR-matrix for the generalized Toda system. Commun. Math. Phys.102, 537–548 (1986) · Zbl 0604.58013 · doi:10.1007/BF01221646 |
| [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) · Zbl 0602.17005 · doi:10.1007/BF00400222 |
| [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) · Zbl 0469.22016 · doi:10.1073/pnas.78.6.3308 |
| [24] | [L] Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math.70, 237–249 (1988) · Zbl 0651.17007 · doi:10.1016/0001-8708(88)90056-4 |
| [25] | [M] Matsuo, A.: Free field representation of quantum affine algebra \(U_q \widehat{(\mathfrak{s}\mathfrak{l}{\text{)}}}\) . Preprint, 1992 |
| [26] | [Re] Reshetikhin, N.Yu.: Private communication |
| [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) · Zbl 0651.17008 · doi:10.1007/BF01218386 |
| [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) · Zbl 0694.17006 · doi:10.1007/BF01219200 |
| [29] | [RS] Reshetikhin, N.Yu., Semenov-Tian-Shansky, M.A.: Central extensions of quantum current groups. Lett. Math. Phys.19, 133–142 (1990) · Zbl 0692.22011 · doi:10.1007/BF01045884 |
| [30] | [S] Segal, G.: Unitary representation of some infinite dimensional groups. Commun. Math. Phys.80, 301–342 (1981) · Zbl 0495.22017 · doi:10.1007/BF01208274 |
| [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) |
| [32] | [W] Wakimoto, M.: Fock representations of affine Lie algebraA 1 (1) . Commun. Math. Phys.104, 605–609 (1986) · Zbl 0587.17009 · doi:10.1007/BF01211068 |
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