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Representation Theory

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Commutative quantum current operators, semi-infinite construction and functional models

Authors: Jintai Ding and Boris Feigin
Journal: Represent. Theory 4 (2000), 330-341
MSC (2000): Primary 17B37
Published electronically: August 1, 2000
MathSciNet review: 1773865
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We construct the commutative current operator $\bar x^+(z)$ inside $U_q(\hat{\mathfrak{sl}}(2))$. With this operator and the condition of quantum integrability on the quantum currents of $U_q(\hat{\mathfrak{sl}}(2))$, we derive the quantization of the semi-infinite construction of integrable modules of $\hat{\mathfrak{sl}}(2)$ which has been previously obtained by means of the current operator $e(z)$ of $\hat{\mathfrak{sl}}(2)$. The quantization of the functional models for $\hat{\mathfrak{sl}}(2)$ is also given.

References [Enhancements On Off] (What's this?)

  • [DM] J. Ding and T. Miwa Zeros and poles of quantum current operators and the condition of quantum integrability, q-alg/9608001,RIMS-1092.
  • [DI] J. Ding and K. Iohara Generalization and deformation of the quantum affine algebras, Rims-1090, q-alg/9608002.
  • [Dr1] V. G. Drinfeld, Hopf algebra and the quantum Yang-Baxter Equation, Dokl. Akad. Nauk. SSSR, 283, 1985, 1060-1064. MR 87h:58080
  • [Dr2] V.G. Drinfeld, Quantum Groups, ICM Proceedings, New York, Berkeley, 1986, 798-820. MR 89f:17017
  • [Dr3] V. G. Drinfeld, A new realization of Yangian and of quantum affine algebra, Soviet Math. Doklady, 36, 1988, 212-216. MR 88j:17020
  • [FS1] B. L. Feigin and A. V. Stoyanovsky, Quasi-particle models for the representations of Lie algebras and the geometry of the flag manifold, RIMS-942.
  • [FS2] B. L. Feigin and A. V. Stoyanovsky, Functional models of the representations of current algebras and the semi-infinite Schubert cells, Funkts. Anal. Prilozhen., 28, No. 1, 1994, 68-90. MR 95g:17027
  • [FS3] B. L. Feigin and A. V. Stoyanovsky, Realization of the modular functors in the space of differentials and geometric approximation of the moduli space of G-bundles, Funkts. Anal. Prilozhen., 28, No. 4, 1994, 42-65. MR 96k:32039
  • [LP] J. Lepowsky and M. Primc, Structure of standard modules for the affine Lie algebra $A_1^{[1]}$, Contemp. Math. 45, Amer. Math. Soc., Providence, 1985. MR 87g:17021
  • [L] G. Lusztig Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70, 1988, 237-249. MR 89k:17029

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Additional Information

Jintai Ding
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025

Boris Feigin
Affiliation: Landau Institute of Theoretical Physics, Moscow, Russia

Received by editor(s): April 17, 1998
Received by editor(s) in revised form: January 14, 2000
Published electronically: August 1, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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