Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

On the universal weight function for the quantum affine algebra $ U_q(\widehat{\mathfrak{gl}}_N)$


Authors: A. Os'kin, S. Pakuliak and A. Silant'ev
Translated by: the authors
Original publication: Algebra i Analiz, tom 21 (2009), nomer 4.
Journal: St. Petersburg Math. J. 21 (2010), 651-680
MSC (2010): Primary 81R10
DOI: https://doi.org/10.1090/S1061-0022-2010-01110-5
Published electronically: May 20, 2010
MathSciNet review: 2584212
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The investigation is continued of the universal weight function for the quantum affine algebra $ U_q(\widehat{\mathfrak{gl}}_N)$. Two recurrence relations are obtained for the universal weight function with the help of the method of projections. On the level of the evaluation representation of $ U_q(\widehat{\mathfrak{gl}}_N)$, two recurrence relations are reproduced, which were calculated earlier for the off-shell Bethe vectors by combinatorial methods. One of the results of the paper is a description of two different but isomorphic currents or ``new'' realizations of the algebra $ U_q(\widehat{\mathfrak{gl}}_N)$, corresponding to two different Gauss decompositions of the fundamental $ \mathrm{L}$-operators.


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

  • 1. V. Chari and A. Pressley, Quantum affine algebras and their representations, Representations of Groups (Banff, AB, 1994), CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 59-78. MR 1357195 (96j:17009)
  • 2. V. G. Drinfel'd, New realization of Yangians and quantum affine algebras, Dokl. Akad. Nauk SSSR 296 (1987), no. 1, 13-17; English transl., Soviet Math. Dokl. 36 (1988), 212-216. MR 0914215 (88j:17020)
  • 3. J. Ding and I. B. Frenkel, Isomorphism of two realizations of quantum affine algebra $ U_q(\widehat{\mathfrak{gl}}_N)$, Comm. Math. Phys. 156 (1993), 277-300. MR 1233847 (94i:17020)
  • 4. J. Ding, S. Khoroshkin, and S. Pakuliak, Integral presentations for the universal $ R$-matrix, Lett. Math. Phys. 53 (2000), no. 2, 121-141. MR 1804187 (2002f:17018)
  • 5. J. Ding and S. Khoroshkin, Weyl group extension of quantized current algebras, Transform. Groups 5 (2000), 35-59. MR 1745710 (2001f:17019)
  • 6. J. Ding, S. Pakulyak, and S. Khoroshkin, Factorization of the universal $ R$-matrix for $ U_q(\widehat{sl}_2)$, Teor. Mat. Fiz. 124 (2000), no. 2, 179-214; English transl., Theor. and Math. Phys. 124 (2000), no. 2, 1007-1037. MR 1821111 (2002b:17011)
  • 7. B. Enriquez, On correlation functions of Drinfeld currents and shuffle algebras, Transform. Groups 5 (2000), no. 2, 111-120. MR 1762114 (2001c:17026)
  • 8. B. Enriquez, S. Khoroshkin, and S. Pakuliak, Weight functions and Drinfeld currents, Comm. Math. Phys. 276 (2007), 691-725. MR 2350435 (2010a:17025)
  • 9. B. Enriquez and V. Rubtsov, Quasi-Hopf algebras associated with $ \mathfrak{sl}_2$ and complex curves, Israel J. Math. 112 (1999), 61-108. MR 1715010 (2000i:17021)
  • 10. S. Pakulyak and S. Khoroshkin, Weight function for the quantum affine algebra $ U_q(\widehat{\mathfrak{sl}}_3)$, Teor. Mat. Fiz. 145 (2005), no. 1, 3-34; English transl., Theor. and Math. Phys. 145 (2005), no. 1, 1373-1399. MR 2213305 (2007c:17016)
  • 11. S. Khoroshkin, S. Pakuliak, and V. Tarasov, Off-shell Bethe vectors and Drinfeld currents, J. Geom. Phys. 57 (2007), 1713-1732. MR 2319317 (2009a:17023)
  • 12. S. Khoroshkin and S. Pakuliak, A computation of universal weight function for quantum affine algebra $ U_q(\widehat{\mathfrak{gl}}_N)$, J. Math. Kyoto Univ. 48 (2008), 277-321. MR 2436738 (2009f:17023)
  • 13. P. Kulish and N. Reshetikhin, Diagonalisation of $ GL(N)$ invariant transfer matrices and quantum $ N$-wave system (Lee model), J. Phys. A 16 (1983), L591-L596. MR 0727044 (84m:82034)
  • 14. J. Brundan and A. Kleshchev, Parabolic presentations of the Yangian $ Y(\mathfrak{gl}_n)$, Comm. Math. Phys. 254 (2005), 191-220. MR 2116743 (2005j:17012)
  • 15. N. Reshetikhin, Jackson-type integrals, Bethe vectors, and solutions to a difference analog of the Knizhnik-Zamolodchikov system, Lett. Math. Phys. 26 (1992), 153-165. MR 1199739 (94g:17031)
  • 16. N. Reshetikhin and M. Semenov-Tian-Shansky, Central extensions of quantum current groups, Lett. Math. Phys. 19 (1990), 133-142. MR 1039522 (91k:17013)
  • 17. A. N. Varchenko and V. O. Tarasov, Jackson integral representations of solutions of the quantized Knizhnik-Zamolodchikov equation, Algebra i Analiz 6 (1994), no. 2, 90-137; English transl., St. Petersburg Math. J. 6 (1995), no. 2, 275-313. MR 1290820 (96f:81052a)
  • 18. -, Combinatorial formulae for nested Bethe vectors, Preprint math.QA/0702277.

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 81R10

Retrieve articles in all journals with MSC (2010): 81R10


Additional Information

A. Os'kin
Affiliation: Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region 141980, Russia
Email: aoskin@theor.jinr.ru

S. Pakuliak
Affiliation: Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region 141980, and Institute of Theoretical and Experimental Physics, Moscow 117259, Russia
Email: pakuliak@theor.jinr.ru

A. Silant'ev
Affiliation: Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region 141980, Russia, and Départment de Mathématiques, Université d’Angers, 2 Bd. Lavoisier, Angers 49045, France
Email: silant@tonton.univ-angers.fr

DOI: https://doi.org/10.1090/S1061-0022-2010-01110-5
Keywords: Hierarchical Bethe ansatz, off-shell Bethe vectors, L-operator, current representation
Received by editor(s): July 22, 2008
Published electronically: May 20, 2010
Additional Notes: The work of the second author was supported in part by RFBR, grant no. 05-01-01086, and by the grant NSh-8065.2006.2 for support of leading scientific schools
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society