On the universal weight function for the quantum affine algebra 𝑈_{𝑞}(̂𝔤𝔩_{𝔑})

Os′kin, A.; Pakuliak, S.; Silant′ev, A.

doi:10.1090/S1061-0022-2010-01110-5

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
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
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.

1. Vyjayanthi Chari and Andrew 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, A new realization of Yangians and of quantum affine algebras, Dokl. Akad. Nauk SSSR 296 (1987), no. 1, 13–17 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 2, 212–216. MR 914215 (88j:17020)
3. Jin Tai Ding and Igor B. Frenkel, Isomorphism of two realizations of quantum affine algebra 𝑈_{𝑞}(𝔤𝔩(𝔫)), Comm. Math. Phys. 156 (1993), no. 2, 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), http://dx.doi.org/10.1023/A:1026730817516
5. J. Ding and S. Khoroshkin, Weyl group extension of quantized current algebras, Transform. Groups 5 (2000), no. 1, 35–59. MR 1745710 (2001f:17019), http://dx.doi.org/10.1007/BF01237177
6. D. Ding, S. Pakulyak, and S. Khoroshkin, Factorization of the universal ℛ-matrix for 𝒰_{𝓆}(̂𝓈𝓁₂), Teoret. Mat. Fiz. 124 (2000), no. 2, 179–214 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 124 (2000), no. 2, 1007–1037. MR 1821111 (2002b:17011), http://dx.doi.org/10.1007/BF02551074
7. B. Enriquez, On correlation functions of Drinfeld currents and shuffle algebras, Transform. Groups 5 (2000), no. 2, 111–120. MR 1762114 (2001c:17026), http://dx.doi.org/10.1007/BF01236465
8. B. Enriquez, S. Khoroshkin, and S. Pakuliak, Weight functions and Drinfeld currents, Comm. Math. Phys. 276 (2007), no. 3, 691–725. MR 2350435 (2010a:17025), http://dx.doi.org/10.1007/s00220-007-0351-y
9. Benjamin Enriquez and Vladimir Rubtsov, Quasi-Hopf algebras associated with 𝔰𝔩₂ and complex curves, Israel J. Math. 112 (1999), 61–108. MR 1715010 (2000i:17021), http://dx.doi.org/10.1007/BF02773478
10. S. Z. Pakulyak and S. M. Khoroshkin, The weight function for the quantum affine algebra 𝑈_{𝑞}(̂𝔰𝔩₃), Teoret. Mat. Fiz. 145 (2005), no. 1, 3–34 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 145 (2005), no. 1, 1373–1399. MR 2213305 (2007c:17016), http://dx.doi.org/10.1007/s11232-005-0167-x
11. S. Khoroshkin, S. Pakuliak, and V. Tarasov, Off-shell Bethe vectors and Drinfeld currents, J. Geom. Phys. 57 (2007), no. 8, 1713–1732. MR 2319317 (2009a:17023), http://dx.doi.org/10.1016/j.geomphys.2007.02.005
12. Sergey Khoroshkin and Stanislav Pakuliak, A computation of universal weight function for quantum affine algebra 𝑈_{𝑞}(̂𝔤𝔩_{𝔑}), J. Math. Kyoto Univ. 48 (2008), no. 2, 277–321. MR 2436738 (2009f:17023)
13. P. P. Kulish and N. Yu. Reshetikhin, Diagonalisation of 𝐺𝐿(𝑁) invariant transfer matrices and quantum 𝑁-wave system (Lee model), J. Phys. A 16 (1983), no. 16, L591–L596. MR 727044 (84m:82034)
14. Jonathan Brundan and Alexander Kleshchev, Parabolic presentations of the Yangian 𝑌(𝔤𝔩_{𝔫}), Comm. Math. Phys. 254 (2005), no. 1, 191–220. MR 2116743 (2005j:17012), http://dx.doi.org/10.1007/s00220-004-1249-6
15. N. Reshetikhin, Jackson-type integrals, Bethe vectors, and solutions to a difference analog of the Knizhnik-Zamolodchikov system, Lett. Math. Phys. 26 (1992), no. 3, 153–165. MR 1199739 (94g:17031), http://dx.doi.org/10.1007/BF00420749
16. N. Yu. Reshetikhin and M. A. Semenov-Tian-Shansky, Central extensions of quantum current groups, Lett. Math. Phys. 19 (1990), no. 2, 133–142. MR 1039522 (91k:17013), http://dx.doi.org/10.1007/BF01045884
17. A. N. Varchenko and V. O. Tarasov, Jackson integral representations for solutions of the Knizhnik-Zamolodchikov quantum equation, Algebra i Analiz 6 (1994), no. 2, 90–137 (Russian, with Russian summary); 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.