Geodesic
Generating Series for Nested Bethe Vectors
Symmetry, integrability and geometry: methods and applications, Tome 4 (2008) Cet article a éte moissonné depuis la source Math-Net.Ru

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We reformulate nested relations between off-shell $U_q(\widehat{\mathfrak{gl}}_N)$ Bethe vectors as a certain equation on generating series of strings of the composed $U_q(\widehat{\mathfrak{gl}}_N)$ currents. Using inversion of the generating series we find a new type of hierarchical relations between universal off-shell Bethe vectors, useful for a derivation of Bethe equation. As an example of application, we use these relations for a derivation of analytical Bethe ansatz equations [Arnaudon D. et al., Ann. Henri Poincaré 7 (2006), 1217–1268, math-ph/0512037] for the parameters of universal Bethe vectors of the algebra $U_q(\widehat{\mathfrak{gl}}_2)$.
Keywords: Bethe ansatz; current algebras; quantum integrable models. 
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     author = {Sergey Khoroshkin and Stanislav Pakuliak},
     title = {Generating {Series} for {Nested} {Bethe} {Vectors}},
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     year = {2008},
     volume = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a80/}
}
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Sergey Khoroshkin; Stanislav Pakuliak. Generating Series for Nested Bethe Vectors. Symmetry, integrability and geometry: methods and applications, Tome 4 (2008). http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a80/

[1] Arnaudon D., Crampè N., Doikou A., Frappat L., Ragoucy E., “Spectrum and Bethe ansatz equations for the $U_q{\left({gl(\mathcal N)}\right)}$ closed and open spin chains in any representation”, Ann. Henri Poincaré, 7 (2006), 1217–1268 ; math-ph/0512037 | DOI | MR | Zbl

[2] Drinfel'd V. G., “New realization of Yangians and quantum affine algebras”, Soviet Math. Dokl., 36 (1988), 212–216 | MR

[3] Ding J. T., Frenkel I. B., “Isomorphism of two realizations of quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$”, Comm. Math. Phys., 156 (1993), 277–300 | DOI | MR | Zbl

[4] Ding J., Khoroshkin S., “Weyl group extension of quantized current algebras”, Transform. Groups, 5 (2000), 35–59 ; math.QA/9804139 | DOI | MR

[5] Enriquez B., “On correlation functions of Drinfeld currents and shuffle algebras”, Transform. Groups, 5 (2000), 111–120 ; math.QA/9809036 | DOI | MR | Zbl

[6] Enriquez B., Khoroshkin S., Pakuliak S., “Weight functions and Drinfeld currents”, Comm. Math. Phys., 276 (2007), 691–725 ; math.QA/0610398 | DOI | MR | Zbl

[7] Enriquez B., Rubtsov V., “Quasi-Hopf algebras associated with $\mathfrak{sl}_2$ and complex curves”, Israel J. Math., 112 (1999), 61–108 ; q-alg/9608005 | DOI | MR

[8] Frappat L., Khoroshkin S., Pakuliak S., Ragoucy É., Bethe ansatz for the universal weight function, arXiv:0810.3135

[9] Khoroshkin S., Pakuliak S., “The weight function for the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_3)$”, Theor. and Math. Phys., 145 (2005), 1373–1399 ; math.QA/0610433 | DOI | MR | Zbl

[10] Khoroshkin S., Pakuliak S., Tarasov V., “Off-shell Bethe vectors and Drinfeld currents”, J. Geom. Phys., 57 (2007), 1713–1732 ; math.QA/0610517 | DOI | MR | Zbl

[11] Khoroshkin S., Pakuliak S., “A computation of an universal weight function for the quantum affine algebra $U_q(\widehat{gl}_N)$”, J. Math. Kyoto Univ., 48 (2008), 277–322 ; arXiv:0711.2819 | MR

[12] Kulish P., Reshetikhin N., “Diagonalization of $GL(N)$ invariant transfer matrices and quantum $N$-wave system (Lee model)”, J. Phys. A: Math. Gen., 16 (1983), L591–L596 | DOI | MR | Zbl

[13] Mukhin E., Tarasov V., Varchenko A., “Bethe eigenvectors of higher transfer matrices”, J. Stat. Mech. Theory Exp., 2006:8 (2006), P08002, 44 pp., ages ; math.QA/0605015 | DOI | MR

[14] Oskin A., Pakuliak S., Silantyev A., On the universal weight function for the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, arXiv:0711.2821

[15] Reshetikhin N., Semenov-Tian-Shansky M., “Central extentions of quantum current groups”, Lett. Math. Phys., 19 (1990), 133–142 | DOI | MR | Zbl

[16] Tarasov V., Varchenko A., “Jackson integrals for the solutions to Knizhnik–Zamolodchikov equation”, St. Petersburg Math. J., 2:2 (1995), 275–313 | MR

[17] Tarasov V., Varchenko A., Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque, 246, 1997, 135 ; q-alg/9703044 | MR | Zbl

[18] Tarasov V., Varchenko A., Combinatorial formulae for nested Bethe vectors, math.QA/0702277 | MR