Geodesic
Bethe Vectors of Quantum Integrable Models with $\mathrm{GL}(3)$ Trigonometric $R$-Matrix
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

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We study quantum integrable models with $\mathrm{GL}(3)$ trigonometric $R$-matrix and solvable by the nested algebraic Bethe ansatz. Using the presentation of the universal Bethe vectors in terms of projections of products of the currents of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_3)$ onto intersections of different types of Borel subalgebras, we prove that the set of the nested Bethe vectors is closed under the action of the elements of the monodromy matrix.
Keywords: nested algebraic Bethe ansatz; Bethe vector; current algebra. 
@article{SIGMA_2013_9_a57,
     author = {Samuel Belliard and Stanislav Pakuliak and Eric Ragoucy and Nikita A. Slavnov},
     title = {Bethe {Vectors} of {Quantum} {Integrable} {Models} with $\mathrm{GL}(3)$ {Trigonometric} $R${-Matrix}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2013},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a57/}
}
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Samuel Belliard; Stanislav Pakuliak; Eric Ragoucy; Nikita A. Slavnov. Bethe Vectors of Quantum Integrable Models with $\mathrm{GL}(3)$ Trigonometric $R$-Matrix. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a57/

[1] Belavin A. A., Drinfel'd V. G., “Solutions of the classical {Y}ang–{B}axter equation for simple {L}ie algebras”, Funct. Anal. Appl., 16 (1982), 159–180 | DOI | MR

[2] Belliard S., Pakuliak S., Ragoucy E., “Universal {B}ethe ansatz and scalar products of {B}ethe vectors”, SIGMA, 6 (2010), 094, 22 pp., arXiv: 1012.1455 | DOI | MR | Zbl

[3] Belliard S., Pakuliak S., Ragoucy E., Slavnov N. A., “The algebraic {B}ethe ansatz for scalar products in $\mathrm{SU}(3)$-invariant integrable models”, J. Stat. Mech. Theory Exp., 2012 (2012), P10017, 25 pp., arXiv: 1207.0956 | DOI | MR

[4] Belliard S., Pakuliak S., Ragoucy E., Slavnov N. A., “Bethe vectors of $\mathrm{GL}(3)$-invariant integrable models”, J. Stat. Mech. Theory Exp., 2013 (2013), P02020, 24 pp., arXiv: 1210.0768 | DOI | MR

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

[6] Drinfel'd V. G., “A new realization of {Y}angians and of quantum affine algebras”, Sov. Math. Dokl., 36 (1988), 212–216 | MR

[7] Enriquez B., Khoroshkin S., Pakuliak S., “Weight functions and {D}rinfeld currents”, Comm. Math. Phys., 276 (2007), 691–725 | DOI | MR | Zbl

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

[9] Faddeev L. D., Sklyanin E. K., Takhtajan L. A., “Quantum inverse problem, I”, Theoret. and Math. Phys., 40 (1979), 688–706 | DOI | MR

[10] Frappat L., Khoroshkin S., Pakuliak S., Ragoucy E., “Bethe ansatz for the universal weight function”, Ann. Henri Poincaré, 10 (2009), 513–548, arXiv: 0810.3135 | DOI | MR | Zbl

[11] Izergin A. G., “Partition function of a six-vertex model in a finite volume”, Sov. Phys. Dokl., 32 (1987), 878–879 | MR | Zbl

[12] Khoroshkin S., Pakuliak S., “A computation of universal weight function for quantum affine algebra {$U_q(\widehat{\mathfrak{gl}}_N)$}”, J. Math. Kyoto Univ., 48 (2008), 277–321, arXiv: 0711.2819 | MR | Zbl

[13] Khoroshkin S., Pakuliak S., “Generating series for nested {B}ethe vectors”, SIGMA, 4 (2008), 081, 23 pp., arXiv: 0810.3131 | DOI | MR | Zbl

[14] Khoroshkin S., Pakuliak S., Tarasov V., “Off-shell {B}ethe vectors and {D}rinfeld currents”, J. Geom. Phys., 57 (2007), 1713–1732, arXiv: math.QA/0610517 | DOI | MR | Zbl

[15] Khoroshkin S. M., Tolstoy V. N., “On {D}rinfeld's realization of quantum affine algebras”, J. Geom. Phys., 11 (1993), 445–452 | DOI | MR | Zbl

[16] Kitanine N., Maillet J. M., Terras V., “Form factors of the {$XXZ$} {H}eisenberg spin-{$\frac 12$} finite chain”, Nuclear Phys. B, 554 (1999), 647–678, arXiv: math-ph/9807020 | DOI | MR | Zbl

[17] Kulish P. P., Reshetikhin N. Yu., “Generalized {H}eisenberg ferromagnet and the {G}ross–{N}eveu model”, Soviet Phys. JETP, 53 (1981), 108–114

[18] Kulish P. P., Reshetikhin N. Yu., “On {${\rm GL}_{3}$}-invariant solutions of the {Y}ang–{B}axter equation and associated quantum systems”, J. Sov. Math., 34 (1982), 1948–1971 | DOI

[19] Kulish P. P., Reshetikhin N. Yu., “Diagonalisation of $\mathrm{GL}(N)$ invariant transfer matrices and quantum {$N$}-wave system ({L}ee model)”, J. Phys. A: Math. Gen., 16 (1983), L591–L596 | DOI | MR | Zbl

[20] Maillet J. M., Terras V., “On the quantum inverse scattering problem”, Nuclear Phys. B, 575 (2000), 627–644, arXiv: hep-th/9911030 | DOI | MR | Zbl

[21] Os'kin A., Pakuliak S., Silantyev A., “On the universal weight function for the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{N})$”, St. Petersburg Math. J., 21 (2010), 651–680, arXiv: 0711.2821 | DOI | MR

[22] Reshetikhin N. Yu., “Calculation of the norm of {B}ethe vectors in models with $\mathrm{SU}(3)$ symmetry”, J. Math. Sci., 46 (1986), 1694–1706 | DOI | MR

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

[24] Slavnov N. A., “Calculation of scalar products of wave functions and form-factors in the framework of the algebraic {B}ethe ansatz”, Theoret. and Math. Phys., 79 (1989), 502–508 | DOI | MR

[25] Tarasov V., Varchenko A., “Combinatorial formulae for nested Bethe vectors”, SIGMA, 9 (2013), 048, 28 pp., arXiv: math.QA/0702277 | DOI | MR

[26] Varchenko A. N., Tarasov V. O., “Jackson integral representations for solutions of the {K}nizhnik–{Z}amolodchikov quantum equation”, St. Petersburg Math. J., 6 (1995), 275–313, arXiv: hep-th/9311040 | MR