Geodesic
Universal Bethe Ansatz and Scalar Products of Bethe Vectors
Symmetry, integrability and geometry: methods and applications, Tome 6 (2010) Cet article a éte moissonné depuis la source Math-Net.Ru

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An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_3)$ is given. This result is obtained in the framework of the universal Bethe ansatz, using presentation of the universal Bethe vectors in terms of the total currents of a “new” realization of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_3)$.
Keywords: Bethe ansatz; quantum affine algebras. 
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     author = {Samuel Belliard and Stanislav Pakuliak and Eric Ragoucy},
     title = {Universal {Bethe} {Ansatz} and {Scalar} {Products} of {Bethe} {Vectors}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a93/}
}
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Samuel Belliard; Stanislav Pakuliak; Eric Ragoucy. Universal Bethe Ansatz and Scalar Products of Bethe Vectors. Symmetry, integrability and geometry: methods and applications, Tome 6 (2010). http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a93/

[1] Bethe H., “Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette”, Z. Phys., 71 (1931), 205–226 | DOI

[2] Takhtajan L. A., Faddeev L. D., “The quantum method of the inverse problem and the Heisenberg $XYZ$ model”, Russian Math. Surveys, 34:5 (1979), 11–68 | DOI

[3] Sklyanin E. K., Faddeev L. D., “Quantum mechanical approach to completely integrable field theory models”, Sov. Phys. Dokl., 23 (1978), 902–904

[4] Sklyanin E. K., Takhtajan L. A., Faddeev L. D., “Quantum inverse problem method. I”, Theoret. and Math. Phys., 40 (1980), 688–706 | DOI | Zbl

[5] Faddeev L. D., “How algebraic Bethe ansatz works for integrable models”, Symétries Quantiques (Les Houches, 1995), eds. A. Connes et al., North-Holland, Amsterdam, 1998, 149–219, arXiv: hep-th/9605187 | MR | Zbl

[6] Korepin V. E., “Calculation of norms of Bethe wave functions”, Comm. Math. Phys., 86 (1982), 391–418 | DOI | MR | Zbl

[7] Izergin A. G., Korepin V. E., “The quantum inverse scattering method approach to correlation functions”, Comm. Math. Phys., 94 (1984), 67–92 | DOI | MR | Zbl

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

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

[10] Kojima T., Korepin V. E., Slavnov N. A., “Determinant representation for dynamical correlation functions of the quantum nonlinear Schrödinger equation”, Comm. Math. Phys., 188 (1997), 657–689, arXiv: hep-th/9611216 | DOI | MR | Zbl

[11] Izergin A. G., Kitanine N., Maillet J. M., Terras V., “Spontaneous magnetization of the $XXZ$ Heisenberg spin-1/2 chain”, Nuclear Phys. B, 554 (1999), 679–696, arXiv: solv-int/9812021 | DOI | MR | Zbl

[12] Kitanine N., Maillet J. M., Terras V., “Correlation functions of the $XXZ$ Heisenberg spin-1/2 chain in a magnetic field”, Nuclear Phys. B, 567 (2000), 554–582, arXiv: math-ph/9907019 | DOI | MR | Zbl

[13] Kitanine N., Maillet J. M., Slavnov N. A., Terras V., “Master equation for spin-spin correlation functions of the $XXZ$ chain”, Nuclear Phys. B, 712 (2005), 600–622, arXiv: hep-th/0406190 | DOI | MR | Zbl

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

[15] Belliard S., Ragoucy E., “The nested Bethe ansatz for “all” closed spin chains”, J. Phys. A: Math. Theor., 41 (2008), 295202, 33 pp., arXiv: 0804.2822 | DOI | MR | Zbl

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

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

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

[19] 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

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

[21] 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

[22] Arnaudon D., Crampé 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, arXiv: math-ph/0512037 | DOI | MR | Zbl

[23] Slavnov N. A., “The algebraic Bethe ansatz and quantum integrable systems”, Russian Math. Surveys, 62 (2007), 727–766 | DOI | MR | Zbl

[24] 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

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

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

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

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