SIGMA 9 (2013), 058, 23 pages      arXiv:1304.7602      https://doi.org/10.3842/SIGMA.2013.058
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric $R$-Matrix

Samuel Belliard ^a, Stanislav Pakuliak ^{b, c, d}, Eric Ragoucy ^e and Nikita A. Slavnov ^f
a) Université Montpellier 2, Laboratoire Charles Coulomb, UMR 5221, F-34095 Montpellier, France
^b) Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow reg., Russia
^c) Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Moscow reg., Russia
^d) Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia
^e) Laboratoire de Physique Théorique LAPTH, CNRS and Université de Savoie, BP 110, 74941 Annecy-le-Vieux Cedex, France
^f) Steklov Mathematical Institute, Moscow, Russia

Received May 27, 2013, in final form September 27, 2013; Published online October 07, 2013

Abstract
We study quantum integrable models with 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.

Key words: nested algebraic Bethe ansatz; Bethe vector; current algebra.

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References

1. Belavin A.A., Drinfel'd V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl. 16 (1982), 159-180.
2. Belliard S., Pakuliak S., Ragoucy E., Universal Bethe ansatz and scalar products of Bethe vectors, SIGMA 6 (2010), 094, 22 pages, arXiv:1012.1455.
3. Belliard S., Pakuliak S., Ragoucy E., Slavnov N.A., The algebraic Bethe ansatz for scalar products in ${\rm SU}(3)$-invariant integrable models, J. Stat. Mech. Theory Exp. 2012 (2012), P10017, 25 pages, arXiv:1207.0956.
4. Belliard S., Pakuliak S., Ragoucy E., Slavnov N.A., Bethe vectors of ${\rm GL}(3)$-invariant integrable models, J. Stat. Mech. Theory Exp. 2013 (2013), P02020, 24 pages, arXiv:1210.0768.
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.
6. Drinfel'd V.G., A new realization of Yangians and of quantum affine algebras, Sov. Math. Dokl. 36 (1988), 212-216.
7. Enriquez B., Khoroshkin S., Pakuliak S., Weight functions and Drinfeld currents, Comm. Math. Phys. 276 (2007), 691-725.
8. 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.
9. Faddeev L.D., Sklyanin E.K., Takhtajan L.A., Quantum inverse problem. I, Theoret. and Math. Phys. 40 (1979), 688-706.
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.
11. Izergin A.G., Partition function of a six-vertex model in a finite volume, Sov. Phys. Dokl. 32 (1987), 878-879.
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.
13. Khoroshkin S., Pakuliak S., Generating series for nested Bethe vectors, SIGMA 4 (2008), 081, 23 pages, arXiv:0810.3131.
14. Khoroshkin S., Pakuliak S., Tarasov V., Off-shell Bethe vectors and Drinfeld currents, J. Geom. Phys. 57 (2007), 1713-1732, math.QA/0610517.
15. Khoroshkin S.M., Tolstoy V.N., On Drinfeld's realization of quantum affine algebras, J. Geom. Phys. 11 (1993), 445-452.
16. Kitanine N., Maillet J.M., Terras V., Form factors of the $XXZ$ Heisenberg spin-[1/2] finite chain, Nuclear Phys. B 554 (1999), 647-678, math-ph/9807020.
17. Kulish P.P., Reshetikhin N.Yu., Generalized Heisenberg ferromagnet and the Gross-Neveu model, Soviet Phys. JETP 53 (1981), 108-114.
18. Kulish P.P., Reshetikhin N.Yu., On ${\rm GL}_3$-invariant solutions of the Yang-Baxter equation and associated quantum systems, J. Sov. Math. 34 (1982), 1948-1971.
19. Kulish P.P., Reshetikhin N.Yu., Diagonalisation of ${\rm GL}(N)$ invariant transfer matrices and quantum $N$-wave system (Lee model), J. Phys. A: Math. Gen. 16 (1983), L591-L596.
20. Maillet J.M., Terras V., On the quantum inverse scattering problem, Nuclear Phys. B 575 (2000), 627-644, hep-th/9911030.
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.
22. Reshetikhin N.Yu., Calculation of the norm of Bethe vectors in models with ${\rm SU}(3)$ symmetry, J. Math. Sci. 46 (1986), 1694-1706.
23. Reshetikhin N.Yu., Semenov-Tian-Shansky M.A., Central extensions of quantum current groups, Lett. Math. Phys. 19 (1990), 133-142.
24. 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.
25. Tarasov V., Varchenko A., Combinatorial formulae for nested Bethe vectors, SIGMA 9 (2013), 048, 28 pages, math.QA/0702277.
26. Varchenko A.N., Tarasov V.O., Jackson integral representations for solutions of the Knizhnik-Zamolodchikov quantum equation, St. Petersburg Math. J. 6 (1995), 275-313, hep-th/9311040.