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 SIGMA, 2013, Volume 9, 058, 23 pages (Mi sigma841)

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

Samuel Belliarda, Stanislav Pakuliakbcd, Eric Ragoucye, Nikita A. Slavnovf

a Université Montpellier 2, Laboratoire Charles Coulomb, UMR 5221, F-34095 Montpellier, France
b Institute of Theoretical and Experimental Physics, 117259  Moscow, Russia
c Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Moscow reg., Russia
d Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow reg., 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

Abstract: 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.

 Funding Agency Grant Number Russian Foundation for Basic Research 11-01-00980-a11-01-00440 National Research University Higher School of Economics 12-09-0064 Agence Nationale de la Recherche 2010-BLAN-0120-02 Ministry of Education and Science of the Russian Federation SS-4612.2012.1 Work of S.P. was supported in part by RFBR grant 11-01-00980-a and grant of Scientific Foundation of NRU HSE 12-09-0064. E.R. was supported by ANR Project DIADEMS (Programme Blanc ANR SIMI1 2010-BLAN-0120-02). N.A.S. was supported by the Program of RAS Basic Problems of the Nonlinear Dynamics, RFBR-11-01-00440, SS-4612.2012.1.

DOI: https://doi.org/10.3842/SIGMA.2013.058

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Bibliographic databases:

ArXiv: 1304.7602
Document Type: Article
MSC: 81R50; 17B80
Received: May 27, 2013; in final form September 27, 2013; Published online October 7, 2013
Language: English

Citation: Samuel Belliard, Stanislav Pakuliak, Eric Ragoucy, Nikita A. Slavnov, “Bethe Vectors of Quantum Integrable Models with $\mathrm{GL}(3)$ Trigonometric $R$-Matrix”, SIGMA, 9 (2013), 058, 23 pp.

Citation in format AMSBIB
\Bibitem{BelPakRag13} \by Samuel~Belliard, Stanislav~Pakuliak, Eric~Ragoucy, Nikita~A.~Slavnov \paper Bethe Vectors of Quantum Integrable Models with $\mathrm{GL}(3)$ Trigonometric $R$-Matrix \jour SIGMA \yr 2013 \vol 9 \papernumber 058 \totalpages 23 \mathnet{http://mi.mathnet.ru/sigma841} \crossref{https://doi.org/10.3842/SIGMA.2013.058} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3116193} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000325208300001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84885582157} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. Z. Pakulyak, E. Ragoucy, N. A. Slavnov, “Scalar products in models with a $GL(3)$ trigonometric $R$-matrix: Highest coefficient”, Theoret. and Math. Phys., 178:3 (2014), 314–335
2. S. Z. Pakulyak, E. Ragoucy, N. A. Slavnov, “Scalar products in models with the $GL(3)$ trigonometric $R$-matrix: General case”, Theoret. and Math. Phys., 180:1 (2014), 795–814
3. Pakuliak S., Ragoucy E., Slavnov N.A., “Bethe Vectors of Quantum Integrable Models Based Onu(Q) ((Gl)Over-Cap(N))”, J. Phys. A-Math. Theor., 47:10 (2014), 105202
4. M. Vubangsi, M. Tchoffo, L. C. Fai, “Position-dependent effective MASS system in a variable potential: displacement operator method”, Phys. Scr., 89:2 (2014), 025101
5. R. I. Nepomechie, Ch. Wang, “Boundary energy of the open XXX chain with a non-diagonal boundary term”, J. Phys. A-Math. Theor., 47:3 (2014), 032001
6. P. Baseilhac, T. Kojima, “Form factors of the half-infinite XXZ spin chain with a triangular boundary”, J. Stat. Mech.-Theory Exp., 2014, P09004
7. Slavnov N.A., “Scalar Products in Gl(3)-Based Models With Trigonometric R-Matrix. Determinant Representation”, J. Stat. Mech.-Theory Exp., 2015, P03019
8. Hao K., Cao J., Li G.-L., Yang W.-L., Shi K., Wang Yu., “A representation basis for the quantum integrable spin chain associated with the su(3) algebra”, J. High Energy Phys., 2016, no. 5, 119
9. Kozlowski K.K. Ragoucy E., “Asymptotic behaviour of two-point functions in multi-species models”, Nucl. Phys. B, 906 (2016), 241–288
10. A. A. Hutsalyuk, A. Liashyk, S. Z. Pakulyak, E. Ragoucy, N. A. Slavnov, “Current presentation for the super-Yangian double $DY(\mathfrak{gl}(m|n))$ and Bethe vectors”, Russian Math. Surveys, 72:1 (2017), 33–99
11. Jan Fuksa, “Bethe Vectors for Composite Models with $\mathfrak{gl}(2|1)$ and $\mathfrak{gl}(1|2)$ Supersymmetry”, SIGMA, 13 (2017), 015, 17 pp.
12. N. Gromov, F. Levkovich-Maslyuk, G. Sizov, “New construction of eigenstates and separation of variables for $\mathrm{SU}(N)$ quantum spin chains”, J. High Energy Phys., 2017, no. 9, 111
13. A. Hutsalyuk, A. Liashyk, S. Z. Pakuliak, E. Ragoucy, N. A. Slavnov, “Scalar products and norm of Bethe vectors for integrable models based on U-Q ((gl)over-cap(m))”, SciPost Phys., 4:1 (2018), 006
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