Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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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