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

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 Uq(glˆ₃) onto intersections of different typ...

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Дата:	2013
Автори:	Belliard, S., Pakuliak, S., Ragoucy, E., Slavnov, N.A.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2013
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/149345
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matri / S. Belliard, S. Pakuliak, E. Ragoucy, N.A. Slavnov // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 26 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1493452019-02-22T01:22:33Z Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix Belliard, S. Pakuliak, S. Ragoucy, E. Slavnov, N.A. 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 Uq(glˆ₃) 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. 2013 Article Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matri / S. Belliard, S. Pakuliak, E. Ragoucy, N.A. Slavnov // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 26 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R50; 17B80 DOI: http://dx.doi.org/10.3842/SIGMA.2013.058 http://dspace.nbuv.gov.ua/handle/123456789/149345 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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 Uq(glˆ₃) 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.
format	Article
author	Belliard, S. Pakuliak, S. Ragoucy, E. Slavnov, N.A.
spellingShingle	Belliard, S. Pakuliak, S. Ragoucy, E. Slavnov, N.A. Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Belliard, S. Pakuliak, S. Ragoucy, E. Slavnov, N.A.
author_sort	Belliard, S.
title	Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix
title_short	Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix
title_full	Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix
title_fullStr	Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix
title_full_unstemmed	Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix
title_sort	bethe vectors of quantum integrable models with gl(3) trigonometric r-matrix
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149345
citation_txt	Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matri / S. Belliard, S. Pakuliak, E. Ragoucy, N.A. Slavnov // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 26 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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first_indexed	2025-07-12T21:53:28Z
last_indexed	2025-07-12T21:53:28Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 058, 23 pages Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix? Samuel BELLIARD † 1 , Stanislav PAKULIAK †2†3†4, Eric RAGOUCY † 5 , Nikita A. SLAVNOV † 6 †1 Université Montpellier 2, Laboratoire Charles Coulomb, UMR 5221, F-34095 Montpellier, France E-mail: samuel.belliard@univ-montp2.fr †2 Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow reg., Russia E-mail: pakuliak@theor.jinr.ru †3 Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Moscow reg., Russia †4 Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia †5 Laboratoire de Physique Théorique LAPTH, CNRS and Université de Savoie, BP 110, 74941 Annecy-le-Vieux Cedex, France E-mail: eric.ragoucy@lapth.cnrs.fr †6 Steklov Mathematical Institute, Moscow, Russia E-mail: nslavnov@mi.ras.ru Received May 27, 2013, in final form September 27, 2013; Published online October 07, 2013 http://dx.doi.org/10.3842/SIGMA.2013.058 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 Uq(ĝl3) 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 2010 Mathematics Subject Classification: 81R50; 17B80 1 Introduction We consider a quantum integrable model defined by the monodromy matrix T (u) with matrix elements Tij(u), i, j = 1, 2, 3, which satisfies the commutation relation R(u, v) · (T (u)⊗ 1) · (1⊗ T (v)) = (1⊗ T (v)) · (T (u)⊗ 1) · R(u, v), (1.1) with the Uq(ĝl3) trigonometric quantum R-matrix R(u, v) = f(u, v) ∑ 1≤i≤3 Eii ⊗ Eii + ∑ 1≤i<j≤3 (Eii ⊗ Ejj + Ejj ⊗ Eii) + ∑ 1≤i<j≤3 ( ug(u, v)Eij ⊗ Eji + vg(u, v)Eji ⊗ Eij ) . (1.2) ?This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html mailto:samuel.belliard@univ-montp2.fr mailto:pakuliak@theor.jinr.ru mailto:eric.ragoucy@lapth.cnrs.fr mailto:nslavnov@mi.ras.ru http://dx.doi.org/10.3842/SIGMA.2013.058 http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html 2 S. Belliard, S. Pakuliak, E. Ragoucy and N.A. Slavnov Here the rational functions f(u, v) and g(u, v) are f(u, v) = qu− q−1v u− v , g(u, v) = ( q − q−1 ) u− v , and (Eij)lk = δilδjk, i, j, l, k = 1, 2, 3 are 3 × 3 matrices with unit in the intersection of the ith row and the jth column and zero matrix elements elsewhere. The R-matrix (1.2) is called ‘trigonometric’ because its classical limit gives the classical trigonometric r-matrix [1]. The trigonometric R-matrix (1.2) is written in multiplicative variables and depends actually on the ratio u/v of these multiplicative parameters. Due to the commutation relation (1.1) the transfer matrix t(u) = trT (u) = T11(u)+T22(u)+ T33(u) generates a set of commuting integrals of motion and the first step of the algebraic Bethe ansatz [9] is the construction of the set of eigenstates for these commuting operators in terms of the monodromy matrix entries. We assume that these matrix elements act in a quantum space V and that this space possesses a vector \|0〉 ∈ V such that Tij(u)\|0〉 = 0, i > j, Tii(u)\|0〉 = λi(u)\|0〉, λi(u) ∈ C [[ u, u−1 ]] . The eigenstates Ba,b(ū; v̄) of the transfer matrix t(u) in quantum integrable models with GL(3) trigonometric R-matrix depend on two sets of variables ū = {u1, . . . , ua} , v̄ = {v1, . . . , vb} , which are called the Bethe parameters. These eigenstates can be constructed in the framework of the nested Bethe ansatz method formulated in [19] and are given by certain polynomials in the monodromy matrix elements T12(u), T23(u), T13(u) with rational coefficients depending on the Bethe parameters. In pioneer papers on nested Bethe ansatz [17, 18, 19] no explicit formulae for the Bethe vectors were obtained. The method, in its original formulation, allows one to get the Bethe equations by requiring that the Bethe vectors are eigenstates of the transfer matrix. Nevertheless, even when the Bethe parameters are free and do not satisfy any restrictions, the structure of the Bethe vectors (sometimes such Bethe vectors are called off-shell) is rather complicated. More explicit formulae for the off-shell nested Bethe vectors were obtained in [26] in the theory of solutions of the quantum Knizhnik–Zamolodchikov equation. The Bethe vectors were given by certain traces over auxiliary spaces of the products of the monodromy matrices and R-matrices. This presentation allows one to investigate the structure of the nested off-shell Bethe vectors and to obtain the explicit formulae for the nested Bethe vectors when the space V becomes a tensor product of evaluation representations of the Yangian and of the positive Borel subalgebra of the quantum affine algebra Uq(ĝlN ) [25]. Explicit expressions for the off-shell nested Bethe vectors in the GL(N) quantum integrable models in terms of the monodromy matrix elements were obtained in the papers [12, 14, 21], where the realization of these vectors in terms of the current generators of the quantum affine algebra Uq(ĝlN ) [7] was used. This realization uses the notion of projections onto intersections of different types of Borel subalgebras in the quantum affine algebras introduced firstly in [8]. Of course, it also uses the isomorphism between the current [6] and the L-operator formulations of the quantum affine algebras [23] investigated in [5]. Quite analogously one can construct dual off-shell Bethe vectors Ca,b(ū; v̄) defined in the dual space V ∗ with the dual vacuum vector 〈0\| ∈ V ∗: 〈0\|Tij(u) = 0, i < j, 〈0\|Tii(u) = λi(u)〈0\|. They can be also explicitly written as polynomials in the monodromy matrix elements T21(u), T32(u), T31(u) with rational coefficients using the current realization of the quantum affine algebra Uq(ĝl3) [2]. Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix 3 For the class of nested quantum integrable models where the inverse scattering problem can be solved and local operators can be expressed in terms of the monodromy matrix elements [20], one can now address the problem of calculation of the form factors and the correlation functions of local operators. It was done in [16] for the quantum integrable models with GL(2) trigonometric R-matrix, using determinant formulae for the scalar products of the Bethe vectors obtained in [24]. To approach this problem one has to answer the following question. Whether the action of the monodromy matrix elements onto nested off-shell Bethe vectors produces linear combinations of vectors with the same structure. If this is true, then the problem of computing the form factors of local operators can be reduced to the calculation of the scalar products between off-shell and on-shell1 Bethe vectors. Moreover, since right and left Bethe vectors are presented as linear combinations of products of the monodromy matrix elements, the calculation of these scalar products itself can be also reduced to the application of the action formulae of the monodromy matrix elements onto Bethe vectors. The goal of this paper is to give a positive answer to this question and to present and prove the explicit formulae for such an action. We should say that in case of quantum integrable models with GL(2) R-matrix, the question about the action formulae is almost trivial, since the right and left off-shell Bethe vectors in this case are given by the product of the monodromy matrix elements T12(u) and T21(u) respectively. These action formulae can be easily extracted from the RTT relation (1.1) for the monodromy operators. In higher-rank systems, due to the nontrivial structure of the nested Bethe vectors, the application of the RTT relations for the calculation of the action formulae becomes a very complicated combinatorial problem. In the following, to solve it, we will use the presentation of the nested off-shell Bethe vectors in terms of the current generators of the quantum affine algebra Uq(ĝl3) and the relation between the monodromy matrix elements and the current generators given by the Gauss decomposition. 2 Quantum affine algebra Uq(ĝl3) In order to reach the goal of the paper, rather than working with a specific quantum integrable model whose monodromy matrix satisfies the commutation relations (1.1), we deal with a more abstract situation. We consider the universal monodromy matrix which coincides with the L- operator of the positive Borel subalgebra of the quantum affine algebra Uq(ĝl3). There exists an isomorphism [5] between the L-operators [23] and the current [6] formulations of this algebra. The expression of the universal Bethe vectors in terms of the current generators was computed in [12], see also equations (3.5), (3.6) below. Using these data, we will calculate the action of the monodromy matrix elements onto these Bethe vectors using essentially the commutations relations of the algebra Uq(ĝl3) in the current realization. The aim of this section is to introduce these algebraic objects. 2.1 Two realizations of Uq(ĝl3) The quantum affine algebra Uq(ĝl3) is an associative algebra with unit. In the L-operator formulation [23] it is generated by the modes L±ij [n], i, j = 1, 2, 3, n ≥ 0 such that L+ ji[0] = L−ij [0] = 0, 1 ≤ i < j ≤ 3. (2.1) 1These are the Bethe vectors whose parameters satisfy the Bethe equations. 4 S. Belliard, S. Pakuliak, E. Ragoucy and N.A. Slavnov These modes can be gathered into the generating series2 L±(u) = ∑ n≥0 3∑ i,j=1 Eij ⊗ L±ij [n]u∓n ∈ End ( C3 ) ⊗ Uq(b±), (2.2) where Uq(b±) ⊂ Uq(ĝl3) are the positive and negative Borel subalgebras of the quantum affine algebra Uq(ĝl3). These generating series can be called universal monodromy matrices since they satisfy the same as (1.1) commutation relation R(u, v) · ( Lµ(u)⊗ 1 ) · (1⊗ Lν(v)) = (1⊗ Lν(v)) · (Lµ(u)⊗ 1) · R(u, v), (2.3) where µ, ν = ±. The quantum affine algebra Uq(ĝl3) is a Hopf algebra and the Borel subalgebras generated by the modes of the L-operators L±(u) are Hopf subalgebras for the standard coproduct ∆ ( L±ij(u) ) = 3∑ k=1 L±kj(u)⊗ L±ik(u). In what follows we will need another realization of the same algebra, the so-called current realization of the quantum affine algebra Uq(ĝl3) given in [6]. To relate the current and L- operator realizations of the same algebra we introduce, according to [5], the Gauss decomposition of the L-operator L±(u) = 1 F±21(u) F±31(u) 0 1 F±32(u) 0 0 1 k±1 (u) 0 0 0 k±2 (u) 0 0 0 k±3 (u)  1 0 0 E±12(u) 1 0 E±13(u) E±23(u) 1  , (2.4) that is to say L±ab(u) = F±ba(u)k+ b (u) + ∑ b<m≤3 F±ma(u)k+ m(t)E±bm(u), a < b, (2.5) L±bb(u) = k±b (u) + ∑ b<m≤3 F±mb(u)k±m(u)E±bm(u), (2.6) L±ab(u) = k±a (u)E±ba(u) + ∑ a<m≤3 F±ma(u)k±m(u)E±bm(u), a > b. (2.7) It was proved in the paper [5] that, after substitution of the decompositions (2.5)–(2.7) into the commutation relations (2.3), one can obtain for the linear combinations of the Gauss coordinates Fi(t) = F+ i+1 i(t)− F−i+1 i(t), Ei(t) = E+ i i+1(t)− E−i i+1(t) (2.8) and k±i (t) the following commutation relations:( q−1z − qw ) Ei(z)Ei(w) = Ei(w)Ei(z) ( qz − q−1w ) , (2.9) (z − w)Ei(z)Ei+1(w) = Ei+1(w)Ei(z) ( q−1z − qw ) , (2.10) k±i (z)Ei(w) ( k±i (z) )−1 = z − w q−1z − qw Ei(w), (2.11) 2There is also one relation for the zero modes of the diagonal matrix elements of L-operators L+ jj [0]L − jj [0] = 1, j = 1, 2, 3, which is not important for our considerations. Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix 5 k±i+1(z)Ei(w) ( k±i+1(z) )−1 = z − w qz − q−1w Ei(w), (2.12) k±i (z)Ej(w) ( k±i (z) )−1 = Ej(w), if i 6= j, j + 1, (2.13)( qz − q−1w ) Fi(z)Fi(w) = Fi(w)Fi(z) ( q−1z − qw ) , (2.14)( q−1z − qw ) Fi(z)Fi+1(w) = Fi+1(w)Fi(z)(z − w), (2.15) k±i (z)Fi(w) ( k±i (z) )−1 = q−1z − qw z − w Fi(w), (2.16) k±i+1(z)Fi(w) ( k±i+1(z) )−1 = qz − q−1w z − w Fi(w), (2.17) k±i (z)Fj(w) ( k±i (z) )−1 = Fj(w), if i 6= j, j + 1, (2.18) [Ei(z), Fj(w)] = δi,jδ(z/w) ( q − q−1 ) ( k+ i (z)/k+ i+1(z)− k−i (w)/k−i+1(w) ) , (2.19) plus the Serre relations for the currents Ei(z) and Fi(z) which are unimportant for this paper. The commutation relations for the algebra Uq(ĝl3), given in terms of the currents, should be considered as formal series identities describing the infinite set of relations between the modes of these currents. The symbol δ(z) entering these relations is the formal series ∑ n∈Z zn. For any series G(t) = ∑ m∈Z G[m]t−m we denote G(t)(+) = ∑ m>0 G[m]t−m, and G(t)(−) = − ∑ m≤0 G[m]t−m. Using this notation the Ding–Frenkel formulae (2.8) can be inverted F±i+1 i(z) = z ( z−1Fi(z) )(±) , E±i i+1(z) = Ei(z) (±). (2.20) 2.2 Different type Borel subalgebras and ordering of current generators The isomorphism between the L-operator [23] and the current [6] formulations of the quantum affine algebra, proved in [5], allows one to express the modes of the L-operators through the modes of the currents and vice versa using the initial relation (2.1) and the formulae (2.5)–(2.7). On the other hand, it was proved in [15] that the current generators for the quantum affine algebras form the part of the Cartan–Weyl basis in these algebras. There exists a natural ordering in the Cartan–Weyl basis. If the generator eγ corresponds to a positive root γ = α + β, where α and β are roots, then these generators are ordered either in a way eα ≺ eγ ≺ eβ or in the way eβ ≺ eγ ≺ eα. An important property of the Cartan– Weyl basis of a Borel subalgebra of the quantum algebras is that the q-commutator of any two generators from this subalgebra, say eα and eβ, is a linear combination of monomials containing only the products of generator eγi which are ‘between’ eα and eβ: eα ≺ eγi ≺ eβ or eα � eγi � eβ. This property of the Cartan–Weyl basis allows one to describe easily the subalgebras in the quantum affine algebras. For instance, in the example above all generators corresponding to the roots α, γi, β form a subalgebra by definition. The standard positive Borel subalgebra in Uq(ĝl3) generated by the modes of L-operators (2.2) is formed by the Cartan–Weyl generators which are ‘between’ the affine root generator eα0 and non-affine negative simple roots generators e−α1 and e−α2 . Respectively, the negative Borel subalgebra is formed by the generators which are ‘between’ eα1 , eα2 and e−α0 . The ordering on the Borel subalgebra can be extended to the ordering of the whole set of Cartan–Weyl generators corresponding to the positive and negative roots such that the same ordering property is valid. This ordering is called ‘circular’ or ‘convex’ and it allows one to order arbitrary monomials in the whole algebra [7]. 6 S. Belliard, S. Pakuliak, E. Ragoucy and N.A. Slavnov Figure 1. Subalgebras of Uq(gl3). The vertical dotted line separates the standard Borel subalgebras. The horizontal dotted line separates the current Borel subalgebras. The horizontal solid axis indicates the increasing of the current generators modes. Ovals denote different subalgebras in the Uq(gl3) standard and current Borel subalgebras. We consider two types of Borel subalgebras of the algebra Uq(ĝl3). Standard positive and negative Borel subalgebras Uq(b ±) ⊂ Uq(ĝl3) are generated by the modes of the L-operators L(±)(u) respectively. For the generators in these subalgebras we can use the modes of the Gauss coordinates (2.5)–(2.7) E±i i+1(u), F±i+1 i(u), k±j (u), i = 1, 2, j = 1, 2, 3. Another type of Borel subalgebras is related to the current realizations of Uq(ĝl3) given in the previous subsection. The Borel subalgebra UF ⊂ Uq(ĝl3) is generated by modes of the currents Fi[n], k+ j [m], i = 1, 2, j = 1, 2, 3, n ∈ Z and m ≥ 0. The Borel subalgebra UE ⊂ Uq(ĝl3) is generated by the modes of the currents Ei[n], k−j [−m], i = 1, 2, j = 1, 2, 3, n ∈ Z and m ≥ 0. We will consider also a subalgebras U ′F = UF \ {k+ j [0]} and U ′E = UE \ {k−j [0]}.3 Further, we will be interested in the intersections, U−F = U ′F ∩ Uq(b−), U+ F = UF ∩ Uq(b+), U−E = UE ∩ Uq(b−), U+ E = U ′E ∩ Uq(b+), and will describe properties of projections to these intersections. We call UF and UE the current Borel subalgebras. Let Uf ⊂ UF and Ue ⊂ UE be the subalgebras of the current Borel subalgebras generated by the modes of the currents Fi[n] and Ei[n], i = 1, 2, n ∈ Z only. In what follows we will use the subalgebras U+ f ⊂ Uf and U+ e ⊂ Ue defined by the intersections U+ f = U+ F ∩ Uf U+ e = U+ E ∩ Ue. Let U±k be subalgebras in Uq(ĝl3) generated by the modes of the Cartan currents k±j (u). We fix a ‘circular’ ordering ‘≺’ on the generators of U q(gl3) (see [7]), such that: · · · ≺ U−k ≺ U − f ≺ U + f ≺ U + k ≺ U + e ≺ U−e ≺ U−k ≺ · · · . (2.21) The ordering of the subalgebras described above can be pictured in the Fig. 1 in the anti- clockwise direction. We will call an element W ∈ U q(gl3) normal ordered and denote it as :W : if it is presented as linear combinations of products W1 ·W2 ·W3 ·W4 ·W5 ·W6 such that W1 ∈ U−f , W2 ∈ U+ f , W3 ∈ U+ k , W4 ∈ U+ e , W5 ∈ U−e , W6 ∈ U−k . 3In order to obtain the quantum affine algebra Uq(ĝl3) in the framework of the quantum double construction [6] one has to impose the relation k+ j [0]k − j [0] = 1, j = 1, 2, 3. Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix 7 We may consider the standard Borel subalgebras as ordered with respect to the circular ordering (2.21): Uq ( b− ) = U−e · U−k · U − f , Uq ( b+ ) = U+ f · U + k · U + e . An analogous statement is valid for the current Borel subalgebras: UF = U−f · U + f · U + k , UE = U+ e · U−e · U−k . Let us note that the matrix elements in the universal monodromy matrix L+(u) given by the formulae (2.5)–(2.7) are normal ordered, i.e. :L+(u): = L+(u). The problem which we address in this paper, namely the calculation of the action of the monodromy matrix elements onto off-shell Bethe vectors, can be reformulated in the following way. We should put the product of these elements and the element P+ f (F2(vb) · · ·F2(v1) · F1(ua) · · ·F1(u1)) ∈ U+ f into its normal order form, modulo terms which annihilate the right vacuum vector \|0〉. Using the Gauss decompositions (2.5)–(2.7), it could be reduced to the commutation of the Gauss coordina- tes E+ ij (u) with the element P+ f (F2(vb) · · ·F2(v1) · F1(ua) · · ·F1(u1)). However, this way of doing the normal ordering is almost equivalent to the use of the RTT commutation relations and is far too complicated to be useful for our purpose. In fact, in this paper, we will employe a different and more efficient strategy: we will use the method of projections introduced in [8] and exploited in a series of papers (see [12] and references therein) to relate the off-shell Bethe vectors with the current realization of the quantum affine algebras. We refer the reader to the above mentioned papers to find a complete theory of the projections onto intersections of the different types of Borel subalgebras. Here, we will give only some short definitions on projections. In order to do this, we need to equip the algebra Uq(ĝl3) together with its decomposition into current Borel subalgebras by the current Hopf structure ∆(D) (Ei(z)) = Ei(z)⊗ 1 + k−i (z) ( k−i+1(z) )−1 ⊗ Ei(z), ∆(D) (Fi(z)) = 1⊗ Fi(z) + Fi(z)⊗ k+ i (z) ( k+ i+1(z) )−1 , ∆(D) ( k±i (z) ) = k±i (z)⊗ k±i (z). (2.22) According to the general theory [7] we introduce the projection operators P±f : UF ⊂ Uq(ĝl3)→ U±F , P±e : UE ⊂ Uq(ĝl3)→ U±E . They are respectively defined by the prescriptions P+ f (f−f+) = ε(f−)f+, P−f (f−f+) = f−ε(f+), ∀ f− ∈ U−F , ∀ f+ ∈ U+ F , (2.23) P+ e (e+e−) = e+ε(e−), P−e (e−e+) = ε(e+)e−, ∀ e− ∈ U−E , ∀ e+ ∈ U+ E , (2.24) where the counit map ε : Uq(ĝl3)→ C is defined on current generators as follows ε(1) = ε ( k±j (u) ) = 1, ε(Ei(u)) = ε (Fi(u)) = 0. Denote by UF and UE the extensions of the algebras UF and UE formed by infinite sums of monomials which are ordered products ai1 [n1] · · · aik [nk] with n1 ≤ · · · ≤ nk, where ail [nl] is either Fil [nl] or k+ il [nl] and Eil [nl] or k−il [nl], respectively. It can be checked that (1) the action of the projections (2.23) can be extended to the algebra UF ; (2) for any f ∈ UF with ∆(D)(f) = ∑ i f ′i ⊗ f ′′i we have f = ∑ i P−f (f ′′i ) · P+ f (f ′i); (2.25) 8 S. Belliard, S. Pakuliak, E. Ragoucy and N.A. Slavnov (3) the action of the projections (2.24) can be extended to the algebra UE ; (4) for any e ∈ UE with ∆(D)(e) = ∑ i e′i ⊗ e′′i we have e = ∑ i P+ e (e′i) · P−e (e′′i ). (2.26) The formulae (2.25) and (2.26) are the main technical tools to calculate the projections of currents. These formulae allow us to present a product of currents in a normal ordered form using projections and the rather simple current Hopf structure (2.22). The Ding–Frenkel isomorphism between L-operator and current realizations of the quan- tum affine algebra Uq(ĝlN ) [5] identifies the Gauss coordinates and the full currents through formulae (2.8) and (2.20). It is clear that the Gauss coordinates F±i+1 i(u) = P±f (Fi(u)) and E±i i+1 = P±e (Ei(u)) are defined by the corresponding projections of the full currents. But there are also higher Gauss coordinates F±ji(u) and E±ij(u) for j > i + 1 and their relation to the currents was not established in [5]. In [12], special elements from the completed algebras UF and UE were introduced such that their projections yield the corresponding higher Gauss co- ordinates. These elements were called ‘composed’ currents. In the case of the quantum affine algebra Uq(ĝl3), there are only two composed currents F3,1(u) ≡ ( q − q−1 ) F1(u)F2(u), E1,3(u) ≡ ( q − q−1 ) E2(u)E1(u), (2.27) such that P+ f (F3,1(u)) = ( q − q−1 ) F+ 31(u), P+ e (E1,3(u)) = ( q − q−1 ) E+ 13(u). 3 Main results 3.1 Notations To save space and simplify presentation, we use the following convention for the products of the commuting entries of the monodromy matrix Tij(w), vacuum eigenvalues λi(w) and their ratios rk(w) = λk(w)/λ2(w), k = 1, 3. Namely, whenever such an operator or a scalar function depends on a set of variables (for instance, Tij(w̄), λi(ū), rk(v̄)), this means that we deal with the product of the operators or scalar functions with respect to the corresponding set: Tij(w̄) = ∏ wk∈w̄ Tij(wk); λ2(ū) = ∏ uj∈ū λ2(uj); rk(v̄`) = ∏ vj∈v̄ vj 6=v` rk(vj). A similar convention will be used for the products of functions f(u, v) and g(u, v) f(wi, w̄i) = ∏ wj∈w̄ wj 6=wi f(wi, wj); g(ū, v̄) = ∏ uj∈ū ∏ vk∈v̄ g(uj , vk). The notation v̄` for an arbitrary set v̄ means the set v̄ \ {v`}. We will also use the sets w̄<j = {w1, ..., wj−1} and w̄>j = w̄j \ w̄<j with obvious convention for the products. Partitions of sets will be noted as ū⇒ {ūI, ūII}. To simplify further formulae we will introduce a special notation for product of non-com- muting currents: F1(ū) = F1(ua)F1(ua−1) · · ·F1(u1), F2(v̄) = F2(vb) · · ·F2(v2)F2(v1) (3.1) Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix 9 and F1(ūj) = F1(ua) · · ·F1(uj+1)F1(uj−1) · · ·F1(u1), F2(v̄i) = F2(vb) · · ·F2(vi+1)F2(vi−1) · · ·F2(v1). (3.2) These notations are in accordance with the one used for commuting objects, except that now one needs to specify the order as prescribed in (3.1) and (3.2). In various formulae below the Izergin determinant Kk(x̄\|ȳ) appears [11]. It is defined for two sets x̄ and ȳ of the same cardinality #x̄ = #ȳ = k: Kk(x̄\|ȳ) = ∏ 1≤i,j≤k (qxi − q−1yj)∏ 1≤i<j≤k (xi − xj)(yj − yi) · det [ q − q−1 (xi − yj)(qxi − q−1yj) ] . (3.3) Below we also use two modifications of the Izergin determinant K (l) k (x̄\|ȳ) = k∏ i=1 xi · Kk(x̄\|ȳ), K (r) k (x̄\|ȳ) = k∏ i=1 yi · Kk(x̄\|ȳ). (3.4) Some properties of the Izergin determinant and its modifications are gathered into Appendix A. 3.2 Explicit expression for Bethe vectors The right and left off-shell Bethe vectors can be presented using the current realization of the quantum affine algebra Uq(ĝl3) [12] Ba,b(ū; v̄) = β(ū\|v̄) f(v̄, ū) P+ f (F2(vb) · · ·F2(v1) · F1(ua) · · ·F1(u1)) · r3(v̄)\|0〉, (3.5) Ca,b(ū; v̄) = β(ū\|v̄) f(v̄, ū) 〈0\|r3(v̄)P+ e (E1(u1) · · ·E1(ua) · E2(v1) · · ·E2(vb)) , (3.6) where β(ū\|v̄) = ∏ 1≤`<`′≤a f(u`′ , u`) ∏ 1≤`<`′≤b f(v`′ , v`), and P+ f and P+ e are projections onto subalgebras of Uq(ĝl3) generated by the non-negative and positive modes of the simple root currents Fi(u) and Ei(u), i = 1, 2, respectively. These projections onto subalgebras in the positive Borel subalgebra of Uq(ĝl3) were introduced in [8] and their detailed theory was developed in [7]. The formal definition of these projections is given in the present paper through the formulae (2.23) and (2.24). In what follows we will consider the action of the universal monodromy matrix elements expressed in terms of the Gauss coordinates (2.4) or in terms of the current generators of the quantum affine algebra Uq(ĝl3) onto universal off-shell Bethe vectors (3.5). To obtain explicit formulae for this action we do not need to calculate the projection in (3.5), but use a special presentation for this projection found in [10] (see also (4.2) below). Using this presentation we only need the commutation relations of the total currents which are much more simple than the RTT -relations or the relations between Gauss coordinates. Note that the function β(ū\|v̄) removes all the poles and zeros which originate from the product of currents of the same type, while the product of functions f(v̄, ū) removes all the poles which originate from the product of currents of different types. Indeed, the product Fi(u2)Fi(u1) has a simple pole at the point u1 = q2u2 and a simple zero at u1 = u2, while the product F2(v)F1(u) has a simple pole at the point u = v. These ‘analytical’ properties of the product of currents are determined by the commutation relations (2.14), (2.15) and were explained in details in the papers [12, 21] using the notion of ordering of the current generators. 10 S. Belliard, S. Pakuliak, E. Ragoucy and N.A. Slavnov 3.3 Multiple action of Tij(w̄) operators on Bethe vectors Now we give the main result of this paper, namely a complete list of the multiple actions of the operators Tij(w̄) onto the Bethe vectors Ba,b(ū; v̄). Proposition 3.1. Throughout the proposition, we denote {v̄, w̄} = ξ̄, {ū, w̄} = η̄ and #w̄ = n. The multiple actions of the Tij(w̄) operators onto the Bethe vectors Ba,b(ū; v̄) are given by: • Multiple action of T13 T13(w̄)Ba,b(ū; v̄) = λ2(w̄)Ba+n,b+n(η̄; ξ̄). (3.7) • Multiple action of T12 T12(w̄)Ba,b(ū; v̄) = λ2(w̄) ∑ f(ξ̄II, ξ̄I) f(w̄, ξ̄I) K(r) n (w̄\|ξ̄I)Ba+n,b(η̄; ξ̄II). (3.8) The sum is taken over partitions of ξ̄ ⇒ {ξ̄I, ξ̄II} with #ξ̄I = n. • Multiple action of T23 T23(w̄)Ba,b(ū; v̄) = λ2(w̄) ∑ f(η̄I, η̄II) f(η̄I, w̄) K(l) n (η̄I\|w̄)Ba,b+n(η̄II; ξ̄). (3.9) The sum is taken over partitions of η̄ ⇒ {η̄I, η̄II} with #η̄I = n. • Multiple action of T22 T22(w̄)Ba,b(ū; v̄) = λ2(w̄) ∑ f(ξ̄II, ξ̄I)f(η̄I, η̄II) f(w̄, ξ̄I)f(η̄I, w̄) K(r) n (w̄\|ξ̄I)K(l) n (η̄I\|w̄)Ba,b(η̄II; ξ̄II). (3.10) The sum is taken over partitions of: η̄ ⇒ {η̄I, η̄II} with #η̄I = n; ξ̄ ⇒ {ξ̄I, ξ̄II} with #ξ̄I = n. • Multiple action of T11 T11(w̄)Ba,b(ū; v̄) = λ2(w̄) ∑ r1(η̄I) f(ξ̄II, η̄I) f(ξ̄II, ξ̄I)f(η̄II, η̄I) f(w̄, ξ̄I)f(ξ̄I, η̄I) K(r) n (w̄\|ξ̄I)K(r) n (ξ̄I\|η̄I)Ba,b(η̄II; ξ̄II). (3.11) The sum is taken over partitions of: η̄ ⇒ {η̄I, η̄II} with #η̄I = n; ξ̄ ⇒ {ξ̄I, ξ̄II} with #ξ̄I = n. • Multiple action of T33 T33(w̄)Ba,b(ū; v̄) = λ2(w̄) ∑ r3(ξ̄I) f(ξ̄I, η̄II) f(ξ̄I, ξ̄II)f(η̄I, η̄II) f(ξ̄I, η̄I)f(η̄I, w̄) K(l) n (η̄I\|w̄)K(l) n (ξ̄I\|η̄I)Ba,b(η̄II; ξ̄II). (3.12) The sum is taken over partitions of: η̄ ⇒ {η̄I, η̄II} with #η̄I = n; ξ̄ ⇒ {ξ̄I, ξ̄II} with #ξ̄I = n. • Multiple action of T21 T21(w̄)Ba,b(ū; v̄) = λ2(w̄) ∑ r1(η̄I) f(η̄II, η̄I)f(η̄II, η̄III)f(η̄III, η̄I)f(ξ̄II, ξ̄I) f(ξ̄, η̄I)f(w̄, ξ̄I)f(η̄II, w̄) × K(l) n (η̄II\|w̄)K(r) n (ξ̄I\|η̄I)K(r) n (w̄\|ξ̄I) Ba−n,b(η̄III; ξ̄II). (3.13) The sum is taken over partitions of: η̄ ⇒ {η̄I, η̄II, η̄III} with #η̄I = #η̄II = n; ξ̄ ⇒ {ξ̄I, ξ̄II} with #ξ̄I = n. Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix 11 • Multiple action of T32 T32(w̄)Ba,b(ū; v̄) = λ2(w̄) ∑ r3(ξ̄I) f(ξ̄I, ξ̄II)f(ξ̄I, ξ̄III)f(ξ̄III, ξ̄II)f(η̄I, η̄II) f(ξ̄I, η̄)f(η̄I, w̄)f(w̄, ξ̄II) × K(l) n (η̄I\|w̄)K(l) n (ξ̄I\|η̄I)K(r) n (w̄\|ξ̄II) Ba,b−n(η̄II; ξ̄III). (3.14) The sum is taken over partitions of: ξ̄ ⇒ {ξ̄I, ξ̄II, ξ̄III} with #ξ̄I = #ξ̄II = n; η̄ ⇒ {η̄I, η̄II} with #η̄I = n. • Multiple action of T31 T31(w̄)Ba,b(ū; v̄) = λ2(w̄) ∑ r1(η̄II) r3(ξ̄I)K (l) n (ξ̄I\|η̄I)K(r) n (ξ̄II\|η̄II)K(l) n (η̄I\|w̄)K(r) n (w̄\|ξ̄II) × f(η̄I, η̄II)f(η̄I, η̄III)f(η̄III, η̄II)f(ξ̄I, ξ̄II)f(ξ̄I, ξ̄III)f(ξ̄III, ξ̄II) f(ξ̄I, η̄)f(ξ̄III, η̄II)f(ξ̄II, η̄II)f(η̄I, w̄)f(w̄, ξ̄II) Ba−n,b−n(η̄III; ξ̄III). (3.15) The sum is taken over partitions of: ξ̄ ⇒ {ξ̄I, ξ̄II, ξ̄III} with #ξ̄I = #ξ̄II = n; η̄ ⇒ {η̄I, η̄II, η̄III} with #η̄I = #η̄II = n. Note that the product of the rational functions f(ξ̄I, η̄)f(ξ̄III, η̄II)f(ξ̄II, η̄II) in the denominator of the r.h.s. of (3.15) can be equally rewritten as f(ξ̄, η̄II)f(ξ̄I, η̄I)f(ξ̄I, η̄III). The proof of formulae (3.7)–(3.15) will be divided into two steps. First, we will prove these for- mulae using the current approach and presentation of the off-shell Bethe vectors in the form (3.5) for the action of only one monodromy element, that is #w̄ = n = 1. Then we will use an induc- tion to prove these formulae for n > 1. 4 Proofs In what follows we will identify the monodromy matrix T (u) with the L-operator L+(u) ∈ Uq(b+) from the positive Borel subalgebra of the quantum affine algebra Uq(ĝl3). 4.1 The case #w̄ = 1 As we have already mentioned our first goal is the proof of the action formulae (3.7)–(3.15) for the single action of the monodromy matrix elements onto off-shell Bethe vectors. In this subsection, we perform this calculation using only the commutation relations of Uq(ĝl3) current generators. 4.1.1 Necessary commutation relations Since the essential part of the off-shell Bethe vectors is concentrated in the projection of full current products, we may consider first the action of monodromy elements onto the projection of a special product of the full currents. According to the properties of the projections (2.25) we can present the projection P+ f (F2(v̄) F1(ū)) in the form P+ f (F2(v̄)F1(ū)) = F2(v̄)F1(ū)− ∑ P−f ( F ′′ ) · P+ f ( F ′ ) , (4.1) where the elements F ′ and F ′′ are defined by the coproduct (2.22) ∆(D) (F2(v̄)F1(ū)) = ∑ F ′ ⊗F ′′, and in the r.h.s. of (4.1) the number of currents entering the elements F ′ is less than the total number of currents in the original product F2(v̄)F1(ū). Then we may continue replacing P+ f (F ′) 12 S. Belliard, S. Pakuliak, E. Ragoucy and N.A. Slavnov by the r.h.s. of (4.1) up to the trivial identity P+ f (Fi(w)) = Fi(w) − P−f (Fi(w)) to obtain the presentation of P+ f (F2(v̄)F1(ū)) as a linear combination of terms which are ordered products of negative projections of the currents and the full currents. The idea of calculation of the action of the monodromy elements is to act on this sum first and then apply the projection P+ f to the result. It will be shown below that a lot of terms in this sum disappear. Then, it is easy to control the surviving terms. Let I be the right ideal of Uq(ĝl3) generated by all elements of the form Fi[n] · Uq(ĝl3) for i = 1, 2 and n < 0. We will denote equalities modulo elements in the ideal I by the symbol ‘∼I ’. Note that this ideal is annihilated by the projection P+ f . A useful presentation of the off-shell Bethe vector was proved in the paper [10] using the notion of q-deformed symmetrization (see Corollary 3.6 in that paper). We rewrite this pre- sentation replacing deformed symmetrization by usual symmetrization (with multiplication by a scalar factor). We have4 [10, 13] P+ f (F2(v̄)F1(ū)) = F2(v̄) · F1(ū)− b∑ i=1 P−f [F3,2(vi)] · F2(v̄i) · F1(ū) f(vi, v̄>i) f(v̄>i, vi) − a∑ i=1 P−f [F2,1(ui)] · F2(v̄) · F1(ūi)f(v̄, ui) f(ui, ū>i) f(ū>i, ui) (4.2) − ∑ 1≤i≤b 1≤j≤a P−f [F3,1(uj)] q − q−1 · F2(v̄i) · F1(ūj)g(vi, uj)vif(v̄i, uj) f(vi, v̄>i) f(v̄>i, vi) f(uj , ū>j) f(ū>j , uj) + W, where the elements W are such that P+ f (Tij(w) ·W) = 0. Recall that v̄i and ūj are the sets v̄ \ {vi} and ū \ {uj}. This fact will be checked further using an equivalence Tij(w) · P−f [Fk,l(u)] ∼I δi,k ( q − q−1 )k−l−1 g(w, u)uTlj(w), (4.3) also proved in [10]. Here and in (4.2) the notation Fk,l(u), 1 ≤ l < k ≤ 3 is used to denote the simple and ‘composed’ currents (see (2.27) and discussion on the ‘analytical’ properties of the composed currents in [10, 12]): F2,1(u) ≡ F1(u), F3,2(u) ≡ F2(u), F3,1(u) ≡ ( q − q−1 ) F1(u)F2(u). The equivalence (4.3) allows one to prove easily that P+ f (Tij(w) ·W) = 0 since the elements of W can be presented in general as ∑ P−f (Fc1,k) · P − f (Fc2,l) ·W′ with c1 > k and c2 > l. For example, for k = l = 1 and according to (4.3) the action Tij · P−f (Fc1,1) · P−f (Fc2,1) ·W′ is proportional to δi,c1δ1,c2 = 0 since c2 > 1. This means that the action of the elements of the monodromy elements onto universal off-shell Bethe vectors is defined only by the four terms presented in (4.2). Then, the calculation of this action will be reduced to the commutation of Gauss coordinates entering the monodromy elements (2.4) and the full currents, which is relatively simple. The calculation of the action of the monodromy matrix elements onto the Bethe vector P+ f (F2(v̄)F1(ū)) is decomposed in several steps. First we use formula (4.3) to get rid of the negative projection of the currents and obtain products of the monodromy elements and the full currents. Then we use the explicit expressions of the monodromy matrix elements (2.5)–(2.7) through the Gauss coordinates to calculate the commutation of the Gauss coordinates E+ ij(w), 4The reasons for existence of the presentation (4.2) were explained in the paper [13], where the whole infinite set of the hierarchical relations between Uq(ĝlN ) off-shell Bethe vectors was described in terms of the generating series. Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix 13 k+ i (w) and the full currents, calculating this commutation modulo certain ideals J and K which will be described below. In the next step, we apply the projection P+ f to the result of this calculation to restore the structure of the off-shell Bethe vectors, using formula (3.5). Finally, we rewrite the resulting sum of Bethe vectors as a sum over partitions. To proceed further, we need to know the commutation relations between the Gauss coordi- nates E+ ij(w) and the full currents Fi(u). To identify P+ f (F2(v̄)F1(ū)) with the off-shell Bethe vector we have to act with this element on the right weight singular vector \|0〉. Thus, we can perform the calculations modulo the right ideal J composed from elements Uq(ĝl3) · Ei[n] for i = 1, 2 and n ≥ 0. Moreover, the commutation relations of E+ ij(u) with the full currents Fi(u) produce terms containing the negative Cartan currents k−(u) which can be neglected since they vanish after application of the projection P+ f . We note K the ideal formed by such elements and equalities modulo elements of the ideals J and K will be denoted by ‘∼J ’ and ‘∼K ’ respectively. In what follows we need to express the Gauss coordinate E+ 13(w) through the current gener- ators. From the RLL-relation (2.3) one can obtain the relation (v − u)[L−21(u), L+ 32(v)] = ( q − q−1 ) ( uL+ 22(v)L−31(u)− vL−22(u)L+ 31(v) ) . (4.4) According to the definition (2.2), L−ij(u) are series with respect to non-negative powers of the spectral parameter u. The coefficient at u0 in (4.4) yields the following relation( q − q−1 ) L−22[0]L+ 31(v) = −[L−21[0], L+ 32(v)]. (4.5) Next we use the explicit expression of the L-operator matrix elements in terms of the Gauss coordinates (2.7) and the inverted Ding–Frenkel formulae (2.20) to observe that L−21[0] = −k−2 [0]E1[0], L−22[0] = k−2 [0], L+ 3i(v) = k+ 3 (v)E+ i3(v), i = 1, 2. (4.6) Let us remind that by definition the Gauss coordinate E+ 23(w) coincides with the projection of the simple root currents E2(w) (see (2.20)) E+ 23(v) = P+ e (E2(v)) = ∑ n>0 E2[n]v−n = ∮ dt v E2(t) 1− t/v , i = 1, 2. (4.7) Substituting the relations (4.6) into (4.5) and using the commutation relations E2(t)k−2 [0] = qk−2 [0]E2(t) and k+ 3 (v)E1[0] = E1[0]k+ 3 (v) that follow from (2.11) and (2.13) respectively we obtain finally E+ 13(w) = 1 q − q−1 ∮ dt w(1− t/w) (E1[0]E2(t)− qE2(t)E1[0]) . (4.8) In (4.7) and (4.8) the symbol ∮ dt g(t) means the term g−1 of the formal series g(t) = ∑ n∈Z gnt −n and the rational function 1 1−t/v is understood as a series ∑ n≥0 (t/v)n. Then from (2.19) we observe that [E+ i i+1(w), Fj(u)] ∼K δijg(w, u)uψ+ i (u), [Ei[0], Fj(u)] ∼K δij ( q − q−1 ) ψ+ i (u), ψ+ i (u) = k+ i (u)/k+ i+1(u), i = 1, 2. Using also one more relation E1[0]ψ+ 2 (w)− qψ+ 2 (w)E1[0] = ( q − q−1 ) ψ+ 2 (w)E+ 12(w), 14 S. Belliard, S. Pakuliak, E. Ragoucy and N.A. Slavnov which follows from (2.12) and (2.13), we may conclude that the action of the Gauss coordina- tes E+ ij(u) onto the product of the full currents F2(v̄)F1(ū) is given by the equalities E+ 13(w) · F2(v̄)F1(ū) ∼K,J ∑ 1≤i≤b 1≤j≤a F2(v̄i)F1(ūj)ψ + 2 (vi)ψ + 1 (uj) × g(w, vi)vig(vi, uj)ujf(vi, ūj) f(ū<j , uj) f(uj , ū<j) f(v̄<i, vi) f(vi, v̄<i) , (4.9) E+ 12(w) · F2(v̄)F1(ū) ∼K,J a∑ j=1 F2(v̄)F1(ūj)ψ + 1 (uj)g(w, uj)uj f(ū<j , uj) f(uj , ū<j) , (4.10) E+ 23(w) · F2(v̄)F1(ū) ∼K,J b∑ i=1 F2(v̄i)F1(ū)ψ+ 2 (vi)g(w, vi)vif(vi, ū) f(v̄<i, vi) f(vi, v̄<i) . (4.11) Now that we have established the action of the Gauss coordinates on products of the full current, we can compute the action of the monodromy operators on Bethe vectors. 4.1.2 Calculation of the action • The action of T13(w). Let us specialize the vector Ba+1,b+1(w, ū; v̄, w′) given by the expres- sion (3.5) at the coinciding points w′ = w. We have Ba+1,b+1(w, ū; v̄, w′)\|w′=w = β(ū\|v̄) f(v̄, ū) f(v̄, w′)f(w, ū) f(v̄, w)f(w′, ū) r3(v̄)r3(w′) × w′ − w qw′ − q−1w P+ f ( F2(vb) · · ·F2(v1)F2(w′) · F1(w)F1(ua) · · ·F1(u1) )∣∣∣∣ w′=w \|0〉. (4.12) Using the commutation relations (2.15), the r.h.s. of (4.12) can be written as Ba+1,b+1(w, ū; v̄, w) = β(ū\|v̄) f(v̄, ū) r3(v̄)r3(w) × P+ f (F2(vb) · · ·F2(v1)F1(w) · F2(w)F1(ua) · · ·F1(u1)) \|0〉. (4.13) On the other hand, the action of the elements T13(w), according to the property (4.3), is given only by the first term in the r.h.s. of (4.2), namely by the product of the full currents F2(v̄) · F1(ū), so that using the explicit form T13(w) = F+ 31(w)k+ 3 (w) we can write T13(w)Ba,b(ū; v̄) = β(ū\|v̄) f(v̄, ū) r3(v̄) × P+ f ( F+ 31(w)k+ 3 (w)F2(vb) · · ·F2(v1) · F1(ua) · · ·F1(u1) ) \|0〉. (4.14) Taking into account the relation between the Gauss coordinate F+ 31(w) and the projection of the composed current F3,1(w) = ( q − q−1 ) F1(w)F2(w) [12] P+ f (F3,1(w)) = ( q − q−1 ) F+ 31(w) or F+ 31(w) = P+ f (F1(w)F2(w)) , the property of the projection operator P+ f ( P+ f (A) ·B ) = P+ f (A ·B) , (4.15) and the commutation relation F1(w)F2(w)k3(w) · F2(v) = F2(v) · F1(w)F2(w)k3(w), Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix 15 we conclude that the r.h.s. of (4.14) is equal to the r.h.s. of (4.13) up to multiplication by λ2(w) and hence the relation (3.7) is proved for n = 1. • The action of T12(w). Again, due to (4.3), the action of the monodromy matrix element T12(w) onto the Bethe vector (3.5) is determined by the product of the full currents F2(v̄)·F1(ū). Taking into account that T12(w) = F+ 21(w)k+ 2 (w) + F+ 31(w)k+ 3 (w)E+ 23(w) = F+ 21(w)k+ 2 (w) + T13(w)E+ 23(w), using (4.11) and the commutation relations of the Cartan currents k+ 2 (w) with the full currents given by (2.16) and (2.17) we obtain T12(w)Ba,b(ū; v̄) = λ2(w)f(v̄, w)Ba+1,b(w, {ū; v̄}) + T13(w) b∑ i=1 K (r) 1 (w\|vi)f(v̄i, vi)Ba,b−1(ū; v̄i). (4.16) In (4.16) we replace the function g(w, vi)vi by the function K (r) 1 (w\|vi) using (3.4) and (A.1). In the first term of the r.h.s. of (4.16) we used again the property of the projection (4.15) and the commutation relation F1(w)k+ 2 (w) · F2(v) = F2(v) · F1(w)k+ 2 (w). Using the action of T13(w) onto the off-shell Bethe vector (just calculated above) we may rewrite (4.16) in the form T12(w)Ba,b(ū; v̄) = λ2(w)f(v̄, w)Ba+1,b({w, ū}; v̄) + λ2(w) b∑ i=1 K (r) 1 (w\|vi)f(v̄i, vi)Ba+1,b({w, ū}; {w, v̄i}), (4.17) which can be rewritten in the form (3.8) as a sum over partitions of the set ξ = {w, v̄} ⇒ {ξ̄I, ξ̄II}, for #ξ̄I = 1 since K (l,r) 1 (w\|ξ̄I) f(w, ξ̄I) ∣∣∣∣∣ ξ̄I={w} = 1. (4.18) The action (3.8) for n = 1 is proved. • The action of T23(w). According to (4.3) the action of the monodromy matrix element T23(w) = F+ 32(w)k+ 3 (w) will be defined by the first and third terms of the r.h.s. of (4.2) which produce two terms in the action: T23(w)Ba,b(ū; v̄) = λ2(w)f(w, ū)Ba,b+1(ū; {v̄, w}) − T13(w) a∑ j=1 g(w, uj)ujf(uj , ūj)Ba−1,b(ūj ; v̄), or T23(w)Ba,b(ū; v̄) = λ2(w)f(w, ū)Ba,b+1(ū; {v̄, w}) + λ2(w) a∑ j=1 K (l) 1 (uj \|w)f(uj , ūj)Ba,b+1({ūj , w}; {v̄, w}). Due to (4.18), they can be rewritten as the sum over partition of the set η = {w, ū} ⇒ {η̄I, η̄II}, for #η̄I = 1. The action (3.9) for n = 1 is proved. 16 S. Belliard, S. Pakuliak, E. Ragoucy and N.A. Slavnov • The action of T22(w). The action of the matrix element T22(w) = k+ 2 (w) + F+ 32(w)k+ 3 (w)E+ 23(w) onto the off-shell Bethe vector (3.5) is determined according to (4.3) by the first and the third terms in (4.2) and using (4.11) we obtain T22(w)Ba,b(ū; v̄) = λ2(w)f(w, ū)f(v̄, w)Ba,b(ū; v̄) + λ2(w)f(w, ū) b∑ i=1 g(w, vi)vif(v̄i, vi)Ba,b(ū; {v̄i, w}) + a∑ j=1 g(uj , w)ujf(uj , ūj)T12(w)Ba−1,b(ūj ; v̄). (4.19) Using now the explicit formula (4.17) for the action of the monodromy matrix element T12(w) onto the off-shell Bethe vector we may rewrite (4.19) in the form T22(w)Ba,b(ū; v̄) = λ2(w)f(w, ū)f(v̄, w)Ba,b(ū; v̄) + λ2(w)f(w, ū) b∑ i=1 K (r) 1 (w\|vi)f(v̄i, vi)Ba,b(ū; {v̄i, w}) + λ2(w)f(v̄, w) a∑ j=1 K (l) 1 (uj \|w)f(uj , ūj)Ba,b({ūj , w}; v̄) + λ2(w) ∑ 1≤i≤b 1≤j≤a K (l) 1 (uj \|w)K (r) 1 (w\|vi)f(v̄i, vi)f(uj , ūj)Ba,b({ūj , w}; {v̄i, w}), (4.20) which can be presented as sum over partitions (3.10) of the sets η = {w, ū} ⇒ {η̄I, η̄II} and ξ = {w, v̄} ⇒ {ξ̄I, ξ̄II} for #η̄I = #ξ̄I = 1. (4.21) The action (3.10) for n = 1 is proved. • The action of T11(w). The action of the matrix element T11(w) = k+ 1 (w) + F+ 21(w)k+ 2 (w)E+ 12(w) + F+ 31(w)k+ 3 (w)E+ 13(w) = k+ 1 (w) + F+ 21(w)k+ 2 (w)E+ 12(w) + T13(w)E+ 13(w) as well as the matrix elements T12(w) and T13(w) is determined due to (4.3) by the first term in (4.2). Using formulae (4.9) and (4.10) we obtain T11(w)Ba,b(ū; v̄) = λ2(w)r1(w)f(ū, w)Ba,b(ū; v̄) + λ2(w) a∑ j=1 r1(uj)K (r) 1 (w\|uj) f(ūj , uj)f(v̄, w) f(v̄, uj) Ba,b({ūj , w}; v̄) + λ2(w) ∑ 1≤i≤b 1≤j≤a r1(uj)K (r) 1 (w\|vi)K(r) 1 (vi\|uj) f(ūj , uj)f(v̄i, vi) f(v̄, uj) Ba,b({ūj , w}; {v̄i, w}). (4.22) The expression (4.22) can be written as the sum (3.11) over partitions (4.21), because the term corresponding to the partition ξII = {v̄i, w} and ηI = {w} vanishes due to presence of the factor f(ξII, ηI) in the denominator of (3.11). The other three types of partitions ξI = ηI = {w}; Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix 17 ξI = {w}, ηI = {uj}; ξI = {vi}, ηI = {uj} yield exactly the three terms in (4.22) due to (4.18). The action (3.11) for n = 1 is proved. • The action of T33(w). According to (4.3) this action will be determined by the first, the second and the forth terms in (4.2). Using these relations, the definition of the universal off-shell Bethe vector (3.5) and the fact that T33(w) = k+ 3 (w) we obtain T33(w)Ba,b(ū; v̄) = λ2(w)r3(w)f(w, v̄)Ba,b(ū; v̄) + λ2(w) b∑ i=1 r3(vi)K (l) 1 (vi\|w) f(vi, v̄i)f(w, ū) f(vi, ū) Ba,b(ū; {v̄i, w}) + λ2(w) ∑ 1≤i≤b 1≤j≤a r3(vi)K (l) 1 (uj \|w)K (l) 1 (vi\|uj) f(vi, v̄i)f(uj , ūj) f(vi, ū) Ba,b({ūj , w}; {v̄i, w}). (4.23) The expression (4.23) can be written as the sum (3.12) over partitions (4.21), because the term corresponding to the partition ηII = {ūj , w} and ξI = {w} vanishes due to the presence of the factor f(ξI, ηII) in the denominator of (3.12). As above, the other three types of partitions ξI = ηI = {w}; ξI = {vi}, ηI = {w}; ξI = {vi}, ηI = {uj} yield exactly the three terms in (4.23), due to (4.18). The action (3.12) for n = 1 is proved. Before continuing with the action of the lower-triangular monodromy matrix entries T21(w), T32(w) and T31(w) onto the off-shell Bethe vectors, let us run a check of the formulae (4.20), (4.22) and (4.23). It is easy to see that these formulae lead to the Bethe equations when one requires that the vector Ba,b(ū; v̄) is an eigenvector of the transfer matrix. Indeed (T11(w) + T22(w) + T33(w))Ba,b(ū; v̄) = τ(w; ū, v̄)Ba,b(ū; v̄), where τ(w; ū, v̄) = λ1(w)f(ū, w) + λ2(w)f(w, ū)f(v̄, w) + λ3(w)f(w, v̄), provided the Bethe equations r1(uj) = f(uj , ūj) f(ūj , uj) f(v̄, uj), r3(vi) = f(v̄i, vi) f(vi, v̄i) f(vi, ū) are satisfied. The coefficient in front of Ba,b({ūj , w}; {v̄i, w}) vanishes due to the trivial identity K (r) 1 (w\|vi)K(r) 1 (vi\|uj) + K (l) 1 (uj \|w)K (r) 1 (w\|vi) + K (l) 1 (uj \|w)K (l) 1 (vi\|uj) = 0. We now compute the action of the lower-triangular monodromy matrix elements onto off- shell Bethe vectors. Let us repeat once again the strategy of our calculation, for example, in the case of the action of the element T21(w) = k+ 2 (w)E+ 12(w) + F+ 32(w)k+ 3 (w)E+ 13(w). The calculation of the action in our approach means to normal order the product T21(w) · P+ f (F2(vb) · · ·F2(v1) · F1(ua) · · ·F1(u1)) . (4.24) It is done in the context of circular ordering of the Cartan–Weyl or current generators of the quantum affine algebra Uq(ĝl3) described in subsection 2.2, and after this ordering one needs to keep only those terms that belong to the subalgebra U+ F . According to the presentation (4.2) and the equivalence (4.3), the r.h.s. of (4.24) can be written as follows P+ f T21(w) · F2(v̄)F1(ū)− a∑ j=1 g(w, uj)ujf(v̄, uj)T11(w) · F2(v̄)F1(ūj) f(uj , ū>j) f(ū>j , uj) , (4.25) 18 S. Belliard, S. Pakuliak, E. Ragoucy and N.A. Slavnov where first we calculate the ordering under projection in (4.25) modulo elements from the ideal J and then apply projection only to those terms which do not belong to this ideal. We can simply remove all the elements from the ideal J in (4.25) before taking the projection, since by definition J \|0〉 = 0. Once it is done, we multiply (4.24) and (4.25) by the product β(ū\|v̄)r3(v̄)f−1(v̄, ū) and act by both of these elements onto right vacuum vector \|0〉 according to the definition (3.5) to recover the action T21(w) onto Ba,b(ū; v̄). Due to the fact that the matrix elements T1`(w), ` = 1, 2, 3, act effectively only on the first term in (4.2) we may formally write T1`(w) · P+ f (F2(v̄) · F1(ū)) = P+ f (T1`(w) · F2(v̄) · F1(ū)) understanding this equality in the sense described above. It means that recovering the Bethe vectors in (4.25), we may first interchange the projection P+ f and the action of T11(w), then restore the Bethe vector from the projection and finally use the already calculated action of the monodromy matrix element T11(w) onto Ba,b(ū; v̄) given by (4.22). This will slightly simplify the whole calculation, although we cannot do the same trick for the calculation of the remaining matrix elements Tij(w), i 6= 1. To calculate the action of these matrix elements onto the off-shell Bethe vectors, we have to use an explicit expression in terms of the Gauss coordinates and the commutation relations of the Gauss coordinates with the full currents. • The action of T21(w). Taking these rules into account and using (4.9) and (4.10) we may calculate T21(w)Ba,b(ū; v̄) = λ2(w) ( a∑ j=1 K (r) 1 (w\|uj)r1(uj) f(w, ūj)f(ūj , uj)f(v̄, w) f(v̄, uj) Ba−1,b(ūj ; v̄) + ∑ 1≤i≤b 1≤j≤a K (r) 1 (w\|vi)K(r) 1 (vi\|uj)r1(uj) f(w, ūj)f(ūj , uj)f(v̄i, vi) f(v̄, uj) Ba−1,b(ūj ; {v̄i, w}) ) + T11(w) a∑ j=1 K (l) 1 (uj \|w)f(uj , ūj)Ba−1,b(ūj ; v̄). (4.26) Then, using (4.22) the expression (4.26) can be written in the form (3.13) with a sum over partitions of the sets η̄ = {ū, w} ⇒ {η̄I, η̄II, η̄III} and ξ̄ = {v̄, w} ⇒ {ξ̄I, ξ̄II} such that #η̄I = #η̄II = #ξ̄I = 1. Note that in doing so, one possible partition ξ̄I = {vi}, ξ̄II = {v̄i, w}, η̄I = {w}, η̄II = {uj}, η̄III = {ūj} yields a zero contribution, due to the factor f−1(ξ̄II, η̄I). The action (3.13) for n = 1 is proved. • The action of T32(w). Repeating the same arguments we may present the intermediate result for the action of this matrix element T32(w)Ba,b(ū; v̄) = λ2(w) ( b∑ i=1 K (r) 1 (w\|vi)r3(w) f(w, v̄i)f(v̄i, vi)f(w, ū) f(w, ū) Ba,b−1(ū; v̄i) + b∑ i=1 K (l) 1 (vi\|w)r3(vi) f(vi, v̄i)f(v̄i, w)f(w, ū) f(vi, ū) Ba,b−1(ū; v̄i) + ∑ 1≤i 6=i′≤b K (l) 1 (vi\|w)K (r) 1 (w\|vi′)r3(vi) f(vi, v̄i)f(v̄i,i′ , vi′)f(w, ū) f(vi, ū) Ba,b−1(ū; {v̄i,i′ , w})  + T12(w) ∑ 1≤j≤a 1≤i≤b K (l) 1 (uj \|w)K (l) 1 (vi\|uj)r3(vi) f(vi, v̄i)f(uj , ūj) f(vi, ū) Ba−1,b−1(ūj ; v̄i). (4.27) Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix 19 Using (4.17) we may present (4.27) in the form (3.14) as sum over partitions of the sets η̄ = {ū, w} ⇒ {η̄I, η̄II} and ξ̄ = {v̄, w} ⇒ {ξ̄I, ξ̄II, ξ̄III} such that #ξ̄I = #ξ̄II = #η̄I = 1. The action (3.14) for n = 1 is proved. • The action of T31(w). The action of the matrix element T31(w) can be calculated analo- gously. The intermediate result of this action is T31(w)Ba,b(ū; v̄) =λ2(w)  ∑ 1≤j≤a 1≤i≤b K (r) 1 (vi\|uj)K(r) 1 (w\|vi)r1(uj)r3(w) f(ūj , uj)f(w, v̄i)f(v̄i, vi) f(v̄, uj) Ba−1,b−1(ūj ; v̄i) + ∑ 1≤j≤a 1≤i≤b K (l) 1 (vi\|w)K (r) 1 (w\|uj)r1(uj)r3(vi) f(ūj , uj)f(w, ūj)f(vi, v̄i)f(v̄i, w) f(vi, uj)f(vi, ūj)f(v̄i, uj) Ba−1,b−1(ūj ; v̄i) + ∑ 1≤j≤a 1≤i 6=i′≤b K (l) 1 (vi\|w)K (r) 1 (vi′ \|uj)K (r) 1 (w\|vi′)r1(uj)r3(vi) × f(ūj , uj)f(w, ūj)f(vi, v̄i)f(v̄i,i′ , vi′) f(vi, uj)f(vi, ūj)f(v̄i, uj) Ba−1,b−1(ūj ; {v̄i,i′ , w})  + T11(w) ∑ 1≤j≤a 1≤i≤b K (l) 1 (vi\|uj)K(l) 1 (uj \|w)r3(vi) f(uj , ūj)f(vi, v̄i) f(vi, ū) Ba−1,b−1(ūj ; v̄i). Using (4.22) we conclude that the final result of the action of the monodromy matrix elements T31(w) can be written in the form (3.15) as sum over partitions of the sets η̄ = {ū, w} ⇒ {η̄I, η̄II, η̄III} and ξ̄ = {v̄, w} ⇒ {ξ̄I, ξ̄II, ξ̄III} such that #ξ̄I = #ξ̄II = #η̄I = #η̄II = 1. The action (3.15) for n = 1 is proved. 4.2 The general case #w̄ = n We have proved the formulae of the multiple actions (3.7)–(3.15) for #w̄ = 1. Then the general case #w̄ = n can be considered via an induction over n. We assume that the equations (3.7)– (3.15) are valid for #w̄ = n− 1 and act successively: first by Tij(w̄n) and then by Tij(wn). The induction for (3.7) is trivial. The proofs of the other formulae require the use of lemma A.1. Consider, for instance, the multiple action of T23(w̄). It is convenient to write (3.9) in the following form: T23(w̄n)Ba,b(ū; v̄) = (−q)1−nλ2(w̄n) ∑ {w̄n,ū}⇒{η̄I,η̄II} f(η̄I, η̄II)K (r) n−1(w̄nq −2\|η̄I)Ba,b+n−1(η̄II; ξ̄). (4.28) Here we have got rid of the poles of K (l) n−1(η̄I\|w̄n) at ηi = wj transforming it into K (r) n−1(w̄nq −2\|η̄I) via (A.2). Thus, the action of T23(w̄n) produces the sum over partitions of the set {w̄n, ū} into subsets η̄I and η̄II. Applying the operator T23(wn) to (4.28) we obtain T23(w̄)Ba,b(ū; v̄) = (−q)−nλ2(w̄) ∑ {w̄n,ū}⇒{η̄I,η̄II} f(η̄I, η̄II)K (r) n−1(w̄nq −2\|η̄I) × ∑ {wn,η̄II}⇒{η̄i,η̄ii} f(η̄i, η̄ii)K (r) 1 (wnq −2\|η̄i)Ba,b+n(η̄ii; ξ̄). (4.29) 20 S. Belliard, S. Pakuliak, E. Ragoucy and N.A. Slavnov Here we have an additional sum over partitions of the set {wn, η̄II} into subsets η̄i and η̄ii. In fact, one can say that we have the sum over partitions of the set {w̄, ū} into three subsets η̄I, η̄i, and η̄ii with one additional constraint wn /∈ η̄I. Obviously f(η̄I, η̄II) = f(η̄I, η̄II)f(η̄I, wn) f(η̄I, wn) = f(η̄I, η̄i)f(η̄I, η̄ii) f(η̄I, wn) . (4.30) It is easy to see that the function in the r.h.s. of (4.30) is a projector of the product f(η̄I, η̄II) onto partitions η̄I, η̄i, and η̄ii, such that wn /∈ η̄I: f(η̄I, η̄i)f(η̄I, η̄ii) f(η̄I, wn) = { f(η̄I, η̄II), if wn /∈ η̄I, 0, if wn ∈ η̄I. (4.31) Then the sum (4.29) takes the form T23(w̄)Ba,b(ū; v̄) = (−q)−nλ2(w̄) ∑ {w̄,ū}⇒{η̄I,η̄i,η̄ii} K (r) n−1(w̄nq −2\|η̄I)K(r) 1 (wnq −2\|η̄i) × f(η̄i, η̄ii)f(η̄I, η̄i)f(η̄I, η̄ii) f(η̄I, wn) Ba,b+n(η̄ii; ξ̄). Setting {η̄I, η̄i} = η̄0 and transforming K (r) 1 (wnq −2\|η̄i) via (A.2) we obtain T23(w̄)Ba,b(ū; v̄) = (−q)1−nλ2(w̄) ∑ {w̄,ū}⇒{η̄0,η̄ii} f(η̄0, η̄ii) f(η̄0, wn) Ba,b+n(η̄ii; ξ̄) × ∑ η̄0⇒{η̄I,η̄i} K (l) 1 (η̄i\|wn)K (r) n−1(w̄nq −2\|η̄I)f(η̄I, η̄i). (4.32) The sum over partitions η̄0 ⇒ {η̄I, η̄i} in the last line of (4.32) can be computed via (A.5), what gives us T23(w̄)Ba,b(ū; v̄) = λ2(w̄) ∑ {w̄,ū}⇒{η̄0,η̄ii} f(η̄0, η̄ii)f(w̄nq −2, η̄0) f(η̄0, wn) K(l) n (η̄0\|w̄)Ba,b+n(η̄ii; ξ̄). It remains to use f(w̄nq −2, η̄0) = f−1(η̄0, w̄n), and we arrive at (3.9) with #w̄ = n. All other formulae of multiple actions are proved in exactly the same manner. Successive ac- tion of Tij(w̄n) and Tij(wn) gives a sum over partitions with constraints. Introducing appropriate projectors as in (4.31) we get rid of these constraints. Then certain sums over partitions can be computed via Lemma A.1. The details of these calculations, however, are rather cumbersome, therefore we do not give them here. 5 Conclusion In this paper, we provided the explicit formulae for the monodromy matrix elements acting onto the off-shell nested Bethe vectors. Hopefully these formulae will help to calculate the form factors of local operators, in the framework of the approach developed in [3]. As in the case of rational SU(3)-symmetric quantum integrable models [22], it will also lead to a formula for the scalar products of the off-shell nested Bethe vectors in quantum integrable models with GL(3) Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric R-Matrix 21 trigonometric R-matrix. Indeed, the off-shell Bethe vectors given by formulae (3.5) and (3.6) can be rewritten through the elements of the monodromy matrix5 (see also [12, 21]): Ba,b(ū; v̄) = ∑ K (r) k (v̄I\|ūI) λ2(ūII)λ2(v̄) f(v̄II, v̄I)f(ūI, ūII) f(v̄, ū) T13(v̄I)T23(v̄II)T12(ūII)\|0〉, (5.1) Ca,b(ū; v̄) = ∑ K (l) k (v̄I\|ūI) λ2(ūII)λ2(v̄) f(v̄II, v̄I)f(ūI, ūII) f(v̄, ū) 〈0\|T21(ūII)T32(v̄II)T31(v̄I), (5.2) where the sum goes over all partitions of the sets ū ⇒ {ūI, ūII} and v̄ ⇒ {v̄I, v̄II} such that #ūI = #v̄I = k, k = 0, . . . ,min(a, b). The proof of the formulae (5.1) and (5.2) will be given elsewhere. In principle, one can use these formulae to prove the relations (3.7)–(3.15) using multiple exchange relations and the properties of the Izergin determinant as it was done in [4] for the GL(3)-invariant integrable models associated with rational R-matrix. However, we showed in this paper that the use of current presentation provides a simpler way to perform the calculation. Combining the explicit presentations (5.1) and (5.2) with the multiple actions calculated in the present paper, we can hope to tackle the problem of computing form factors and scalar products. This strategy was applied successfully to the case of GL(3)-invariant integrable models associated with rational R-matrix, giving some hope for the trigonometric case. A Properties of the Izergin determinant The following properties of the Izergin determinant easily follows from the definition (3.3). Initial condition: K1(x̄\|ȳ) = g(x, y). (A.1) Rescaling of the arguments: Kn(αx̄\|αȳ) = α−nKn(x̄\|ȳ). Reduction: Kn(x̄, zq−2\|ȳ, z) = −q z Kn(x̄\|ȳ) and Kn(x̄, z\|ȳ, zq2) = − 1 qz Kn(x̄\|ȳ). Inverse order of arguments: Kn ( x̄q−2\|ȳ ) = (−q)nf−1(ȳ, x̄)Kn(ȳ\|x̄) and Kn ( x̄\|ȳq2 ) = (−q)−nf−1(ȳ, x̄)Kn(ȳ\|x̄). Residues in the poles at xj = yk: Kn(x̄\|ȳ)\|xn→yn = g(xn, yn)f(yn, ȳn)f(x̄n, xn)Kn−1(x̄n\|ȳn) + reg, where reg means regular part. Using these properties of Kn one can easily derive similar properties for its modifications K(l,r), in particular, K(r) n (x̄q−2\|ȳ) = (−q)nf−1(ȳ, x̄)K(l) n (ȳ\|x̄) and K(l) n (x̄\|ȳq2) = (−q)−nf−1(ȳ, x̄)K(r) n (ȳ\|x̄). (A.2) One more important property of Kn(x̄\|ȳ) is a summation formula. 5Observe that, up to the replacement K (l,r) k → Kk, these formulae have the same structure as the formulae for Bethe vectors in rational GL(3)-invariant models. 22 S. Belliard, S. Pakuliak, E. Ragoucy and N.A. Slavnov Lemma A.1 (main lemma). Let γ̄, ᾱ and β̄ be three sets of complex variables with #α = m1, #β = m2, and #γ = m1 +m2. Then∑ Km1(γ̄I\|ᾱ)Km2(β̄\|γ̄II)f(γ̄II, γ̄I) = (−q)−m1f(γ̄, ᾱ)Km1+m2({ᾱq−2, β̄}\|γ̄). (A.3) The sum is taken with respect to all partitions of the set γ̄ ⇒ {γ̄I, γ̄II} with #γ̄I = m1 and #γ̄II = m2. Due to (A.2) the equation (A.3) can be also written in the form∑ Km1(γ̄I\|ᾱ)Km2(β̄\|γ̄II)f(γ̄II, γ̄I) = (−q)m2f(β̄, γ̄)Km1+m2 ( γ̄\| { ᾱ, β̄q2 }) . (A.4) An analog of this lemma was proved in [3, Appendix A]. The proof of (A.3) coincides with the one given in [3]. The equations (A.3), (A.4) yield similar identities involving K(l,r), for instance,∑ K(l) m1 (γ̄I\|ᾱ)K(r) m2 (β̄\|γ̄II)f(γ̄II, γ̄I) = (−q)m2f(β̄, γ̄)K (l) m1+m2 ( γ̄\| { ᾱ, β̄q2 }) . (A.5) Acknowledgements Work of S.P. was supported in part by RFBR grant 11-01-00980-a and grant of Scientific Foun- dation 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. 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. 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J. 6 (1995), 275–313, hep-th/9311040. http://dx.doi.org/10.1016/0393-0440(93)90070-U http://dx.doi.org/10.1016/S0550-3213(99)00295-3 http://dx.doi.org/10.1016/S0550-3213(99)00295-3 http://arxiv.org/abs/math-ph/9807020 http://dx.doi.org/10.1007/BF01095104 http://dx.doi.org/10.1088/0305-4470/16/16/001 http://dx.doi.org/10.1016/S0550-3213(00)00097-3 http://arxiv.org/abs/hep-th/9911030 http://dx.doi.org/10.1090/S1061-0022-2010-01110-5 http://arxiv.org/abs/0711.2821 http://dx.doi.org/10.1007/BF01099200 http://dx.doi.org/10.1007/BF01045884 http://dx.doi.org/10.1007/BF01045884 http://dx.doi.org/10.1007/BF01016531 http://dx.doi.org/10.3842/SIGMA.2013.048 http://arxiv.org/abs/math.QA/0702277 http://arxiv.org/abs/hep-th/9311040 1 Introduction 2 Quantum affine algebra Uq(gl"0362gl3) 2.1 Two realizations of Uq(gl"0362gl3) 2.2 Different type Borel subalgebras and ordering of current generators 3 Main results 3.1 Notations 3.2 Explicit expression for Bethe vectors 3.3 Multiple action of Tij() operators on Bethe vectors 4 Proofs 4.1 The case #=1 4.1.1 Necessary commutation relations 4.1.2 Calculation of the action 4.2 The general case #=n 5 Conclusion A Properties of the Izergin determinant References

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

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