Generating Series for Nested Bethe Vectors

We reformulate nested relations between off-shell Uq(^glN) Bethe vectors as a certain equation on generating series of strings of the composed Uq(^glN) currents. Using inversion of the generating series we find a new type of hierarchical relations between universal off-shell Bethe vectors, useful fo...

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Datum:	2008
Hauptverfasser:	Khoroshkin, S., Pakuliak, S.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2008
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1490052019-02-20T01:23:51Z Generating Series for Nested Bethe Vectors Khoroshkin, S. Pakuliak, S. We reformulate nested relations between off-shell Uq(^glN) Bethe vectors as a certain equation on generating series of strings of the composed Uq(^glN) currents. Using inversion of the generating series we find a new type of hierarchical relations between universal off-shell Bethe vectors, useful for a derivation of Bethe equation. As an example of application, we use these relations for a derivation of analytical Bethe ansatz equations [Arnaudon D. et al., Ann. Henri Poincaré 7 (2006), 1217-1268, math-ph/0512037] for the parameters of universal Bethe vectors of the algebra Uq(^gl2). 2008 Article Generating Series for Nested Bethe Vectors / S. Khoroshkin, S. Pakuliak // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 18 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 17B37; 81R50 http://dspace.nbuv.gov.ua/handle/123456789/149005 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 reformulate nested relations between off-shell Uq(^glN) Bethe vectors as a certain equation on generating series of strings of the composed Uq(^glN) currents. Using inversion of the generating series we find a new type of hierarchical relations between universal off-shell Bethe vectors, useful for a derivation of Bethe equation. As an example of application, we use these relations for a derivation of analytical Bethe ansatz equations [Arnaudon D. et al., Ann. Henri Poincaré 7 (2006), 1217-1268, math-ph/0512037] for the parameters of universal Bethe vectors of the algebra Uq(^gl2).
format	Article
author	Khoroshkin, S. Pakuliak, S.
spellingShingle	Khoroshkin, S. Pakuliak, S. Generating Series for Nested Bethe Vectors Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Khoroshkin, S. Pakuliak, S.
author_sort	Khoroshkin, S.
title	Generating Series for Nested Bethe Vectors
title_short	Generating Series for Nested Bethe Vectors
title_full	Generating Series for Nested Bethe Vectors
title_fullStr	Generating Series for Nested Bethe Vectors
title_full_unstemmed	Generating Series for Nested Bethe Vectors
title_sort	generating series for nested bethe vectors
publisher	Інститут математики НАН України
publishDate	2008
url	http://dspace.nbuv.gov.ua/handle/123456789/149005
citation_txt	Generating Series for Nested Bethe Vectors / S. Khoroshkin, S. Pakuliak // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 18 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT khoroshkins generatingseriesfornestedbethevectors AT pakuliaks generatingseriesfornestedbethevectors
first_indexed	2025-07-12T20:51:18Z
last_indexed	2025-07-12T20:51:18Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 081, 23 pages Generating Series for Nested Bethe Vectors? Sergey KHOROSHKIN † and Stanislav PAKULIAK ‡† † Institute of Theoretical & Experimental Physics, 117259 Moscow, Russia E-mail: khor@itep.ru ‡ Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, Russia E-mail: pakuliak@theor.jinr.ru Received September 14, 2008; Published online November 24, 2008 Original article is available at http://www.emis.de/journals/SIGMA/2008/081/ Abstract. We reformulate nested relations between off-shell Uq(ĝlN ) Bethe vectors as a certain equation on generating series of strings of the composed Uq(ĝlN ) currents. Using inversion of the generating series we find a new type of hierarchical relations between univer- sal off-shell Bethe vectors, useful for a derivation of Bethe equation. As an example of application, we use these relations for a derivation of analytical Bethe ansatz equations [Arnaudon D. et al., Ann. Henri Poincaré 7 (2006), 1217–1268, math-ph/0512037] for the parameters of universal Bethe vectors of the algebra Uq(ĝl2). Key words: Bethe ansatz; current algebras; quantum integrable models 2000 Mathematics Subject Classification: 17B37; 81R50 1 Introduction Hierarchical (nested) Bethe ansatz (NBA) was designed [12] to solve quantum integrable mo- dels with glN symmetries. The cornerstone of NBA is a procedure which relates Bethe vectors for the model with glN symmetry to analogous objects with glN−1 symmetry. This hierarchi- cal procedure is implicit, it allows to obtain Bethe equations for the parameters of the Bethe vectors while the explicit construction of these vectors itself remains rather non-trivial prob- lem. Authors of the papers [16, 17] proposed a closed expression for off-shell Bethe vectors as matrix elements of the monodromy operator built of products of fundamental L-operators. However a calculation of this expression in every particular representation is still a nontrivial problem. Such a calculations was done in [18] on the level of the evaluation homomorphism of Uq(ĝlN ) → Uq(glN ). The construction of [16, 17] yields the off-shell Bethe vectors in terms of matrix elements of the monodromy matrix which satisfies the corresponding quantum Yang–Baxter equation and generate a Borel subalgebra of quantum affine algebra Uq(ĝlN ) [15] for trigonometric R-matrix or the Yangian Y (glN ) for the rational R-matrix. Those quantum affine algebras as well as doubles of Yangians possess another “new” realization introduced in [2]. In this realization the corresponding algebra is described in terms of generating series (currents) and an isomorphism between different realizations of these infinite-dimensional algebras was observed in [3]. Using this isomorphism one may try to look for the expressions for the universal off-shell Bethe vectors in terms of the modes of the currents. This program was realized in [9, 10], where explicit formulas for the off-shell Bethe vectors in terms of the currents were found. A significant part of this approach to the construction of Bethe vectors is a method of projection introduced in [7] and developed in the recent paper [6]. This method operates with projections of Borel subalgebra to its intersections with Borel subalgebras of a different type. It was proved in [6] that universal off-shell Bethe vectors can be identified with the projections of products of the ?This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/Kac-Moody algebras.html mailto:khor@itep.ru mailto:pakuliak@theor.jinr.ru http://www.emis.de/journals/SIGMA/2008/081/ http://arxiv.org/abs/math-ph/0512037 http://www.emis.de/journals/SIGMA/Kac-Moody_algebras.html 2 S. Khoroshkin and S. Pakuliak Drinfeld currents to the intersections of Borel subalgebras of different types. In the latter paper it was checked in a rather general setting that the Bethe vectors obtained from the projections of the currents satisfy the same comultiplication rule as the Bethe vectors constructed in terms of the fundamental L-operators. This approach was used in [11, 14] for further generalization of the results obtained in [18]. A deduction of general expressions for off-shell Bethe vectors [11, 14] is based on hierarchical relations between projections of products of the currents (see Proposition 4.2 in [11]). However these relations do not help in the investigation of the action of integrals of motions on off-shell Bethe vectors. The situation is quite different in classical approach of nested Bethe ansatz. The corresponding hierarchical relations allow to compute the action of integrals of motion and derive the Bethe equations [12, 13], but can be hardly used in the computation of the explicit expressions of Bethe vectors. This signifies an existence of two types of hierarchical relations for off-shell Bethe vectors and thus of two different presentations of them. The goal of this paper is to observe these two type of the hierarchical relations within the approach of the method of projections. The new important objects which appear in the application of the method of projections to the investigation of the off-shell Bethe vectors are so called strings and their projections. Strings are special ordered products of composed current introduced in [4]. Our hierarchical relations express Uq(ĝlN ) Bethe vectors via products of special strings and Uq(ĝlN−1) Bethe vectors. On the other hand the basic point of application of the method of projection is the ordered decomposition of the product of the currents. The factors of this decomposition are projections of the products of the currents (3.12). The crucial observation is that this decomposition and hierarchical relations for the opposite projections of the products of currents have a similar structure. Combinations of relations of these two different type can be used to obtain a new type hierarchical relations for off-shell Bethe vectors. In order to solve the latter problem we collect all off-shell Bethe vectors into multi-variable generating series. We also introduce the generating series for the products of the currents and for the strings and their projections. We rewrite the hierarchical relations for Uq(ĝlN ) Bethe vectors as a simple relation on the product of the generating series of the projections of the strings and the generating series of the Uq(ĝlN−1) Bethe vectors. A similar construction can be performed for off-shell Bethe vectors related to opposite Borel subalgebra. However the product entering into these relations is not usual. It contains a q-symmetrization with a special functional weight. This ?-product is associative and the generating series are invertible with respect to this product. Finally the new type of the hierarchical relations reduces to inversion of the generating series of opposite projections of the strings. This inversion is effectively performed. To do this we intro- duce some combinatorial language of tableaux filled by the Bethe parameters (see Section 4.5). Applications of the new type hierarchical relations to the investigation of the properties of quantum integrals of motion are given in [8]. Here we demonstrate how they work in the simplest case of the universal Uq(ĝl2) Bethe vectors. As a result we get universal Bethe equations of the analytical Bethe ansatz [1]. In contrast to the usual Bethe equations these equations refer to Cartan currents instead of the eigenvalues of the diagonal elements of monodromy matrix on highest weight vectors. In the Appendix we collect the basic defining relations for the Uq(ĝlN ) composed currents. 2 Generating series and ?-product 2.1 A q-symmetrization Let t̄ = {t1, . . . , tn} be a set of formal variables. Let G(t̄) be a Laurent series taking values in Uq(ĝlN ). Consider the permutation group Sn and its action on the formal series of n variables Generating Series for Nested Bethe Vectors 3 t̄ = {t1, . . . , tn} defined for the elementary transpositions σi,i+1 as follows π(σi,i+1)G(t1, . . . , ti, ti+1, . . . , tn) = q−1 − qti/ti+1 q − q−1ti/ti+1 G(t1, . . . , ti+1, ti, . . . , tn), (2.1) where the rational series 1 1−x is understood as a series ∑ n≥0 xn and q is a deformation parameter of Uq(ĝlN ). Summing the action over the group of permutations we obtain the operator Sym t̄ =∑ σ∈Sn π(σ) acting as follows: Sym t̄ G(t̄) = 1 n! ∑ σ∈Sn ∏ `<`′ σ(`)>σ(`′) q−1 − qtσ(`′)/tσ(`) q − q−1tσ(`′)/tσ(`) G(σt). (2.2) The product is taken over all pairs (`, `′), such that conditions ` < `′ and σ(`) > σ(`′) are satisfied simultaneously. We call operator Symt a q-symmetrization. One can check that the operation given by (2.2) is a projector Sym t̄ Sym t̄ (·) = Sym t̄ (·). (2.3) Fix any positive integer N > 1. Let l̄ = {l1, . . . , lN−1} and r̄ = {r1, . . . , rN−1} be the sets of non-negative integers satisfying a set of inequalities la ≤ ra, a = 1, . . . , N − 1. (2.4) Denote by [l̄, r̄] a set of segments which contain positive integers {la + 1, la + 2, . . . , ra − 1, ra} including ra and excluding la. The length of each segment is equal to ra − la. For a given set [l̄, r̄] of segments we denote by t̄[l̄,r̄] the sets of variables t̄[l̄,r̄] = { t1l1+1, . . . , t 1 r1 ; t2l2+1, . . . , t 2 r2 ; . . . ; tN−1 lN−1+1, . . . , t N−1 rN−1 } . (2.5) For any a = 1, . . . , N−1 we denote the sets of variables corresponding to the segments [la, ra] = {la + 1, la + 2, . . . , ra} as t̄a[la,ra] = {tala+1, . . . , t a ra }. All the variables in t̄a[la,ra] have the type a. For the segments [la, ra] = [0, na] we use the shorten notations t̄[0̄,n̄] ≡ t̄[n̄] and t̄a[0,na] ≡ t̄a[na]. We also name for a short collections [l̄, r̄] of segments as a segment. Denote by Sl̄,r̄ = Sl1,r1 × · · · × SlN−1,rN−1 a direct product of the groups Sla,ra permuting integers la + 1, . . . , ra. Let G(t̄[l̄,r̄]) be a series depending on the ratios tai /tbj for a < b and tai /taj for i < j. The q-symmetrization over the whole set of variables t̄[l̄,r̄] of the series G(t̄[l̄,r̄]) is defined by the formula Sym t̄[l̄,r̄] G(t̄[l̄,r̄]) = ∑ σ∈Sl̄,r̄ ∏ 1≤a≤N−1 1 (ra − la)! ∏ `<`′ σa(`)>σa(`′) q−1 − q taσa(`′)/taσa(`) q − q−1taσa(`′)/taσa(`) G(σ t̄[l̄,r̄]), (2.6) where the set σ t̄[l̄,r̄] is defined as σ t̄[l̄,r̄] = { t1σ1(l1+1), . . . , t 1 σ1(r1); t 2 σ2(l2+1), . . . , t 2 σ2(r2); . . . ; t N−1 σN−1(lN−1+1) , . . . , tN−1 σN−1(rN−1) } .(2.7) We say that the series G(t̄[l̄,r̄]) is q-symmetric, if it is invariant under the action π of each group Sla,ra with respect to the permutations of the variables tala+1, . . . , tra for a = 1, . . . , N − 1: Sym t̄[l̄,r̄] G(t̄[l̄,r̄]) = G(t̄[l̄,r̄]). (2.8) 4 S. Khoroshkin and S. Pakuliak Due to (2.3) the q-symmetrization G(t̄[l̄,r̄]) = Sym t̄[l̄,r̄] Q(t̄[l̄,r̄]) of any series Q(t̄[l̄,r̄]) is a q-sym- metric series. The rational function N−1∏ a=1 ∏ i<j 1− tai /taj q − q−1 tai /taj (2.9) which is understood as a series with respect to tai /taj is an example of a q-symmetric series. 2.2 Generating series Let ui, i = 1, . . . , N − 1 be formal parameters. We denote the set of these parameters as ū = {u1, . . . , uN−1}. Define a generating series A(ū) = 1 + ∑ n1,...,nN−1≥0 n̄6=0̄ A(t̄[n̄]) un1 1 un2 2 · · ·unN−1 N−1 , (2.10) where the coefficients A(t̄[n̄]) are arbitrary q-symmetric series (Sym t̄[n̄] ( A(t̄[n̄]) ) = A(t̄[n̄])) of the formal variables t̄n̄ numbered by the multi-index n̄ = {n1, . . . , nN−1}: t̄[n̄] = { t11, . . . , t 1 n1 ; t21, . . . , t 2 n2 ; . . . ; tN−2 1 , . . . , tN−2 nN−2 ; tN−1 1 , . . . , tN−1 nN−1 } . (2.11) We call generating series of this type q-symmetric generating series. Note that the multi-index n̄ of the coefficients of a q-symmetric generating series is uniquely defined by the set of formal variables t̄[n̄], thus the coefficients A(t̄[n̄]) are used without any additional index. However once it will be convenient for us to use instead the notation An̄ ≡ A(t̄[n̄]) (see proof of Proposition 5). For two generating series A(ū) and B(ū) we define ?-product as a generating series C(ū) = A(ū) ? B(ū) = ∑ n1,...,nN−1≥0 C(t̄[n̄]) un1 1 un2 2 · · ·unN−1 N−1 (2.12) with coefficients C(t̄[n̄]) = ∑ 0≤sN−1≤nN−1 · · · ∑ 0≤s1≤n1 Sym t̄[n̄] ( Zs̄(t̄[n̄])A(t̄[s̄,n̄]) · B(t̄[0̄,s̄]) ) , (2.13) where A(t̄[s̄,n̄]) = A ( t1s1+1, . . . , t 1 n1 ; t2s2+1, . . . , t 2 n2 ; . . . ; tN−1 sN−1+1, . . . , t N−1 nN−1 ) (2.14) and Zs̄(t̄[n̄]) = N−2∏ a=1 ∏ sa<`≤na 0<`′≤sa+1 q − q−1ta`/ta+1 `′ 1− ta`/ta+1 `′ . (2.15) Proposition 1. The ?-product is associative, namely, for three arbitrary q-symmetric series A(ū), B(ū) and C(ū) of the form (2.10) (A(ū) ? B(ū)) ? C(ū) = A(ū) ? (B(ū) ? C(ū)) . (2.16) Generating Series for Nested Bethe Vectors 5 Proof. We check an equality (2.16) first in the simplest case of the generating series depending on one generating parameter u. Equating the coefficients at the n-th power of this parameter we obtain from (2.16) an equality Sym t̄  ∑ n≥m≥0 ∑ n≥s≥m A(ts+1, . . . , tn) · B(tm+1, . . . , ts) · C(t1, . . . , tm)  = Sym t̄  ∑ n≥s′≥0 ∑ s′≥m′≥0 A(ts′+1, . . . , tn) · B(tm′+1, . . . , ts′) · C(t1, . . . , tm′)  , (2.17) where the property (2.8) of the q-symmetric generating series was used and t̄ is a set {t1, . . . , tn}. An equality (2.17) is an obvious identity if one replaces the ordering of the summations. It is clear that in the general case the arguments remain the same and the appearing of the series (2.15) does not change these arguments. � Proposition 2. For any generating series A(ū) there exist an unique q-symmetric series B(ū) such that B(ū) ?A(ū) = A(ū) ? B(ū) = 1. Proof. Since A(ū) has the form of a Taylor series with the free term equal to 1, we can always reconstruct uniquely the inverse series solving recursively the equations for the coefficients of the series B(ū). By the construction the coefficients of this series will be also q-symmetric. � 3 Universal nested Bethe vectors for Uq(ĝlN) Quantum affine algebras in the current realization [2] provide examples of the q-symmetric ge- nerating series. We will construct these generating series for the quantum affine algebra Uq(ĝlN ) and show that ?-products of these generating series provide hierarchical relations for NBA. We now recall the current realization of the algebra Uq(ĝlN ). The quantum affine algebra Uq(ĝlN ) is generated by the modes of the currents Ei(z) = ∑ n∈Z Ei[n]z−n, Fi(z) = ∑ n∈Z Fi[n]z−n, k±j (z) = ∑ n≥0 kj [±n]z∓n, (3.1) where i = 1, . . . , N − 1 and j = 1, . . . , N subject to the commutation relations given in the Appendix A. The generating series Fi(z), Ei(z) and k±(z) are called total and Cartan currents respectively. We consider two types of Borel subalgebras of the algebra Uq(ĝlN ). Generators of the standard Borel subalgebras Uq(b±) ⊂ Uq(ĝlN ) can be expressed in terms of the modes of the currents (3.1). To do this one has to introduce the composed currents Ea,b(z) and Fb,a(z) for a < b − 1 and 1 ≤ a < b ≤ N (see Appendix A for the definition of the currents Fb,a(z)). The Borel subalgebra Uq(b+) is generated by the modes of the currents: Ei[m], m > 0; Fi[n], k+ j [n], n ≥ 0 and Ea,b[1], a < b−1. Dual standard Borel subalgebra Uq(b−) is generated by the modes of the currents: Fi[m], m < 0; Ei[n], k+ j [n], n ≤ 0 and Fb,a[−1], a < b − 1. Here i = 1, . . . , N − 1 and j = 1, . . . , N . The reader can find description of the standard Borel subalgebras in terms of the modes of the Uq(ĝl3) currents in the paper [9]. This decomposition of the algebra Uq(ĝlN ) is related to the standard realization of this algebra in terms of pair of the dual L-operators, where generators of the standard Borel subalgebras serve as the modes of the Gauss coordinates of the corresponding L-operators. 6 S. Khoroshkin and S. Pakuliak Another type of Borel subalgebras is related to the current realization of Uq(ĝlN ) and was introduced in [2]. The Borel subalgebra UF ⊂ Uq(ĝlN ) is generated by the modes Fi[n], k+ j [m], i = 1, . . . , N − 1, j = 1, . . . , N , n ∈ Z and m ≥ 0. The Borel subalgebra UE ⊂ Uq(ĝlN ) is generated by the modes Ei[n], k−j [−m], i = 1, . . . , N − 1, j = 1, . . . , N , n ∈ Z and m ≥ 0. We also consider a subalgebra U ′F ⊂ UF , generated by the elements Fi[n], k+ j [m], i = 1, . . . , N − 1, j = 1, . . . , N , n ∈ Z and m > 0, and a subalgebra U ′E ⊂ UE generated by the elements Ei[n], k−j [−m], i = 1, . . . , N − 1, j = 1, . . . , N , n ∈ Z and m > 0. We call these subalgebras of Uq(ĝlN ) the current Borel subalgebras. Further, we will be interested in the intersections, U−f = U ′F ∩ Uq(b−), U+ F = UF ∩ Uq(b+) (3.2) and will describe properties of projections to these intersections. The current Borel subalgebras are Hopf subalgebras of Uq(ĝlN ) with respect to the current Hopf structure for the algebra Uq(ĝlN ) defined in [2]: ∆(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 , (3.3) ∆(D) ( k±i (z) ) = k±i (z)⊗ k±i (z). The quantum affine algebra Uq(ĝlN ) with ommited central charge and gradation operator can be identified with the quantum double of its current Borel subalgebra constructed using the comultiplication (3.3). One may check that the intersections U−f and U+ F are subalgebras. It was proved in [10] that these subalgebras are coideals with respect to Drinfeld coproduct (3.3) ∆(D)(U+ F ) ⊂ Uq(ĝlN )⊗ U+ F , ∆(D)(U−f ) ⊂ U−f ⊗ Uq(ĝlN ), and the multiplication m in Uq(ĝlN ) induces an isomorphism of vector spaces m : U−f ⊗ U+ F → UF . According to the general theory presented in [6] we define projection operators P+ : UF ⊂ Uq(ĝlN ) → U+ F and P− : UF ⊂ Uq(ĝlN ) → U−f by the prescriptions P+(f− f+) = ε(f−) f+, P−(f− f+) = f− ε(f+), for any f− ∈ U−f , f+ ∈ U+ F ,(3.4) where ε is the counit map: ε : Uq(ĝlN ) → C. Denote by UF an extension of the algebra UF formed by linear combinations of series, given as infinite sums of monomials ai1 [n1] · · · aik [nk] with n1 ≤ · · · ≤ nk, and n1 + · · · + nk fixed, where ail [nl] is either Fil [nl] or k+ il [nl]. It was proved in [6] that (1) the projections (3.4) 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 ′′i ) · P+(f ′i). (3.5) Generating Series for Nested Bethe Vectors 7 3.1 Generating series for universal Bethe vectors It was proved in the papers [10, 11] that the projection of the product of the Uq(ĝlN ) currents can be identified with universal Bethe vectors (UBV). In this paper we show that the hierarchical relations for UBV can be presented in a compact form using ?-product of certain q-symmetric generating series. Then the formal inversion of generating series allows to obtain another form of hierarchical relations and to investigate further (see [8]) special properties of UBV when their parameters satisfy the universal Bethe equations appeared in the framework of the analytical Bethe ansatz [1]. In this paper we will demonstrate these properties for the Uq(ĝl2) universal Bethe vectors. Products of the Uq(ĝlN ) currents yield examples of the q-symmetric generating series. We consider the generating series of the product of the currents FN (ū) = ∑ n1,...,nN−1≥0 FN (t̄[n̄])u n1 1 un2 2 · · ·unN−1 N−1 , (3.6) where each term FN (t̄[n̄]) means the following normalized product of the currents FN (t̄[n̄]) = FN−1(tN−1 nN−1 ) · · ·FN−1(tN−1 1 ) · · ·F1(t1n1 ) · · ·F1(t11) nN−1! · · ·n1! . (3.7) We set FN 0̄ ≡ 1. More generally, following the convention (2.14), for a segment [l̄, r̄] and the related collection t̄[l̄,r̄] of variables, see (2.5), we set FN (t̄[l̄,r̄]) = FN−1(tN−1 rN−1 ) · · ·FN−1(tN−1 lN−1+1) · · ·F1(t1r1 ) · · ·F1(t1l1+1) (rN−1 − lN−1)! · · · (r1 − l1)! . The superscript N in the notation FN (t̄[n̄]) of the coefficients of the q-symmetric generating series signifies that these coefficients belong to the subalgebra UF ⊂ Uq(ĝlN ). Further on we will consider smaller algebras Uq(ĝlj+1), j = 1, . . . , N−1 embedded into Uq(ĝlN ) in two different way. Embedding τj : Uq(ĝlj+1) ↪→ Uq(ĝlN ) is defined by removing the currents Fj+1(t), . . . , FN−1(t), Ej+1(t), . . . , EN−1(t) and the Cartan currents k±j+2(t), . . . , k ± N (t). Embedding τ̃j : Uq(ĝlj+1) ↪→ Uq(ĝlN ) is defined by removing the currents F1(t), . . . , FN−j−1(t), E1(t), . . . , EN−j−1(t) and the Cartan currents k±1 (t), . . . , k±N−j−1(t). Note, that the embedding τN−1 = τ̃N−1 is an identical map. Using these embeddings, for any segment [l̄, r̄] such that li = ri for all i = N − j + 1, N − j + 2, . . . , N − 1 we define a series FN−j+1(t̄[l̄,r̄]) = FN−j(t N−j rN−j ) · · ·FN−j(tN−1 lN−j+1) · · ·F1(t1r1 ) · · ·F1(t1l1+1) (rN−j − lN−j)! · · · (r1 − l1)! and for any segment [l̄, r̄] such that li = ri for all i = 1, 2, . . . , j − 1 a series F̃N−j+1(t̄[l̄,r̄]) = FN−1(tN−1 rN−1 ) · · ·FN−1(tN−1 lN−1+1) · · ·Fj(t1rj ) · · ·Fj(t1lj+1) (rN−1 − lN−1)! · · · (rj − lj)! . These series are gathered into q-symmetric generating functions FN−j+1(u1, . . . , uN−j) and F̃N−j+1(uj , . . . , uN−1). Subscripts a = 1, . . . , N − 1 of the formal parameters ua in the de- finition of these generating series denote the indices of the simple roots of the algebra glN . For example, the notation FN−1(u1, . . . , uN−2) means the generating series which coefficients take values in the subalgebra Uq(ĝlN−1) embedded into Uq(ĝlN ) by means of the map τN−2. On the 8 S. Khoroshkin and S. Pakuliak other hand, the notation F̃N−1(u2, . . . , uN−1) means the generating series taking value in the subalgebra Uq(ĝlN−1) embedded into Uq(ĝlN ) by means of the map τ̃N−2. Further on we use the special notation for the ordered product of the noncommutative entries. Symbols ←−∏ a Aa and −→∏ a Aa will mean ordered products of noncommutative entries Aa, such that Aa is on the right (resp., on the left) from Ab for b > a: ←−∏ j≥a≥i Aa = AjAj−1 · · ·Ai+1Ai, −→∏ i≤a≤j Aa = AiAi+1 · · ·Aj−1Aj . Using these notations we write the n̄-th term of the generating series (3.6) as follows FN (t̄[n̄]) = ←−∏ N−1≥a≥1 1 na!  ←−∏ na≥`≥1 Fa(ta` )  . (3.8) Besides of the generating series of products of the total currents we consider also the gener- ating series of the projections of the products of the currents P± ( FN (ū) ) = ∑ n1,...,nN−1≥0 P± ( FN (t̄[n̄]) ) un1 1 un2 2 · · ·unN−1 N−1 , (3.9) where P± ( FN (t̄[n̄]) ) = P± ( FN−1(tN−1 nN−1 ) · · ·FN−1(tN−1 1 ) · · ·F1(t1n1 ) · · ·F1(t11) ) nN−1! · · ·n1! . (3.10) In the same manner we define series P± ( FN−j+1(ū) ) = P± ( FN−j+1(u1, . . . , uN−j) ) and P± ( F̃N−j+1(ū) ) = P± ( F̃N−j+1(uj , . . . , uN−1) ) . The series P+ ( FN (ū) ) is the generating series of all possible universal off-shell Bethe vectors. Our goal is to show that the hierarchical relations of the nested Bethe vectors imply the facto- rization property of this generating series with respect to the ?-product of certain q-symmetric generating series. An associativity of this product allows to obtain a new presentation for the universal Bethe vectors. We call any expression ∑ i f (i) − · f (i) + , where f (i) − ∈ U−f and f (i) + ∈ U+ F (normal) ordered. Proposition 3. The q-symmetric generating series (3.6) can be written using a ?-product in a normal ordered form FN (ū) = P− ( FN (ū) ) ? P+ ( FN (ū) ) , (3.11) where the q-symmetric generating series P± ( FN (ū) ) are defined by (3.9). Proof. Using the property of the projections (3.5) an equality of the series FN (t̄[n̄]) = ∑ 0≤sN−1≤nN−1 · · · ∑ 0≤s1≤n1 Sym t̄[n̄] ( Zs̄(t̄[n̄])P − (FN (t̄[s̄,n̄]) ) · P+ ( FN (t̄[0̄,s̄]) )) (3.12) was proved in [11] (see, Proposition 4.1 therein). That proof was based on the comultiplication property (3.3) and the commutation relation between currents. Using a definition of the ?-pro- duct and considering the coefficients in front of monomial un1 1 · · ·unN−1 N−1 in the both sides of the equality (3.11) we obtain the formal series equality (3.12). � Generating Series for Nested Bethe Vectors 9 4 Generating series of strings and nested Bethe ansatz 4.1 Composed currents and the strings For two sets of variables {t11, . . . , t1k} and {t21, . . . , t2k} we introduce the series V (t2k, . . . , t 2 1; t 1 k, . . . , t 1 1) = k∏ m=1 t1m/t2m 1− t1m/t2m k∏ m′=m+1 q − q−1t1m′/t2m 1− t1m′/t2m = k∏ m=1 t1m/t2m 1− t1m/t2m m−1∏ m′=1 q − q−1t1m/t2m′ 1− t1m/t2m′ . (4.1) Fix j = 1, . . . , N − 1 and a collection of non-negative integers s̄j = {s1, . . . , sj} satisfying the admissibility condition: 0 = s0 ≤ s1 ≤ s2 ≤ · · · ≤ sj−1 ≤ sj . (4.2) We define a series depending on the set of the variables t̄[s̄j ] = {t11, . . . , t1s1 ; t21, . . . , t 2 s2 ; . . . ; tj1, . . . , t j sj } of the form X(t̄[s̄j ]) = j−1∏ a=1 V (ta+1 sa , . . . , ta+1 1 ; tasa , . . . , ta1). (4.3) When j = 1 we set X(·) = 1. Define an ordered normalized product of the composed currents, which we call a string of the type j:1 Sj+1(t̄[s̄j ]) = X(t̄[s̄j ]) ←−∏ j≥a≥1  1 (sa − sa−1)! ←−∏ sa≥`>sa−1 Fj+1,a(t j `)  (4.4) taking values in the subalgebra UF ⊂ Uq(ĝlj+1) embedded into Uq(ĝlN ) by the map τj . The composed currents Fj,i(t) corresponding to non-simple roots of the algebra glN belongs to the completion UF and their definition is given in the Appendix A (see (A.3), (A.4), (A.5)). More generally, let m̄ = {m1, . . . ,mN−1} and n̄ = {n1, . . . , nN−1} be a pair of collections of nonnegative integers such that na −ma = 0 for a = j + 1, . . . , N − 1 and na −ma = sa for any a = 1, . . . , j. Then for the set of variables t̄[m̄,n̄] = {t1m1+1, . . . , t 1 n1 ; t2m2 , . . . , t2n2 ; . . . ; tjmj+1, . . . , t j nj }, we set X(t̄[m̄,n̄]) = j−1∏ a=1 V (ta+1 ma+1+sa , . . . , ta+1 ma+1+1; t a na , . . . , tama+1), Sj+1(t̄[m̄,n̄]) = X(t̄[m̄,n̄]) ←−∏ j≥a≥1  1 (sa − sa−1)! ←−∏ sa+mj≥`>sa−1+mj Fj+1,a(t j `)  . 1Here the definition of the string differs from those used in [11, 14] by the combinatorial factor being a rational function of parameter ta i . 10 S. Khoroshkin and S. Pakuliak We define the q-symmetric generating series of the strings of the type j by the formula Sj+1(u1, . . . , uj) = ∑ sj≥sj−1≥···≥s1≥0 Sym t̄ [s̄j ] ( Sj+1(t̄[s̄j ]) ) us1 1 us2 2 · · ·usj−1 j−1 u sj j . (4.5) Here superscript j+1 signifies that this generating series takes values in the subalgebra Uq(ĝlj+1) embedded into Uq(ĝlN ) by the map τj . The subscripts of the parameters u1, . . . , uj signify that this subalgebra is generated by the Uq(ĝlN ) currents corresponding to the simple roots with indices 1, . . . , j. Let P+ ( FN (u1, . . . , uN−1) ) be the generating series of the universal off-shell Bethe vectors for the algebra Uq(ĝlN ) and P+ ( FN−1(u1, . . . , uN−2) ) be the analogous series for the smaller algebra Uq(ĝlN−1) embedded into Uq(ĝlN ) by the map τN−2. We have the following Proposition 4. Hierarchical relations between universal weight functions for algebras Uq(ĝlN ) and Uq(ĝlN−1) can be written as the following equality on generating series: P+ ( FN (u1, . . . , uN−1) ) = P+ ( SN (u1, . . . , uN−1) ) ? P+ ( FN−1(u1, . . . , uN−2) ) . (4.6) Proof. Taking the coefficients in front of the monomial un1 1 un2 2 · · ·unN−1 N−1 we obtain an equality of the formal series P+ ( FN (t̄[n̄]) ) = ∑ nN−1=sN−1≥···≥s1≥0 Sym t̄[n̄] ( Zs̄(t̄[n̄]) P+ ( SN (t̄[n̄−s̄,n̄]) ) · P+ ( FN−1(t̄[n̄−s̄]) ) ) . (4.7) An equality (4.7) coincides with the statement of the Proposition 4.2 of the paper [11] up to renormalization of the universal weight function by the combinatorial factor. � Corollary 1. Generating series of the Uq(ĝlN ) universal weight functions can be written using ordering ?-product of the generating series of the strings P+ ( FN (u1, . . . , uN−1) ) = P+ ( SN (u1, . . . , uN−1) ) ? P+ ( SN−1(u1, . . . , uN−2) ) ? · · · ? P+ ( S3(u1, u2) ) ? P+ ( S2(u1) ) . (4.8) In (4.8) we assume that the universal Bethe vectors for the algebra Uq(ĝl1) are equal to 1. Recall that the generating series P+ ( Sj(u1, . . . , uj−1) ) belongs to subalgebra UF ⊂ Uq(ĝlj) embedded into Uq(ĝlN ) by the map τj−1 which removes the currents Fa(t), Ea(t) and k±a+1(t) with a = j, j + 1, . . . , N − 1. 4.2 Hierarchical relations for the negative projections One of the results of the papers [11, 14] is the hierarchical relation for the positive projections of the product of the currents. Let us recall shortly the main idea of this calculation. In order to calculate the projection P+ ( FN−1(tN−1 nN−1 ) · · ·FN−1(tN−1 1 ) · · ·F1(t1n1 ) · · ·F1(t11) ) we separate all factors Fa(ta` ) with a < N − 1 and apply to this product the ordering procedure based on the property (3.12). We obtain under total projection the q-symmetrization of terms xiP −(yi)P+(zi), where xi are expressed via modes of FN−1(t) and yi, zi via modes of Fa(t) with a < N − 1. Then we used the property of the projection that P+ ( xiP −(yi)P+(zi) ) = P+ ( xiP −(yi) ) · P+(zi) Generating Series for Nested Bethe Vectors 11 and reorder the product of xi and P− (yi) under positive projection to obtain the string build from the composed currents (cf. equation (4.7)). We will use an analogous strategy to calculate the negative projection of the same product of the currents P− ( FN−1(tN−1 nN−1 ) · · ·FN−1(tN−1 1 ) · · ·F2(t2n2 ) · · ·F2(t21)F1(t1n1 ) · · ·F1(t11) ) . (4.9) Now we separate all factors Fa(ta` ) with a > 1 and apply to this product the ordering rule (3.12). Again, we obtain under total negative projection the q-symmetrization of terms P−(xi)P+(yi)zi, where zi are expressed via modes of F1(t) and yi, zi via modes of Fa(t) with a > 1. Using now the property of the projections P− ( P−(xi)P+(yi)zi ) = P− (xi) · P− ( P+(yi)zi ) and reordering the product of P+(yi) and zi under negative projection we obtain the desired hierarchical relations for the negative projection of the currents product. We will not repeat these calculations since they are analogous to the ones presented in [11], but will formulate the final answer of these hierarchical relations. For two sets of variables {t11, . . . , t1l } and {t21, . . . , t2l } we introduce the series Ṽ (t2l , . . . , t 2 1; t 1 l , . . . , t 1 1) = l∏ m=1 1 1− t1m/t2m l∏ m′=m+1 q − q−1t1m′/t2m 1− t1m′/t2m = l∏ m=1 1 1− t1m/t2m m−1∏ m′=1 q − q−1t1m/t2m′ 1− t1m/t2m′ . (4.10) Fix k = 1, . . . , N−1 and collection of the non-negative integers s̃k = {sk, . . . , sN−1} satisfying the admissibility condition sk ≥ sk+1 ≥ sk+2 ≥ · · · ≥ sN−1 ≥ sN = 0. (4.11) We define a series depending on the set of the variables t̄[s̃k] = {tk1, . . . , tksk ; tk+1 1 , . . . , tk+1 sk+1 ; . . . ; tN−1 1 , . . . , tN−1 sN−1 } of the form X̃(t̄[s̃k]) = N−2∏ a=k Ṽ (ta+1 sa+1 , . . . , ta+1 1 ; tasa , . . . , tasa−sa+1+1). (4.12) When k = N − 1 we set X̃(·) = 1. Define an ordered normalized product of the composed currents, which we call a dual string of the type N − k: S̃N−k+1(t̄[s̃k]) = X̃(t̄[s̃k]) ←−∏ N≥a>k  1 (sa−1 − sa)! ←−∏ sk−sa≥`>sk−sa−1 Fa,k(tk` )  . (4.13) Note that the notion of the dual string is different from the notion of the inverse string used in [11]. More generally, let m̄ = {m1, . . . ,mN−1} and n̄ = {n1, . . . , nN−1} be a pair of collections of nonnegative integers such that na − ma = 0 for a = 1, . . . , k − 1 and na − ma = sa for any a = k, . . . , N − 1. Then for the collection of variables t̄[m̄,n̄] = { tkmk+1, . . . , t k nk ; tk+1 mk+1 , . . . , tk+1 nk+1 ; . . . ; tN−1 mN−1+1, . . . , t N−1 nN−1 } , 12 S. Khoroshkin and S. Pakuliak we set X̃(t̄[m̄,n̄]) = N−2∏ a=k Ṽ (ta+1 na+1 , . . . , ta+1 ma+1+1; t a na , . . . , tana−sa+1+1), S̃N−k+1(t̄[m̄,n̄]) = X̃(t̄[m̄,n̄]) ←−∏ N≥a>k  1 (sa−1 − sa)! ←−∏ nk−sa≥`>nk−sa−1 Fa,k(tk` )  . Doing the calculations described above we obtain the recurrence relations for the negative projections P− ( FN (t̄[n̄]) ) = ∑ n1=s1≥···≥sN−1≥0 Sym t̄[n̄] ( Zs̄(t̄[n̄]) ·P− ( FN−1(t̄[s̄,n̄]) ) ·P− ( S̃N (t̄[s̄]) )) . (4.14) We define the generating series of the dual strings of the type N − j by the formula S̃N−k+1(uk, . . . , uN−1) = ∑ sk≥sk+1≥···≥sN−1≥0 Sym t̄ [s̃k] ( S̃N−k+1(t̄[s̃k]) ) usk k u sk+1 k+1 · · ·u sN−1 N−1 taking values in the subalgebra UF ⊂ Uq(ĝlN−k+1) embedded into Uq(ĝlN ) by the map τ̃N−k which removes the currents Fa(t), Ea(t) and k±a (t) for a = 1, . . . , k − 1. The recurrence rela- tions (4.14) can be written as the ?-product of the generating series P− ( FN (u1, . . . , uN−1) ) = P− ( F̃N−1(u2, . . . , uN−1) ) ? P− ( S̃N (u1, . . . , uN−1) ) . (4.15) Generating series of the negative projections of the product of the currents can be written using ordering ?-product of the generating series of the dual strings P− ( FN (u1, . . . , uN−1) ) = P− ( S̃2(uN−1) ) ? P− ( S̃3(uN−2, uN−1) ) ? · · · ? P− ( S̃N−1(u2, . . . , uN−1) ) ? P− ( S̃N (u1, . . . , uN−1) ) . (4.16) 4.3 Other type of the hierarchical relations A special ordering property of the universal Bethe vectors when their parameters ta` satisfy the universal Bethe equations [1] was investigated in [8]. This property leads to the fact that the ordering of the product of the universal transfer matrix and the universal nested Bethe vectors is proportional to the same Bethe vector modulo the terms which belong to some ideal in the algebra if the parameters of this vector satisfy the universal Bethe equations. We will demonstrate this property for the Uq(ĝl2) universal Bethe vectors in the Section 4.7. A cornerstone of this ordering property lies in a new hierarchical relations for the universal Bethe vectors, which can be proved using the technique of the generating series. Here we give the detailed proof of the relation which particular form was used in the paper [8]. Using normal ordering relation (3.11) and (4.15) we may write the generating series of the product of the currents in the form FN (u1, . . . , uN−1) = P− ( FN (u1, . . . , uN−1) ) ? P+ ( FN (u1, . . . , uN−1) ) (4.17) = P− ( F̃N−1(u2, . . . , uN−1) ) ? P− ( S̃N (u1, . . . , uN−1) ) ? P+ ( FN (u1, . . . , uN−1) ) . On the other hand these generating series may be presented as the factorized product FN (u1, . . . , uN−1) = F̃N−1(u2, . . . , uN−1) · F2(u1) = F̃N−1(u2, . . . , uN−1) ? F2(u1). (4.18) Generating Series for Nested Bethe Vectors 13 Applying the ordering relation (3.11) to the series F̃N−1(u2, . . . , uN−1) again we obtain an alternative to (4.17) expression for the generating series FN (u1, . . . , uN−1): FN (u1, . . . , uN−1) = F̃N−1(u2, . . . , uN−1) ? F2(u1) = P− ( F̃N−1(u2, . . . , uN−1) ) ? P+ ( F̃N−1(u2, . . . , uN−1) ) ? F2(u1). (4.19) Equating the right hand sides of (4.17) and (4.19) we obtain the identity P− ( S̃N (u1, . . . , uN−1) ) ? P+ ( FN (u1, . . . , uN−1) ) = P+ ( F̃N−1(u2, . . . , uN−1) ) ? F2(u1) (4.20) or P+ ( FN (u1, . . . , uN−1) ) = ( P− ( S̃N (u1, . . . , uN−1) ))−1 ? P+ ( FN−1(u2, . . . , uN−1) ) ? F2(u1). (4.21) The identity (4.21) relates the universal off-shell Bethe vectors for the algebra Uq(ĝlN ) and for the smaller algebra Uq(ĝlN−1). They can be considered as an universal formulation of the relation used in the pioneer paper [12] for the obtaining the nested Bethe equations. The equality (4.21) between generating series contains many hierarchical relations between UBV. In order to get some particular identities between these UBV one has to invert explicitly the generating series P− ( S̃N (u1, . . . , uN−1) ) . This will be done in the next subsection. 4.4 Inverting generating series of the strings Let E(ū) be the generating series of negative projections of dual strings of the type N − 1: E(ū) = P− ( S̃N (u1, . . . , uN−1) ) and D(ū) be the inverse series: D(ū) ? E(ū) = 1. By the construction, see (4.13), the coefficients E(t̄[m̄]) = P− ( Sym t̄[m̄] ( S̃N (t̄[m̄]) )) (4.22) of the generating series E(ū) are nonzero only if the admissibility conditions m1 ≥ m2 ≥ · · · ≥ mN−1 ≥ 0 for the set {m̄} = {m1,m2, . . . ,mN−1} are satisfied. We have to find coefficients of the generating series D(ū) such that D(ū) ? E(ū) = 1. The latter equality is equivalent to the system of equations Sym t̄[n̄] ( ∑ n1≥m1≥0 · · · ∑ nN−1≥mN−1≥0 m1≥···≥mN−1 Zm̄(t̄[n̄])D(t̄[m̄,n̄]) · E(t̄[m̄]) ) = 0. (4.23) for the unknown functional coefficients D(t̄[k̄]) for all possible fixed values of n1, . . . , nN−1. Proposition 5. The coefficients D(t̄[k̄]) are nonzero only if admissibility conditions k1 ≥ k2 ≥ · · · ≥ kN−1 ≥ 0 are satisfied. Proof. can be performed recursively by considering first the cases for n1 > 0 and n2 = · · · = nN−1 = 0. For these values of n̄ the series Zm̄(t̄[n̄]) = 1 and coefficients of the series in (4.23) depend only on the variables of the first type t11, . . . t 1 n1 . The relation (4.23) takes in this case the form Sym t̄[n]  ∑ n1≥m1≥0 D(t1m1+1, . . . , t 1 n1 ) · E(t11, . . . , t 1 m1 )  = 0, (4.24) 14 S. Khoroshkin and S. Pakuliak 1 2 3 4 Figure 1. Example of the connected diagram for n = 16, p = 4, χ1 = 6, χ2 = 3, χ3 = 5, χ4 = 2. Its solution can be written in the form D(t̄[n]) = Sym t̄[n] n1−1∑ p=0 (−1)p+1 ∑ n1=kp+1>kp>···>k1>k0=0 ←−∏ p+1≥r≥1 E(t1kr−1+1, . . . , t 1 kr )  . (4.25) After this we consider the relation (4.23) for arbitrary n1≥ 0, n2 = 1 and n3 = · · ·= nN−1 = 0. Avoiding writing the dependence on the ‘t’ parameters, that is using notations Ek̄ instead of E(t[k̄]), the relation (4.23) takes the form Sym t̄[n̄] ( n1∑ m1=0 Dn1−m1,1,0 · Em1,0,0 ) + Sym t̄[n̄] ( n1∑ m1=1 Zm1,1,0 · Dn1−m1,0,0 · Em1,1,0 ) = 0. (4.26) The rational series Zm1,0,0 disappears in the first sum of (4.26) by the same reason as in (4.24). Let us consider the relation (4.26) for n1 = 0. The second sum is absent and the first sum contains only one terms D0,1,0 · E0,0,0 = 0 which is equal to zero. This proves that the coefficient D0,1,0 = 0 vanishes identically. Now the first sum in the relation (4.26) is terminated at m1 = n1−1 and this relation allows to find all coefficients Dn1,1,0 starting from D1,1,0 = −Z1,1,0E1,1,0. Considering the relation (4.23) for n1 ≥ 0, n2 = 2 and n3 = · · · = nN−1 = 0 we prove first that D0,2,0 = D1,2,0 = 0 and then can find all coefficients Dn1,2,0 starting from D2,2,0. It is clear now that the coefficients Dn1,n2,0 are non-zero only if n1 ≥ n2. Continuing we prove the statement of the proposition. � 4.5 Inversion and combinatorics To invert explicitly the generating series of the projection of the strings we have to introduce certain combinatorial data. First of all, according to Proposition 5 we fix a sequence of non- negative integers n1 ≥ n2 ≥ · · · ≥ nN−1 and the corresponding set of the variables t̄[n̄]. Choose any positive integer n and p = 1, . . . , n. A diagram χ of size \|χ\| = n and height p = h(χ) is an ordered decomposition of n into a sum of p nonnegative integers, n = χ1 + · · ·+ χp. Equivalently, a diagram χ consists of p rows and the i-th row contains χi boxes, An example of such diagram is shown in the Fig. 1. The rows of the diagrams are numbered from the bottom to the top. If χi = 0 for some i = 1, . . . , p then the diagram contains several disconnected pieces. We will call the diagram χ connected if all χi 6= 0 for i = 1, . . . , p. A tableaux χ̄ with a given diagram χ is a filling of all boxes of χ by the indices {1, 2, . . . , N−1} of the positive roots of the algebra glN with the condition of non-increasing from the left to the right along the rows. If an index a is associated to a box of the tableaux, we say that this box has a ‘type’ a. We will call the tableaux associated to the connected diagrams the connected Generating Series for Nested Bethe Vectors 15 2 2 2 1 1 1 3 1 1 1 1 3 3 2 2 1 Figure 2. Example of connected tableaux in case of N = 4 and its weight {n1, n2, n3} = {16, 8, 3}. 3 2 1 3 1 1 t31, t 2 2, t 1 3 t21, t 1 2 t11 t32, t 2 3, t 1 4 t16 t15 Figure 3. Example of tableaux for N = 4 with associated variables tai . tableaux. For a given tableaux we define its weight n̄(χ̄) as a set of numbers n̄ = n̄(χ̄) = {n1(χ̄), . . . , nN−1(χ̄)} such that na(χ̄) is a number of boxes which have type bigger or equal than a, a = 1, . . . , N − 1. The size and the height of the tableaux is the size and the height of the corresponding diagram. An example of a connected tableaux is given on the Fig. 2. Denote by χ̄i the ith row of the tableaux χ̄. Denote by ci a = ci a(χ̄) the number of type a boxes in the row χ̄i. Set di a = di a(χ̄) = ci N−1 + · · ·+ ci a and hi a = d1 a + · · ·+di a In particular, di 1(χ̄) is the length χi of the row χ̄i and the collection d̄i = {di 1, . . . , d i N−1} is the character n̄(χ̄i) of the row χ̄i considered as a tableaux by itself. Clearly di a ≥ di b when a ≤ b and na = hp a = d1 a + · · ·+ dp a for all a = 1, . . . N − 1. This formula demonstrates in particular that the weight of a tableaux always satisfies the admissibility conditions n1 ≥ n2 ≥ · · · ≥ nN−1. To each connected tableaux χ̄ of the weight n̄ = n̄(χ̄) we associate a decomposition of the set of the variables t̄[n̄] into the union of \|χ\| disjoint subsets, each corresponding to a box of the diagram χ of tableaux χ̄. To each box of the type a we associate one variable of the type 1, one variables of the type 2, etc., one variable of the type a, altogether a variables. We will number variables of the each type starting from the most bottom row and the most right box where variable of this type appear for the first time. Let us give an example of the decomposition and of the ordering for the tableaux shown on the Fig. 3. The most bottom and the right box has the type 1. We associate to this box one variable2 t11 of the same type. Next to the left along the same row box has the type 2. We associate to this box two variables t12 and t21 of the types 1 and 2. Last box in the bottom row has type 3 and we associate to this box three variables t13, t22 and t31. Next box is in the next row and also has the type 3. To this box we associate also three variables t14, t23 and t32. Next two boxes in the third row both have the type 1 and we associate to the most right box in this row one variable t15 and to the last box also one variable t16. In general, for each tableaux χ̄ of the weight n̄(χ̄) and any segment [l̄, r̄], such that r̄− l̄ = n̄ the set of the variables t̄[l̄,r̄] t̄[l̄,r̄] = { t1l1+1, . . . , t 1 r1 ; t2l2+1, . . . , t 2 r2 ; . . . ; tN−1 lN−1+1, . . . , t N−1 rN−1 } . decouples into h(χ̄) groups of variables t̄χ̄i = { t1 l1+hi−1 1 +1 , . . . , t1l1+hi 1 ; . . . ; tN−1 lN−1+hi−1 N−1+1 , . . . , tN−1 lN−1+hi N−1 } . (4.27) 2Recall that superscript a signifies the ‘type’ of the variable ta i and subscript i counts the number of the variables of this type. 16 S. Khoroshkin and S. Pakuliak The variable tak belongs to the subset t̄χ̄i if la + hi−1 a < k ≤ la + hi a. It is located in the (la + hi a + 1 − k)th box of the row χ̄i counting boxes in this row from the left edge of the tableaux. In the same setting we define Zχ̄(t̄[l̄,r̄]) = ∏ 1≤i<j≤h(χ̄) Zχ̄i,χ̄j (t̄χ̄i ; t̄χ̄j ), (4.28) where Zχ̄i,χ̄j (t̄χ̄i ; t̄χ̄j ) = N−2∏ a=1 la+hj a∏ `=la+hj−1 a +1 la+1+hi a+1∏ `′=la+1+hi−1 a+1+1 q − q−1ta`/ta+1 `′ 1− ta`/ta+1 `′ is a rational series defined by the interchanging of the variables of the type a + 1 from the i-th row and variables of the type a from the jth row of the tableaux χ̄. In our example, the group of variables( t32, t 3 1; t 2 3, t 2 2, t 2 1; t 1 6, t 1 5, t 1 4, t 1 3, t 1 2, t 1 1 ) (4.29) decomposes into three groups( ·; ·; t16, t15 ) ( t32; t 2 3; t 1 4 ) ( t31; t 2 2, t 2 1; t 1 3, t 1 2, t 1 1 ) . (4.30) In this example the rational series Zχ̄(t̄[n̄]) is equal to∏ `=5,6 q − q−1 t1`/t23 1− t1`/t23 · ∏ `=4,5,6 `′=1,2 q − q−1 t1`/t2`′ 1− t1`/t2`′ · q − q−1 t23/t31 1− t23/t31 . (4.31) For a given tableaux χ̄ we define the ordered product Eχ̄(t̄[n̄]) = ←−∏ h(χ̄)≥i≥1 E(t̄χ̄i) (4.32) of the negative projections of the strings (4.22). Each factor E(t̄χ̄i) corresponds to the projection of the certain string depending on the set of variables t̄χ̄i (4.27). Decomposition of the tableaux row into boxes shows the structures of this string. Each box of the type a corresponds to the composed current Fa+1,1 which depends on the type 1 variable placed in this box. Variables of other types from the same box enter through rational factors. For the tableaux shown on the Fig. 3 this product reads E(t15, t 1 6) · E(t14; t 2 3; t 3 2) · E(t11, t 1 2, t 1 3; t 2 1, t 2 2; t 3 1) and E(t15, t 1 6) = P− ( F2,1(t16)F2,1(t15) ) , E(t14; t 2 3; t 3 2) = 1 1− t14/t23 1 1− t23/t32 P− ( F4,1(t14) ) , E(t11, t 1 2, t 1 3; t 2 1, t 2 2; t 3 1) = 1 1− t13/t22 1 1− t22/t31 1 1− t12/t21 P− ( F4,1(t13)F3,1(t12)F2,1(t11) ) . Proposition 6. The coefficient D(t̄[n̄]) of the inverse series ( P− ( S̃N (u1, . . . , uN−1) ))−1 are given D(t̄[n̄]) = ∑ χ̄ (−1)h(χ̄)+1Sym t̄[n̄] ( Zχ̄(t̄[n̄]) · Eχ̄(t̄[n̄]) ) (4.33) by the sum over all possible connected tableaux χ̄ such that the weight of tableaux n̄(χ̄) is equal to n̄. Generating Series for Nested Bethe Vectors 17 Proof. For arbitrary non-empty set n̄ 6= 0̄ we substitute expression (4.33) into (4.23) to obtain the relation∑ χ̄ n̄(χ̄)=n̄ (−1)h(χ̄)+1Sym t̄[n̄] ( Zχ̄(t̄[n̄]) · Eχ̄(t̄[n̄]) ) + ∑ m̄ m̄6=0̄ ∑ χ̄′ n̄(χ̄′)=n̄−m̄ (−1)h(χ̄′)+1Sym t̄[n̄] ( Zm̄(t̄[n̄]) · Zχ̄′(t̄[m̄,n̄]) · Eχ̄′(t̄[m̄,n̄]) · E(t̄[m̄]) ) (4.34) which has to be equal 0. After this substitution the product of the series Zχ̄′(t̄[m̄,n̄]) · Eχ̄′(t̄[m̄,n̄]) should be under q-symmetrization over the set of the variables t̄[m̄,n̄]. Since the series Zm̄(t̄n̄) is symmetric with respect to the set of these variables and the series E(t̄m̄) does not depend on the variables t̄[m̄,n̄] we may include these series under q-symmetrization over the variables t̄[m̄,n̄]. Then, since the variables t̄[m̄,n̄] forms a subset of the variables t̄[n̄], the q-symmetrization over variables t̄[m̄,n̄] disappear due to the property (2.3). We will prove the cancellation of the terms in (4.34) in the sums over tableaux of the fixed height. Keep the terms in the summation of the first line of this relation which correspond to the connected tableaux χ̄ such that na(χ̄) = na and h(χ̄) = p + 1. Keep the terms in the summation of the second line of (4.34) which correspond to all connected tableaux χ̄′ such that na(χ̄′) = na −ma and h(χ̄′) = p for fixed p = 0, . . . , n1 − 1. Fix a term in the first sum of (4.34) corresponding to some connected tableaux χ̄ with a weight n̄. Consider the first (bottom) line of the tableaux χ̄. Denote by ma = c1 a + · · ·+ c1 N−1 the nonnegative integers defined by this row. It is clear that this set of integers satisfies the admissibility condition m1 ≥ m2 ≥ · · · ≥ mN−1 and ma ≤ na. In the second double sum of (4.34) choose the term corresponding to this set m̄ and the tableaux χ̄′ defined by the following rule. If we glue from the bottom of the tableaux χ̄′ the row of boxes such that it has a length m1 and the number of boxes of the type a is equal to ma −ma+1 then for the obtained tableaux χ̃ we require ri a(χ̃) = ri a(χ̄) and h(χ̃) = p + 1 for all possible values i and a. We claim that for each fixed tableaux χ̄ there are unique set m̄ such that ma ≤ na and there are a single tableaux χ̄′ which satisfies above conditions. The tableaux χ̃ and χ̄ coincide actually. The product of the coefficients Eχ̄′(t̄[m̄,n̄]) · E(t̄m̄) will be equal to Eχ̄(t̄[n̄]). According to the definitions of the series Zm̄(t̄n̄) (2.15) and Zχ̄′(t̄[m̄,n̄]) (4.28) their product will be equal to the series Zχ̄(t̄[n̄]). The term corresponding to the fixed tableaux χ̄ in the first line of (4.34) and the term from the second line given by m̄ and χ̄′ described above cancel each other since they will enter with different signs: h(χ̄) = h(χ̄′) + 1 = p + 1. � For the example of the tableaux shown in the Fig. 3 the sets t̄[m̄] and t̄[m̄,n̄] are {t31; t22, t21; t13, t12, t11} and {t32; t23; t16, t15, t14} respectively. The series Zm̄(t̄n̄) is ∏ `=4,5,6 `′=1,2 q − q−1t1`/t2`′ 1− t1`/t2`′ · q − q−1t23/t31 1− t23/t31 . (4.35) The series Zχ̄′(t̄[m̄,n̄]) is ∏ `=5,6 q − q−1t1`/t23 1− t1`/t23 . (4.36) The product of (4.35) and (4.36) obviously coincides with (4.31). 18 S. Khoroshkin and S. Pakuliak 4.6 Inversion of generating series for Uq(ĝl2) Quantum affine algebra Uq(ĝl2) in its current realization formed by the modes of the currents3 E(t), F (t) and Cartan currents k±1 (t), k±2 (t). Let us invert explicitly the generating series P− (F(ū)) in the simplest case of one generating parameters ū = u1 ≡ u in (3.11) to show what kind of relations can be obtained for the generating series of the Uq(ĝl2) off-shell Bethe vectors P+ (F(u)): P+ (F(u)) = P− (F(u))−1 ? F(u). (4.37) For any positive n and non-negative p ≤ n we define the set of p + 1 positive integers {k̄p} = {k1, k2, . . . , kp, kp+1} such that 0 = k0 < k1 < k2 < · · · < kp < kp+1 = n. Using this data we define the ordered products Fk̄p (t̄[n]) = ←−∏ p+1≥m≥1 1 (km − km−1)! P− ( F (tkm) · · ·F (tkm−1+1) ) . It is clear that the inverse generating series P− (F(u)) can be written using q-symmetrization of these ordered products Fk̄p (t̄[n]) as follows P− (F(u))−1 = 1 + ∑ n>0 Sym t̄[n] n−1∑ p=0 (−1)p+1 ∑ {k̄p} Fk̄p (t̄[n]) un. (4.38) Then using the definition of the ?-product we can obtain from (4.37) a special presentation for the Uq(ĝl2) universal weight function P+ ( F(t̄[n]) ) = Sym t̄[n]  n∑ s=0 n−s−1∑ p=0 (−1)p+1 ∑ {kp} Fk̄p (t̄[s,n]) · F(t̄[s])  , (4.39) where summation over the set {k̄p} runs over all possible ki such that s = k0 < k1 < k2 < · · · < kp < kp+1 = n. An extreme term in the sum when s = n and the sum over p is absent corresponds to the product of the total currents F (tn) · · ·F (t1). Note that an equality (4.39) can be treated as generalization of Ding–Frenkel relation P+ (F (t)) = F (t)−P− (F (t)) [3] when the positive projection is taken from the product of the currents. 4.7 Universal Bethe ansatz for Uq(ĝl2) Let J be the left ideal of Uq(b+) ⊂ Uq(ĝl2), generated by all element of the form Uq(b+) · E[n], n > 0. As it was mentioned above the standard Borel subalgebra Uq(b+) in terms of the currents generators is formed by the modes F [n], k+ 1,2[n], n ≥ 0 and E[m], m > 0. Let us denote subalgebras generated by these modes as U+ f , U+ k and U+ e , respectively. The multiplication in Uq(b+) implies an isomorphism of the vectors spaces U+ f ⊗ U+ k ⊗ U+ e → Uq(b+). We introduce ordering of the generators in the Borel subalgebra Uq(b+) U+ f ≺ U+ k ≺ U+ e (4.40) 3Since algebra gl2 has only one root we remove index of this simple root in the notation of the currents in case of the current realization of the algebra Uq(ĝl2). Generating Series for Nested Bethe Vectors 19 induced by the circular ordering of the Cartan–Weyl generators in the whole algebra Uq(ĝl2) [6]. We call any element w ∈ Uq(b+) normal ordered and denote it as : W : if it is presented as the linear combination of the elements of the form W1 ·W2 ·W3, where W1 ∈ U+ f , W2 ∈ U+ k , W3 ∈ U+ e . It is convenient to gather the generators of the subalgebras U+ f and U+ e into generating series F+(t) = ∑ n≥0 F [n]t−n, E+(t) = ∑ n>0 E[n]t−n which we call the half-currents. A universal transfer matrix is the following combinations of the Cartan and half-currents T (t) = k+ 1 (t) + F+(t)k+ 2 (t)E+(t) + k+ 2 (t). (4.41) Using the commutation relations in the algebra Uq(b+) ⊂ Uq(ĝl2) one may check that these transfer matrices commute for the different values of the spectral parameters [T (t), T (t′)] = 0 and so generates the infinite set of commuting quantities4. We are interesting in the ordering relations between universal transfer matrix T (t) and the universal Bethe vector P+ ( F(t̄[n]) ) . Note that the universal transfer matrix is ordered according to the ordering (4.40). Proposition 7. A formal series identity is valid in Uq(b+) : T (t) · P+ ( F(t̄[n]) ) : = P+ ( F(t̄[n]) ) · τ(t; t̄[n]) mod J (4.42) modulo elements of the left ideal J if the set {tj} of the Bethe parameters satisfies the set of the universal Bethe equations [1], j = 1, . . . , n: k+ 1 (tj) k+ 2 (tj) = n∏ m6=j qtj − q−1tm q−1tj − qtm (4.43) and τ(t; t̄[n]) = k+ 1 (t) n∏ j=1 q−1t− qtj t− tj + k+ 2 (t) n∏ j=1 qt− q−1tj t− tj (4.44) is an eigenvalue of the universal transfer matrix. Proof. Recall that a universal Bethe vector in the considered case coincides with projection of the product of the currents: P+ ( F(t̄[n]) ) = P+ (F (tn) · · ·F (t1)) /n! and can be presented as factorized product of the linear combinations of the half-currents F+(ti) [6, 9]. A direct way to prove the statement of the Proposition 7 is to use the commutation relations in Uq(b+) between half-currents [E+(t), F+(t′)] = (q − q−1)t′ t− t′ ( k+ 1 (t′)k+ 2 (t′)−1 − k+ 1 (t)k+ 2 (t)−1 ) , k+ 2 (t)F+(t′)k+ 2 (t)−1 = qt− q−1t′ t− t′ F+(t′)− (q − q−1)t′ t− t′ F+(t) and similar for k+ 1 (t)F+(t′)k+ 1 (t)−1 to present the product T (t) · P+ ( F(t̄[n]) ) in the normal ordered form. But this way is not easy even in the simplest case of the algebra Uq(ĝl2). It 4A standard way to prove this commutativity is to note that (4.41) is a trace of the fundamental L-operators for Uq(ĝl2) and the commutativity follows from the Yang–Baxter equation for these L-operators. 20 S. Khoroshkin and S. Pakuliak becomes much more involved in the general case of the algebra Uq(ĝlN ). There is a simple way to avoid these difficulties using the relation (4.39). This relation allows to replace the projection of the product of the currents onto positive Borel subalgebra Uq(b+) by the linear combination of the terms P+ (F (tn) · · ·F (t1)) = F (tn) · · ·F (t1) − nSym t̄ ( P− (F (tn)) · F (tn−1) · · ·F (t1) ) + W, (4.45) where W are the terms which have on the left the product of at least two negative projections of the currents F (t). Let us substitute (4.45) into the product T (t) · P+ ( F(t̄[n]) ) . Although left hand side of (4.45) belongs to the positive Borel subalgebra, each term in the right hand side of the equality (4.45) does not belong to Uq(b+). The product of the universal transfer matrix with these terms will produce the terms such that some of them belong to Uq(b+) and other does not belong to Uq(b+). The latter terms which after ordering do not belong to Uq(b+) can be omitted since we are interesting only in the terms which belong to the positive Borel subalgebras. In particular, one may check that : T (t) · W : 6∈ Uq(b+), where W are the terms in (4.45) not showing explicitly (see [8] for details). Using the commutation relations of the half-currents E+(t) with total currents F (t′) [E+(t), F (t′)] = (q − q−1)t′ t− t′ ( k+ 1 (t′)k+ 2 (t′)−1 − k−1 (t′)k−2 (t′)−1 ) , the commutation relations of the Cartan currents with total currents F (t′) k+ 2 (t)F (t′)k+ 2 (t)−1 = qt− q−1t′ t− t′ F+(t′), k+ 1 (t)F (t′)k+ 1 (t)−1 = q−1t− qt′ t− t′ F+(t′) and the commutation relations of E+(t), k+ 1 (t), k+ 2 (t) with P− (F (t′)) we may check that the only terms which belong to the positive Borel subalgebra Uq(b+) and do not belong to the left ideal J in the normal ordered product : T (t) · P+ (F (tn) · · ·F (t1)) : are : T (t) · P+ (F (tn) · · ·F (t1)) : = P+ (F (tn) · · ·F (t1)) ( n∏ i=1 q−1t− qti t− ti k+ 1 (t) + n∏ i=1 qt− q−1ti t− ti k+ 2 (t) ) + nSym t̄ ( P+ ( F+(t)k+ 2 (t)F (tn) · · ·F (t2) ) (q − q−1)t1 t− t1 k+ 1 (t1)k+ 2 (t1)−1 ) − nSym t̄ ( P+ ( F+(t)k+ 2 (t)F (tn−1) · · ·F (t1) ) (q − q−1)tn t− tn ) . (4.46) In order to prove the statement of the Proposition 7 we have to cancel the last two terms in (4.46). This can be done using the properties of the q-symmetrization that for any formal series G(t1, . . . , tn) on n formal variables ti we have nSym t̄ G(t1, . . . , tn) = n∑ m=1 n∏ j=m+1 q−1tm − qtj qtm − q−1tj Sym t̄\tmG(t1, . . . , tm−1, tm+1, . . . , tn, tm) and nSym t̄G(t1, . . . , tn) = n∑ m=1 m−1∏ j=1 q−1tj − qtm qtj − q−1tm Sym t̄\tm G(tm, t1, . . . , tm−1, tm+1, . . . , tn), where q-symmetrization in the right hand sides of this formal series identities runs over (n− 1) variables t̄ \ tm = {t1, . . . , tm−1, tm+1, . . . , tm}. Using these relations we conclude that last two terms in (4.46) cancel each other provided the relation (4.43) is satisfied. � Generating Series for Nested Bethe Vectors 21 A Current realization of Uq(ĝlN) The commutation relations for the algebra Uq(ĝlN ) in the current realization are given by the following set of the relations (q−1z − qw)Ei(z)Ei(w) = Ei(w)Ei(z)(qz − q−1w), (z − w)Ei(z)Ei+1(w) = Ei+1(w)Ei(z)(q−1z − qw), k±i (z)Ei(w) ( k±i (z) )−1 = z − w q−1z − qw Ei(w), k±i+1(z)Ei(w) ( k±i+1(z) )−1 = z − w qz − q−1w Ei(w), k±i (z)Ej(w) ( k±i (z) )−1 = Ej(w), if i 6= j, j + 1, (qz − q−1w)Fi(z)Fi(w) = Fi(w)Fi(z)(q−1z − qw), (A.1) (q−1z − qw)Fi(z)Fi+1(w) = Fi+1(w)Fi(z)(z − w), k±i (z)Fi(w) ( k±i (z) )−1 = q−1z − qw z − w Fi(w), k±i+1(z)Fi(w) ( k±i+1(z) )−1 = qz − q−1w z − w Fi(w), k±i (z)Fj(w) ( k±i (z) )−1 = Fj(w), if i 6= j, j + 1, [Ei(z), Fj(w)] = δi,j δ(z/w) (q − q−1) ( k+ i (z)/k+ i+1(z)− k−i (w)/k−i+1(w) ) and the Serre relations for the currents Ei(z) and Fi(z) Symz1,z2 (Ei(z1)Ei(z2)Ei±1(w)− (q + q−1)Ei(z1)Ei±1(w)Ei(z2) + Ei±1(w)Ei(z1)Ei(z2)) = 0, Symz1,z2 (Fi(z1)Fi(z2)Fi±1(w)− (q + q−1)Fi(z1)Fi±1(w)Fi(z2) + Fi±1(w)Fi(z1)Fi(z2)) = 0. (A.2) Formulae (A.1) and (A.2) should be considered as formal series identities describing the infinite set of the relations between modes of the currents. The symbol δ(z) entering these relations is a formal series ∑ n∈Z zn. Following [4, 9], we introduce composed currents Fj,i(t) for i < j. The composed currents for nontwisted quantum affine algebras were defined in [4]. According to this paper, the coefficients of the series Fj,i(t) belong to the completion UF of the algebra UF . The completion UF determines analyticity properties of products of currents (and coincide with analytical properties of their matrix coefficients for highest weight representations [5]). One can show that for \|i− j\| > 1, the product Fi(t)Fj(w) is an expansion of a function analytic at t 6= 0, w 6= 0. The situation is more delicate for j = i, i ± 1. The products Fi(t)Fi(w) and Fi(t)Fi+1(w) are expansions of analytic functions at \|w\| < \|q2t\|, while the product Fi(t)Fi−1(w) is an expansion of an analytic function at \|w\| < \|t\|. Moreover, the only singularity of the corresponding functions in the whole region t 6= 0, w 6= 0, are simple poles at the respective hyperplanes, w = q2t for j = i, i + 1, and w = t for j = i − 1. Recall, that the deformation parameter q is a generic complex number, which is neither 0 nor a root of unity. The definition of the composed currents may be written in analytical form Fj,i(t) = − res w=t Fj,a(t)Fa,i(w) dw w = res w=t Fj,a(w)Fa,i(t) dw w (A.3) 22 S. Khoroshkin and S. Pakuliak for any a = i + 1, . . . , j − 1. It is equivalent to the relation Fj,i(t) = ∮ Fj,a(t)Fa,i(w) dw w − ∮ q−1 − qt/w 1− t/w Fa,i(w)Fj,a(t) dw w , Fj,i(t) = ∮ Fj,a(w)Fa,i(t) dw w − ∮ q−1 − qw/t 1− w/t Fa,i(t)Fj,a(w) dw w . (A.4) In (A.4) ∮ dw w g(w) = g0 for any formal series g(w) = ∑ n∈Z gnz−n. Using the relations (A.1) on Fi(t) we can calculate the residues in (A.3) and obtain the following expressions for Fj,i(t), i < j: Fj,i(t) = (q − q−1)j−i−1Fi(t)Fi+1(t) · · ·Fj−1(t). (A.5) For example, Fi+1,i(t) = Fi(t), and Fi+2,i(t) = (q − q−1)Fi(t)Fi+1(t). The last product is well- defined according to the analyticity properties of the product Fi(t)Fi+1(w), described above. In a similar way, one can show inductively that the product in the right hand side of (A.5) makes sense for any i < j. Formulas (A.5) prove that the defining relations for the composed currents (A.3) or (A.4) yields the same answers for all possible values i < a < j. Calculating formal integrals in (A.4) we obtain the following presentations for the composed currents: Fj,i(t) = Fj,a(t)Fa,i[0]− q−1Fa,i[0]Fj,a(t) + (q − q−1) ∑ k<0 Fa,i[k]Fj,a(t) t−k, Fj,i(t) = Fj,a[0]Fa,i(t)− qFa,i(t)Fj,a[0] + (q − q−1) ∑ k≥0 Fa,i(t) Fj,a[k] t−k, which are useful for the calculation of their projections. Acknowledgements The main idea to use generating series for the description of the hierarchical Bethe ansatz ap- peared during authors visit to Max-Planck Institute für Mathematik, Bonn, in January, 2008. Authors acknowledge this scientific center for the hospitality and stimulating scientific atmo- sphere. This work was partially done when the second author (S.P.) visited Laboratoire d’Annecy-Le- Vieux de Physique Théorique in 2006 and 2007. These visits were possible due to the financial support of the CNRS-Russia exchange program on mathematical physics. He thanks LAPTH for the hospitality and stimulating scientific atmosphere. Authors are grateful to Luc Frappat and Éric Ragoucy for many helpful discussions. The authors were supported in part by RFBR grant 08-01-00392 and grant for the support of scientific schools NSh-3036.2008.2. The first author was also supported by the Atomic Energy Agency of the Russian Federation, and by the ANR grant 05-BLAN-0029-01. The second author was also supported in part by RFBR-CNRS grant 07-02-92166-CNRS. References [1] Arnaudon D., Crampè N., Doikou A., Frappat L., Ragoucy E., Spectrum and Bethe ansatz equations for the Uq(gl(N )) closed and open spin chains in any representation, Ann. Henri Poincaré 7 (2006), 1217–1268, math-ph/0512037. [2] Drinfel’d V.G., New realization of Yangians and quantum affine algebras, Soviet Math. Dokl. 36 (1988), 212–216. http://arxiv.org/abs/math-ph/0512037 Generating Series for Nested Bethe Vectors 23 [3] Ding J.T., Frenkel I.B., Isomorphism of two realizations of quantum affine algebra Uq(ĝlN ), Comm. Math. Phys. 156 (1993), 277–300. [4] Ding J., Khoroshkin S., Weyl group extension of quantized current algebras, Transform. Groups 5 (2000), 35–59, math.QA/9804139. [5] Enriquez B., On correlation functions of Drinfeld currents and shuffle algebras, Transform. Groups 5 (2000), 111–120, math.QA/9809036. [6] Enriquez B., Khoroshkin S., Pakuliak S., Weight functions and Drinfeld currents, Comm. Math. Phys. 276 (2007), 691–725, math.QA/0610398. [7] Enriquez B., Rubtsov V., Quasi-Hopf algebras associated with sl2 and complex curves, Israel J. Math. 112 (1999), 61–108, q-alg/9608005. [8] Frappat L., Khoroshkin S., Pakuliak S., Ragoucy É., Bethe ansatz for the universal weight function, arXiv:0810.3135. [9] Khoroshkin S., Pakuliak S., The weight function for the quantum affine algebra Uq(ŝl3), Theor. and Math. Phys. 145 (2005), 1373–1399, math.QA/0610433. [10] Khoroshkin S., Pakuliak S., Tarasov V., Off-shell Bethe vectors and Drinfeld currents, J. Geom. Phys. 57 (2007), 1713–1732, math.QA/0610517. [11] Khoroshkin S., Pakuliak S., A computation of an universal weight function for the quantum affine algebra Uq(ĝlN ), J. Math. Kyoto Univ. 48 (2008), 277–322, arXiv:0711.2819. [12] Kulish P., Reshetikhin N., Diagonalization of GL(N) invariant transfer matrices and quantum N -wave system (Lee model), J. Phys. A: Math. Gen. 16 (1983), L591–L596. [13] Mukhin E., Tarasov V., Varchenko A., Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp. 2006 (2006), no. 8, P08002, 44 pages, math.QA/0605015. [14] Oskin A., Pakuliak S., Silantyev A., On the universal weight function for the quantum affine algebra Uq(ĝlN ), arXiv:0711.2821. [15] Reshetikhin N., Semenov-Tian-Shansky M., Central extentions of quantum current groups, Lett. Math. Phys. 19 (1990), 133–142. [16] Tarasov V., Varchenko A., Jackson integrals for the solutions to Knizhnik–Zamolodchikov equation, St. Petersburg Math. J. 2 (1995), no. 2, 275–313. [17] Tarasov V., Varchenko A., Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997), 1–135, q-alg/9703044. [18] Tarasov V., Varchenko A., Combinatorial formulae for nested Bethe vectors, math.QA/0702277. http://arxiv.org/abs/math.QA/9804139 http://arxiv.org/abs/math.QA/9809036 http://arxiv.org/abs/math.QA/0610398 http://arxiv.org/abs/q-alg/9608005 http://arxiv.org/abs/0810.3135 http://arxiv.org/abs/math.QA/0610433 http://arxiv.org/abs/math.QA/0610517 http://arxiv.org/abs/0711.2819 http://arxiv.org/abs/math.QA/0605015 http://arxiv.org/abs/0711.2821 http://arxiv.org/abs/q-alg/9703044 http://arxiv.org/abs/math.QA/0702277 1 Introduction 2 Generating series and \star-product 2.1 A q-symmetrization 2.2 Generating series 3 Universal nested Bethe vectors for U_q(\widehat{gl}_N) 3.1 Generating series for universal Bethe vectors 4 Generating series of strings and nested Bethe ansatz 4.1 Composed currents and the strings 4.2 Hierarchical relations for the negative projections 4.3 Other type of the hierarchical relations 4.4 Inverting generating series of the strings 4.5 Inversion and combinatorics 4.6 Inversion of generating series for U_q(\widehat{gl}_2) 4.7 Universal Bethe ansatz for U_q(\widehat{gl}_2) A Current realization of U_q(\widehat{gl}_N) References

Generating Series for Nested Bethe Vectors

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