A computation of universal weight function for quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$

Khoroshkin, S.; Pakuliak, S.

doi:10.1215/kjm/1250271413

Cited by 53 publications

(129 citation statements)

References 3 publications

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“…The background for this conjecture was the observation that both quantities satisfy the same coproduct property [8]. It was proved in [12] that calculation of the projection of the product of currents gives nested recurrence relations for the off-shell Bethe vectors similar to those obtained in [18] at the level of the tensor product of the evaluation U q ( p gl N )-modules. This leads to the conclusion that the projection method yields the universal off-shell Bethe vectors for an arbitrary U q ( p gl N )-module generated by a singular weight vector.…”

Section: §1 Introductionmentioning

confidence: 82%

“…A more general formula for the universal off-shell Bethe vectors was obtained in [12] by using the current realization of the quantum affine algebra U q ( p gl N ) and the method of projections introduced in [9] and developed in [8]. Formula (2.18) was obtained in the latter paper after specialization to the evaluation modules.…”

Section: Combinatorial Formulas For Off-shell Bethe Vectorsmentioning

confidence: 99%

“…The Gauss decomposition (3.4)-(3.6) was used 3 in [12] in order to obtain a recurrence relation for the universal weight function (4.6) and to prove the conjecture of [11] that the constructions of the off-shell Bethe vectors by using the L-operator approach [17] and the method of projections [8] coincide for an arbitrary U q ( p gl N )-module generated by arbitrary singular vectors. In this paper, another Gauss decomposition (3.1)-(3.3) will be used in order to get an alternative recurrence relation for the universal weight function (4.5), which will lead to formula (2.20) for the off-shell Bethe vector.…”

Section: Combinatorial Formulas For Off-shell Bethe Vectorsmentioning

confidence: 99%

“…This leads to the conclusion that the projection method yields the universal off-shell Bethe vectors for an arbitrary U q ( p gl N )-module generated by a singular weight vector. In [12], only one type of the recurrence relations that lead to a formula of type (2.18) was considered. Here we generalize the results of [12] and also prove that, in order to get both types of recurrence relations, one needs to use two different isomorphic current realizations of the quantum affine algebra U q ( p gl N ).…”

Section: §1 Introductionmentioning

confidence: 99%

“…In [12], only one type of the recurrence relations that lead to a formula of type (2.18) was considered. Here we generalize the results of [12] and also prove that, in order to get both types of recurrence relations, one needs to use two different isomorphic current realizations of the quantum affine algebra U q ( p gl N ). The origin of these two different realizations lies in two possibilities for introducing the Gauss decompositions of L-operators (see (3.1…”

Section: §1 Introductionmentioning

confidence: 99%

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On the universal weight function for the quantum affine algebra $U_q(\widehat {\mathfrak {gl}}_N)$

Os'kin¹,

Pakuliak

Silant′ev³

2010

St. Petersburg Math. J.

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Abstract. The investigation is continued of the universal weight function for the quantum affine algebra U q ( p gl N ). Two recurrence relations are obtained for the universal weight function with the help of the method of projections. On the level of the evaluation representation of U q ( p gl N ), two recurrence relations are reproduced, which were calculated earlier for the off-shell Bethe vectors by combinatorial methods. One of the results of the paper is a description of two different but isomorphic currents or "new" realizations of the algebra U q ( p gl N ), corresponding to two different Gauss decompositions of the fundamental L-operators. §1. IntroductionThe hierarchical (nested) Bethe ansatz was designed in [13] to construct the eigenvectors of the commuting integrals for quantum integrable models associated with the Lie algebra gl N . It is based on the inductive procedure that relates gl N and gl N −1 Bethe vectors. Since the Bethe vectors for models with gl 2 symmetry are known, this hierarchical procedure yields an implicit description of the Bethe vectors of the models with symmetry of higher rank.If the parameters of these vectors satisfy the Bethe equations, then the corresponding vectors are eigenvectors of a commuting set of operators in some quantum integrable model. In the current paper, we consider Bethe vectors with free parameters, the socalled "off-shell Bethe vectors".Further development of the off-shell Bethe vectors theory was achieved in [17], where they were presented as particular matrix elements of monodromy operators. This construction was used in [18] to obtain explicit formulas for the off-shell Bethe vectors on the tensor product of evaluation modules of U q ( p gl N ). Two different recurrence relations for the off-shell Bethe vectors on the evaluation U q ( p gl N )-modules were obtained in [18]. Iteration of these two relations allows us to obtain different explicit formulas for the off-shell Bethe vectors (see examples (2.18) and (2.20) below). The existence of two types of recurrence relations in the nested Bethe ansatz is a consequence of two different ways of embedding. When these algebras are realized in terms of L-operators, the L-operator of U q ( p gl N −1 ) can be placed either into the top-left or into the down-right corner of the U q ( p gl N ) L-operator. Due to applications to the theory of quantized Knizhnik-Zamolodchikov equations, the off-shell Bethe vectors are called weight functions. We use both names for these objects. We say that a weight function is universal if it is defined in an arbitrary U q ( p gl N )-module generated by an arbitrary singular vector.2010 Mathematics Subject Classification. Primary 81R10.

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Section: §1 Introductionmentioning

confidence: 82%

Section: Combinatorial Formulas For Off-shell Bethe Vectorsmentioning

confidence: 99%

Section: Combinatorial Formulas For Off-shell Bethe Vectorsmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the universal weight function for the quantum affine algebra $U_q(\widehat {\mathfrak {gl}}_N)$

Os'kin¹,

Pakuliak

Silant′ev³

2010

St. Petersburg Math. J.

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Bethe Vectors of gl(3)-Invariant Integrable Models, Their Scalar Products and Form Factors

Ragoucy

2014

Springer Proceedings in Mathematics &Amp; Statistics

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On Bethe vectors in $$ \mathfrak{g}{\mathfrak{l}}_3 $$-invariant integrable models

Liashyk

Slavnov

2018

J. High Energ. Phys.

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We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl 3 -invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.

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A computation of universal weight function for quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$

Abstract: We compute an universal weight function (off-shell Bethe vectors) in any representation with a weight singular vector of the quantum affine algebra U q ( gl N ) applying the method of projections of Drinfeld currents developed in [EKP].

Cited by 53 publications

References 3 publications

On the universal weight function for the quantum affine algebra $U_q(\widehat {\mathfrak {gl}}_N)$

On the universal weight function for the quantum affine algebra $U_q(\widehat {\mathfrak {gl}}_N)$

Bethe Vectors of gl(3)-Invariant Integrable Models, Their Scalar Products and Form Factors

On Bethe vectors in $$ \mathfrak{g}{\mathfrak{l}}_3 $$-invariant integrable models

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