Weight Functions and Drinfeld Currents

Enriquez, Benjamin; Khoroshkin, S.; Pakuliak, S.

doi:10.1007/s00220-007-0351-y

Cited by 65 publications

(185 citation statements)

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“…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%

“…We can check (see [8,11]) that the intersections U − f and U + F are coideals with respect to the coproduct (3.16),…”

Section: Borel Subalgebras and Their Intersections We Consider Two Tmentioning

confidence: 99%

“…An alternative approach to the construction of off-shell Bethe vectors for an arbitrary quantum affine algebra was developed in [8]. This approach involves the current realization of the corresponding symmetry algebra or a "new" realization of the quantum double of the Yangians and quantum affine algebras [2].…”

Section: §1 Introductionmentioning

confidence: 99%

“…It was conjectured in [11] that the projections onto the intersection of different type Borel subalgebras for the product of Drinfeld currents coincide with the off-shell Bethe vectors [17]. 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.…”

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: Combinatorial Formulas For Off-shell Bethe Vectorsmentioning

confidence: 99%