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

Abstract

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.

Current presentation for the super-Yangian double $ DY(\mathfrak{gl}(m\vert n))$ and Bethe vectors

“…Let us prove (C.3) step by step. First of all we observe that the 'negative' projection of the product of the composed currents (C.1) with restrictions (C.2) can be factorized [13,14]…”

“…to stress that one can use either one or another decomposition into series (2.33) of the rational function 1 u−v . It is known [14] that another current realization of the Yangian double DY (gl(m|n)) can be obtained using a different Gauss decomposition of the monodromy matrix, as in (2.17)- (2.19). The commutation relations between Cartan currentsk ± i (u) and the simple root total currentsF i (u),Ê i (u)…”

“…As we already mentioned, the universal Bethe vectors are constructed from the matrix elements of one universal monodromy matrix T + ij and belong to the standard 'positive' Borel subalgebra U + . The goal of this section is to express the universal Bethe vectors through the current generators of the Yangian double DY (gl(m|n)) using the approach developed in the papers [13,14,26].…”

“…The general theory of the relation between Bethe vectors and currents was developed in the paper [26] and then applied in the papers [13,14] to the construction of the hierarchical Bethe vectors for quantum integrable models based on the quantum affine algebra U q ( gl(N )). The main tool used in these papers was the language of projections onto intersections of different type Borel subalgebras.…”

“…For the composed currents E i,j (v) defined by the formula (A.2) we have 14) or using (A.12) we obtain for these composed currents…”