Weight function and nested Bethe ansatz (Q866862)

The paper is devoted to the comparison of two constructions of weight functions for the quantum algebra \(U_q(\widehat{\mathfrak{gl}}_N)\), the first being given in terms of \(L\)-operators and the second being defined in terms of projections according to the approach formulated in [\textit{S. Khoroshkin} and \textit{S. Pakuliak}, ``Weight function for the quantum affine algebra \(U_q(\widehat{\mathfrak{sl}}_3)\)'', Theor. Math. Phys. 145, No. 1, 1373--1399 (2005); \textit{B. Enriquez, S. Khoroshkin} and \textit{S. Pakuliak}, ``Weight functions and Drinfeld current''s, Preprint-ITEP-TH-40/05 (math/0610398)]. To this aim, projections of the product of the Drinfeld's currents onto intersections of the different Borel subalgebras in the current realization of the quantum affine algebra \(U_q(\widehat{\mathfrak{gl}}_N)\) are calculated. The author proves coincidence of the two construction for the symmetric tensor products of the fundamental vector representation of \(U_q(\widehat{ \mathfrak{g l}}_N)\) by resorting to the isomorphism between \(L\)-operator and current formulation of \(U_q(\widehat{ \mathfrak{g l}}_N)\) established in [\textit{J. Ding} and \textit{I. B. Frenkel}, ``Isomorphism of two realizations of quantum affine algebra \(U_q(\widehat{\mathfrak{gl}}_N)\)'', Commun. Math. Phys. 156, No. 2, 277--300 (1993; Zbl 0786.17008)] and by demonstrating that identical recursion relations on the rank \(N\) are satisfied. A slightly extended discussion of the problem can be found in [\textit{S. Khoroshkin, S. Pakuliak} and \textit{V. Tarasov}, ``Off-shell Bethe vectors and Drinfeld currents'', J. Geom. Phys. 57, No. 8, 1713--1732 (2007; Zbl 1148.17010)].