New symmetries of $ {\mathfrak{gl}(N)}$ -invariant Bethe vectors

Abstract

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(N )-invariant R-matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of the Bethe vectors are identical up to normalization and reshuffling of the Bethe parameters. To prove this correspondence we use the current approach. This identity gives new combinatorial relations for the scalar products of the Bethe vectors. The q-deformed case, as well as the superalgebra case, are also evoked in the conclusion. 1 a.

Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame

“…where σ can be arbitrary and where we also restored arbitrariness in the choice of I. Theq i are "on-shell" twisted polynomials solving the QQ-relations/Bethe equations. It would be natural to define "off-shell" Bethe vectors |τ by (3.74) with arbitrary polynomialŝ q i , we however did not check how this compares to the off-shell Bethe vectors studied in the literature [34,35].…”

Bethe Vectors for Orthogonal Integrable Models

“…Using results of the paper[23] one can similarly determine a set of generators of the Bn algebra corresponding to the classical algebra so2n+1.…”

Gauss Coordinates vs Currents for the Yangian Doubles of the Classical Types

“…The main new results of this section are the relations between Gauss coordinates (3.25), (3.37) and (3.49) of T -operators for the Yangian doubles DY (o 2n+1 ), DY (sp 2n ) and DY (o 2n ) respectively. These formulas are direct consequence of the equality (2.11) and the results obtained in [15]. For the sake of completeness we recall these results in the Section 3.1.…”

“…The main result of the paper [15] was an explicit presentation of the isomorphism of the Yangian double DY (gl N ) (2.12) in terms of the Gauss coordinates. Gauss decomposition of the monodromiesT ± (u) has literally the same form as in (3.1) with the Gauss coordinates…”

“…and formulas (3.11)-(3.13) were proved in [15] by reordering Gauss coordinates using the commutation relations between them in the T -operatorsT ± (u) given by (2.9). In the current realization the Yangian double DY (gl N ) possess the automorphism…”