Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric $R$-Matrix

TL;DR

This paper investigates the structure of Bethe vectors in quantum integrable models with GL(3) trigonometric R-matrix, demonstrating their closure under monodromy matrix actions using algebraic Bethe ansatz techniques.

Contribution

It introduces a new approach to express universal Bethe vectors via projections of quantum affine algebra currents, proving their invariance under monodromy matrix elements.

Findings

Bethe vectors form a closed set under monodromy matrix action.

Representation of Bethe vectors via quantum affine algebra currents.

Application to models solvable by nested algebraic Bethe ansatz.

Abstract

We study quantum integrable models with GL(3) trigonometric $R$-matrix and solvable by the nested algebraic Bethe ansatz. Using the presentation of the universal Bethe vectors in terms of projections of products of the currents of the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_3)$ onto intersections of different types of Borel subalgebras, we prove that the set of the nested Bethe vectors is closed under the action of the elements of the monodromy matrix.