# Bethe vectors for orthogonal integrable models

## TL;DR

This paper develops a new algebraic framework for describing Bethe vectors in $ ext{so}_3$ integrable models, deriving action formulas, recursions, and Bethe equations, facilitating scalar product calculations.

## Contribution

It introduces a novel approach using current generators and algebra isomorphisms to analyze Bethe vectors in $ ext{so}_3$ models, expanding the algebraic Bethe ansatz methods.

## Key findings

- Derived explicit action formulas for monodromy matrix elements on Bethe vectors
- Established recursion relations for off-shell Bethe vectors
- Formulated Bethe equations for on-shell Bethe vectors

## Abstract

We consider quantum integrable models associated with $\mathfrak{so}_3$ algebra. We describe Bethe vectors of these models in terms of the current generators of the $\mathcal{D}Y(\mathfrak{so}_3)$ algebra. To implement this approach we use isomorphism between $R$-matrix and Drinfeld current realizations of the Yangians and their doubles for classical types $B$, $C$, and $D$ series algebras. Using these results we derive the actions of the monodromy matrix elements on off-shell Bethe vectors. We show that these action formulas lead to recursions for off-shell Bethe vectors and Bethe equations for on-shell Bethe vectors. The action formulas can also be used for calculating the scalar products in the models associated with $\mathfrak{so}_3$ algebra.