Bethe Vectors in Quantum Integrable Models with Classical Symmetries

TL;DR

This paper provides a clear definition of off-shell Bethe vectors in various quantum integrable models with classical symmetries, demonstrating their properties and their transition to on-shell vectors when Bethe equations are satisfied.

Contribution

It introduces a unified, simple definition of off-shell Bethe vectors for models with classical symmetries and derives key properties from this definition.

Findings

Off-shell Bethe vectors become on-shell when Bethe equations are satisfied.

Derived action formulas for monodromy entries on Bethe vectors.

Established recurrence relations and coproduct formulas for these vectors.

Abstract

The first goal of this paper is to give a precise and simple definition for off-shell Bethe vectors in a generic $g$-invariant integrable model for $g=gl_n$, $o_{2n+1}$, $sp_{2n}$ and $o_{2n}$. We prove from our definition that the off-shell Bethe vectors indeed become on-shell when the Bethe equations are obeyed. Then, we show that some properties for these off-shell Bethe vectors, such as the action formulas of monodromy entries on these vectors, their rectangular recurrence relations and their coproduct formula, are a consequence of our definition.