Higher order Hamiltonians for the trigonometric Gaudin model

TL;DR

This paper derives explicit higher Hamiltonians for the trigonometric Gaudin model by taking the classical limit of Bethe subalgebra elements from the quantum $XXZ$ model, advancing understanding of integrable systems.

Contribution

It provides a new explicit construction of higher Hamiltonians for the trigonometric Gaudin model using the $q o 1$ limit of the Bethe subalgebra.

Findings

Explicit higher Hamiltonians derived for the Gaudin model.

Connection established between $XXZ$ Bethe subalgebra and Gaudin Hamiltonians.

Enhanced tools for studying integrable models and their symmetries.

Abstract

We consider the trigonometric classical $r$-matrix for $\mathfrak{gl}_N$ and the associated quantum Gaudin model. We produce higher Hamiltonians in an explicit form by applying the limit $q\to 1$ to elements of the Bethe subalgebra for the $XXZ$ model.