On the nested algebraic Bethe ansatz for spin chains with simple $\mathfrak{g}$-symmetry

TL;DR

This paper introduces a new framework for the nested algebraic Bethe ansatz applicable to spin chains with any simple Lie algebra symmetry, utilizing residual $U(1)$ charges and block Gauss decomposition.

Contribution

It develops a general approach for the nested algebraic Bethe ansatz for $rak{g}$-symmetric spin chains, extending existing methods to all simple Lie algebras.

Findings

Derivation of nesting of Yangian algebras

Establishment of AB commutation relations

Application to rational spin chains with $rak{g}$-symmetry

Abstract

We propose a new framework for the nested algebraic Bethe ansatz for a closed, rational spin chain with $\mathfrak{g}$-symmetry for any simple Lie algebra $\mathfrak{g}$. Starting the nesting process by removing a single simple root from $\mathfrak{g}$, we use the residual $U(1)$ charge and the block Gauss decomposition of the $R$-matrix to derive many standard results in the Bethe ansatz, such as the nesting of Yangian algebras, and the AB commutation relation.