Hermitian Property and the Simplicity of Spectrum of Bethe Subalgebras in Yangians

Abstract

The image of the Bethe subalgebra $B(C)$ in the tensor product of representations of the Yangian $Y(\mathfrak{gl}_n)$ contains the full set of Hamiltonians of the Heisenberg magnet chain XXX. The main problem in the XXX integrable system is the diagonalization of the operators by which the elements of Bethe subalgebras act on the corresponding representations of the Yangian. The standard approach is the Bethe ansatz. As the first step toward solving this problem, we want to show that the eigenvalues of these operators have multiplicity 1. In this work we obtained several new results on the simplicity of spectra of Bethe subalgebras in Kirillov–Reshetikhin modules in the case of $Y(\mathfrak{g})$, where $\mathfrak{g}$ is a simple Lie algebra.