# Simplicity of spectra for Bethe subalgebras in $Y(\mathfrak{gl}_2)$

## TL;DR

This paper proves that Bethe subalgebras in the Yangian of gl_2 have simple spectra on certain finite-dimensional modules, using determinant computations, and extends these results to algebra limits.

## Contribution

It establishes simplicity of spectra for Bethe subalgebras in Y(gl_2) with real diagonal C and extends the results to algebra degenerations.

## Key findings

- Bethe subalgebras with real diagonal C have simple spectrum on irreducible modules.
- The simplicity holds for modules corresponding to disjoint real strings.
- Results are extended to limits of Bethe algebras.

## Abstract

We consider Bethe subalgebras B(C) in the Yangian $Y(\mathfrak{gl}_2)$ with $C$ regular $2\times 2$ matrix. We study the action of Bethe subalgebras of $Y(\mathfrak{gl}_2)$ on finite-dimensional representations of $Y(\mathfrak{gl}_2)$. We prove that $B(C)$ with real diagonal $C$ has simple spectrum on any irreducible $Y(\mathfrak{gl}_2)$-module corresponding to a disjoint union of real strings. We extend this result to limits of Bethe algebras. Our main tool is the computation of Shapovalov-type determinant for the nilpotent degeneration of $B(C)$.