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

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 $B(C)$.

18 pages

References in corpus (2)

• Crystals and monodromy of Bethe vectors
• Bethe subalgebras in Yangians and the wonderful compactification