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

Inna Mashanova-Golikova

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arxiv 1906.09049 v2 pith:3GYFGEUW submitted 2019-06-21 math.QA math.RT

Simplicity of spectra for Bethe subalgebras in Y(mathfrak{gl}₂)

Inna Mashanova-Golikova This is my paper

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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 $B(C)$.

Simplicity of spectra for Bethe subalgebras in Y(mathfrak{gl}₂)

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