Simplicity of spectra for Bethe subalgebras in Y(\mathfrakgl₂)

2019/06/21 by Mashanova-Golikova, Inna
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1906.09049

Abstract

We consider Bethe subalgebras B(C) in the Yangian Y(\mathfrakgl₂) with C regular 2× 2 matrix. We study the action of Bethe subalgebras of Y(\mathfrakgl₂) on finite-dimensional representations of Y(\mathfrakgl₂). We prove that B(C) with real diagonal C has simple spectrum on any irreducible Y(\mathfrakgl₂)-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).

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