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

DOI10.1007/S40598-020-00171-7arXiv1906.09049MaRDI QIDQ6320830

Publication date: 21 June 2019

Abstract: We consider Bethe subalgebras B(C) in the Yangian

Y (m a t h f r a k {g l}_{2})

with

C

regular

2 i m e s 2

matrix. We study the action of Bethe subalgebras of

Y (m a t h f r a k {g l}_{2})

on finite-dimensional representations of

Y (m a t h f r a k {g l}_{2})

. We prove that

B (C)

with real diagonal

C

has simple spectrum on any irreducible

Y (m a t h f r a k {g l}_{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)

.

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Ring-theoretic aspects of quantum groups (16T20) Yang-Baxter equations (16T25)

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