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Simplicity of Spectra for Bethe Subalgebras in $${\mathrm {Y}}({\mathfrak {gl}}_2)$$

Authors :
Inna Mashanova-Golikova
Source :
Arnold Mathematical Journal. 7:313-339
Publication Year :
2021
Publisher :
Springer Science and Business Media LLC, 2021.

Abstract

We consider Bethe subalgebras B(C) in the Yangian $${\mathrm {Y}}({\mathfrak {gl}}_2)$$ with C regular $$2\times 2$$ matrix. We study the action of Bethe subalgebras of $${\mathrm {Y}}({\mathfrak {gl}}_2)$$ on finite-dimensional representations of $${\mathrm {Y}}({\mathfrak {gl}}_2)$$ . We prove that B(C) with real diagonal C has simple spectrum on any irreducible $${\mathrm {Y}}({\mathfrak {gl}}_2)$$ -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).

Details

ISSN :
21996806 and 21996792
Volume :
7
Database :
OpenAIRE
Journal :
Arnold Mathematical Journal
Accession number :
edsair.doi...........4bc1dd307a8e7f85eda352fdb4976600
Full Text :
https://doi.org/10.1007/s40598-020-00171-7