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).
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Bjorner, A., Brenti, F.: Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)
Halacheva, I., Kamnitzer, J., Rybnikov, L., Weekes, A.: Crystals and monodromy of Bethe vectors. arXiv:1708.05105
Ilin, A., Rybnikov, L.: Degeneration of Bethe subalgebras in the Yangian of \({\mathfrak{gl}}_n\), L. Lett. Math. Phys. 108, 1083–1107 (2018)
Ilin, A., Rybnikov, L.: Bethe subalgebras in Yangians and the wonderful compactification. arXiv:1810.07308
Khoroshkin, S., Tolstoy, V.: Yangian double. Lett. Math. Phys. 36(4), 373–402 (1996)
Molev, A.: Yangians and Classical Lie Algebras. Mathematical Surveys and Monographs, vol. 143. American Mathematical Society, Providence, RI (2007)
Mukhin, E., Tarasov, V., Varchenko, A.: Bethe eigenvectors of higher transfer matrices. J. Stat. Mech. Theory Exp. 8, 1–44 (2006)
Mukhin, E., Tarasov, V., Varchenko, A.: Bethe algebra of homogeneous XXX Heisenberg model. Commun. Math. Phys. 288, 1–39 (2009)
Nazarov, M., Olshanski, G.: Bethe subalgebras in twisted Yangians. Commun. Math. Phys. 178(2), 483–506 (1996)
Shapovalov, N.N.: On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra. Funct. Anal. Appl. 6, 307–312 (1972)
Tarasov, V.: Structure of quantum L-operators for the R-matrix of the XXZ-model. Theor. Math. Phys. 61, 1065–1071 (1984)
Takhtajan, L.A., Faddeev, L.D.: Quantum inverse scattering method and the Heisenberg XYZ-model. Russ. Math. Surv. 34(5), 11–68 (1979)
Acknowledgements
We would like to thank L. Rybnikov for many insighful discussions and profound attention to our work. We would like to thank A. Tsymbaliuk for his deeply careful reading and numerous suggestions for improvement. We also would like to thank V. Vologodsky for many helpful discussions. This research was carried out within the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ‘5-100’. The work was supported by RFBR Grant number 19-31-90124.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mashanova-Golikova, I. Simplicity of Spectra for Bethe Subalgebras in \({\mathrm {Y}}({\mathfrak {gl}}_2)\). Arnold Math J. 7, 313–339 (2021). https://doi.org/10.1007/s40598-020-00171-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40598-020-00171-7