Communications in Mathematical Physics

, Volume 372, Issue 1, pp 343–366 | Cite as

Bethe Subalgebras in Yangians and the Wonderful Compactification

  • Aleksei Ilin
  • Leonid RybnikovEmail author


Let \({\mathfrak {g}}\) be a complex simple Lie algebra. We study the family of Bethe subalgebras in the Yangian \(Y({\mathfrak {g}})\) parameterized by the corresponding adjoint Lie group G. We describe their classical limits as subalgebras in the algebra of polynomial functions on the formal Lie group \(G_1[[t^{-1}]]\). In particular we show that, for regular values of the parameter, these subalgebras are free polynomial algebras with the same Poincaré series as the Cartan subalgebra of the Yangian. Next, we extend the family of Bethe subalgebras to the De Concini–Procesi wonderful compactification \(\overline{G}\supset G\) and describe the subalgebras corresponding to generic points of any stratum in \(\overline{G}\) as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in \({\mathfrak {g}}\). In particular, we describe explicitly all Bethe subalgebras corresponding to the closure of the maximal torus in the wonderful compactification.



We thank the referees for the careful reading of the first version of the text and for many helpful remarks. This research was carried out within the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ‘5-100’. The results of Sect. 4 has been obtained under support of the RSF Grant 19-11-00056. The work of both authors has also been supported in part by the Simons Foundation. The first author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury.


  1. 1.
    De Concini, C., Procesi, C.: Complete symmetric varieties. Invariant theory (Montecatini, 1982). Lecture Notes in Mathematics, vol. 996, pp. 1–44. Springer, Berlin (1983)Google Scholar
  2. 2.
    Drinfeld, V.: Hopf algebras and the quantum Yang–Baxter equation. Sov. Math. Dokl. 32, 254–258 (1985)Google Scholar
  3. 3.
    Drinfeld, V.: Quantum groups. J. Sov. Math. 41(2), 898–915 (1988)CrossRefGoogle Scholar
  4. 4.
    Evens, S., Jones, B.: On the Wonderful Compactification. arXiv:0801.0456
  5. 5.
    Finkelberg, M., Kamnitzer, J., Pham, K., Rybnikov, L., Weekes, A.: Comultiplication for shifted Yangians and quantum open Toda lattice. Adv. Math. 327, 349–389 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a class of representations of the Yangian and moduli space of monopoles. Commun. Math. Phys. 260(3), 511–525 (2005)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Guay, N., Regelskis, V., Wendlandt, C.: Equivalences between three presentations of orthogonal and symplectic Yangians. Lett. Math. Phys. 109(2), 327–379 (2019)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ilin, A., Rybnikov, L.: Degeneration of Bethe subalgebras in the Yangian of \({\mathfrak{gl}}_n\). Lett. Math. Phys. 108(4), 1083–1107 (2018)ADSMathSciNetzbMATHGoogle Scholar
  9. 9.
    Kamnitzer, J., Webster, B., Weekes, A., Yacobi, O.: Yangians and quantizations of slices in the affine Grassmannian. Algebra Number Theory 8(4), 857–893 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kamnitzer, J., Tingley, P., Webster, B., Weekes, A., Yacobi, O.: Highest weights for truncated shifted Yangians and product monomial crystals. Preprint arXiv:1511.09131
  11. 11.
    Maulik, D., Okounkov, A.: Quantum Groups and Quantum Cohomology. arXiv:1211.1287
  12. 12.
    Molev, A.: Feigin–Frenkel center in types B, C and D. Invent. Math. 191, 1–34 (2013)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Molev, A.: Yangians and classical Lie algebras. Mathematical Surveys and Monographs Volume 143 (2007)Google Scholar
  14. 14.
    Nazarov, M., Olshanski, G.: Bethe subalgebras in twisted Yangians. Commun. Math. Phys. 178, 483–506 (1996)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Steinberg, R.: Conjugacy Classes in Algebraic Groups. Lecture Notes in Mathematics, vol. 366. Springer, Berlin (1974)CrossRefGoogle Scholar
  16. 16.
    Takhtajan, L.A., Faddeev, L.D.: Quantum inverse scattering method and the Heisenberg XYZ-model. Russ. Math. Surv. 34(5), 11–68 (1979)CrossRefGoogle Scholar
  17. 17.
    Tarasov, V.: Structure of quantum L-operators for the R-matrix of the XXZ-model. Theor. Math. Phys. 61, 1065–1071 (1984)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wendlandt, C.: The R-matrix presentation for the Yangian of a simple Lie algebra. Commun. Math. Phys. 363(1), 289–332 (2018)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Research University Higher School of EconomicsMoscowRussian Federation
  2. 2.Institute for Information Transmission Problems of RASMoscowRussian Federation

Personalised recommendations