Arnold Mathematical Journal

, Volume 5, Issue 2–3, pp 285–313 | Cite as

Gelfand–Tsetlin Degeneration of Shift of Argument Subalgebras in Types B, C and D

  • Leonid RybnikovEmail author
  • Mikhail Zavalin
Research Contribution


The universal enveloping algebra of any semisimple Lie algebra \(\mathfrak {g}\) contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of \(\mathfrak {g}\). For \(\mathfrak {g}=\mathfrak {gl}_n\) the Gelfand–Tsetlin commutative subalgebra in \(U(\mathfrak {g})\) arises as some limit of subalgebras from this family. We study the analogous limit of shift of argument subalgebras for classical Lie algebras (\(\mathfrak {g}=\mathfrak {sp}_{2n}\) or \(\mathfrak {so}_{n}\)). The limit subalgebra is described explicitly in terms of Bethe subalgebras in twisted Yangians \(Y^-(2)\) and \(Y^+(2)\), respectively. We index the eigenbasis of such limit subalgebra in any irreducible finite-dimensional representation of \(\mathfrak {g}\) by Gelfand–Tsetlin patterns of the corresponding type, and conjecture that this indexing is, in appropriate sense, natural. According to Halacheva et al. (Crystals and monodromy of Bethe vectors. arXiv:1708.05105, 2017) such eigenbasis has a natural \(\mathfrak {g}\)-crystal structure. We conjecture that this crystal structure coincides with that on Gelfand–Tsetlin patterns defined by Littelmann in Cones, crystals, and patterns (Transform Groups 3(2):145–179, 1998).



We thank Alexander Molev for explanations on Yangians. The paper was completed during our stay at University of Tokyo. We are grateful to University of Tokyo and especially to Junichi Shiraishi for hospitality. This research was carried out within the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ’5-100’. Theorem B has been obtained under support of the RSF grant 19-11-00056. The first author has also been supported in part by the Simons Foundation. We thank the referee for extremely helpful remarks on the first version of the paper and for improving and simplifying the proof of Theorem A.


  1. Gerrard, A., MacKay, N., Regelskis, V.: Nested algebraic Bethe ansatz for open spin chains with even twisted Yangian symmetry. ArXiv e-prints, October 2017, arXiv:1710.08409
  2. Halacheva, I., Kamnitzer, J., Rybnikov, L., Weekes, A.: Crystals and monodromy of Bethe vectors. ArXiv e-prints, August 2017, arXiv:1708.05105
  3. Kostant, B.: Lie group representations on polynomial rings. Am. J. Math. 85, 327–404 (1963)MathSciNetCrossRefGoogle Scholar
  4. Littelmann, P.: Cones, crystals, and patterns. Transform. Groups 3(2), 145–179 (1998)MathSciNetCrossRefGoogle Scholar
  5. Mishchenko, A.S., Fomenko, A.T.: Integrability of Euler’s equations on semisimple Lie algebras. Trudy Sem. Vektor. Tenzor. Anal. 19, 3–94 (1979)MathSciNetzbMATHGoogle Scholar
  6. Molev, A., Nazarov, M., Olshanski, G.: Yangians and classical Lie algebras. Uspekhi Mat. Nauk 51(2(308)), 27–104 (1996)MathSciNetCrossRefGoogle Scholar
  7. Molev, A., Olshanski, G.: Centralizer construction for twisted Yangians. Selecta Math. (N.S.) 6(3), 269–317 (2000)MathSciNetCrossRefGoogle Scholar
  8. Molev, A. I.: Gelfand–Tsetlin bases for classical Lie algebras. In: Handbook of Algebra. Vol. 4. Handb. Algebr., pp. 109–170. Elsevier, Amsterdam (2006)Google Scholar
  9. Molev, A., Yakimova, O.: Quantisation and nilpotent limits of Mishchenko-Fomenko subalgebras. ArXiv e-prints, November 2017, arXiv:1711.03917
  10. Nazarov, M., Olshanski, G.: Bethe subalgebras in twisted Yangians. Commun. Math. Phys. 178(2), 483–506 (1996)MathSciNetCrossRefGoogle Scholar
  11. Panyushev, D.I., Yakimova, O.S.: The argument shift method and maximal commutative subalgebras of Poisson algebras. Math. Res. Lett. 15(2), 239–249 (2008)MathSciNetCrossRefGoogle Scholar
  12. Rybnikov, L.G.: Centralizers of some quadratic elements in Poisson–Lie algebras and a method for the translation of invariants. Uspekhi Mat. Nauk 60(2(362)), 173–174 (2005)MathSciNetCrossRefGoogle Scholar
  13. Rybnikov, L.G.: The shift of invariants method and the Gaudin model. Funktsional. Anal. i Prilozhen. 40(3), 30–43, 96 (2006)MathSciNetCrossRefGoogle Scholar
  14. Shuvalov, V.V.: On the limits of Mishchenko–Fomenko subalgebras in Poisson algebras of semisimple Lie algebras. Funktsional. Anal. i Prilozhen. 36(4), 55–64 (2002)MathSciNetCrossRefGoogle Scholar
  15. Tarasov, A.A.: The maximality of some commutative subalgebras in Poisson algebras of semisimple Lie algebras. Uspekhi Mat. Nauk 57(5(347)), 165–166 (2002)MathSciNetCrossRefGoogle Scholar
  16. Vinberg, E.B.: Some commutative subalgebras of a universal enveloping algebra. Izv. Akad. Nauk SSSR Ser. Mat. 54(1), 3–25, 221 (1990)MathSciNetGoogle Scholar

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© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Research University Higher School of Economics, Russian FederationMoscowRussia
  2. 2.Institute for Information Transmission Problems of RASMoscowRussia

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