Letters in Mathematical Physics

, Volume 108, Issue 4, pp 1083–1107 | Cite as

Degeneration of Bethe subalgebras in the Yangian of \(\mathfrak {gl}_n\)

  • Aleksei Ilin
  • Leonid RybnikovEmail author


We study degenerations of Bethe subalgebras B(C) in the Yangian \(Y(\mathfrak {gl}_n)\), where C is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parameterizes all possible degenerations, is the Deligne–Mumford moduli space of stable rational curves \(\overline{M_{0,n+2}}\). All subalgebras corresponding to the points of \(\overline{M_{0,n+2}}\) are free and maximal commutative. We describe explicitly the “simplest” degenerations and show that every degeneration is the composition of the simplest ones. The Deligne–Mumford space \(\overline{M_{0,n+2}}\) generalizes to other root systems as some De Concini–Procesi resolution of some toric variety. We state a conjecture generalizing our results to Bethe subalgebras in the Yangian of arbitrary simple Lie algebra in terms of this De Concini–Procesi resolution.


Yangians Bethe subalgebras Deligne-Mumford compactification Centralizer construction 

Mathematics Subject Classification

17B37 17B63 17B80 



We thank Michael Finkelberg and Alexander Molev for helpful remarks and references. We thank the referee for careful reading of the first version of the text. This work has been funded by the Russian Academic Excellence Project ’5-100’. The work was finished during A.I.’s internship at MIT supported by NRU HSE. A.I. is deeply indebted to MIT and especially to R. Bezrukavnikov and P. Etingof, for providing warm hospitality and excellent working conditions. The work of A.I. and L.R. was supported in part by the Simons Foundation. The work of L.R. has been supported by the Russian Science Foundation under grant 16-11-10160.


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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

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