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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
Article

Abstract

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.

Notes

Acknowledgements

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.

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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

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