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Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties

  • Dmitri I. PanyushevEmail author
  • Oksana S. Yakimova
Article
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Abstract

The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra \({\mathfrak {g}}\), we obtain several results on the completeness of homogeneous Poisson-commutative subalgebras of \({\mathscr {S}}({\mathfrak {g}})\) on coadjoint orbits. This concerns, in particular, Gelfand–Tsetlin and Mishchenko–Fomenko subalgebras. Our results reveal the crucial role of nilpotent orbits and sheets in \({\mathfrak {g}}\simeq {\mathfrak {g}}^{*}\).

Keywords

Integrable systems Moment map Coisotropic actions Coadjoint orbits 

Mathematics Subject Classification

17B63 14L30 17B08 17B20 22E46 

Notes

Acknowledgements

We are grateful to the anonymous referee for the detailed report and important suggestions.

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Copyright information

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

Authors and Affiliations

  1. 1.Institute for Information Transmission Problems of the R.A.S.MoscowRussia
  2. 2.Mathematisches InstitutUniversität zu KölnCologneGermany

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