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Sbornik: Mathematics, 2003, Volume 194, Issue 11, Pages 1585–1598
DOI: https://doi.org/10.1070/SM2003v194n11ABEH000778
(Mi sm778)
 

This article is cited in 9 scientific papers (total in 9 papers)

Geometry of translations of invariants on semisimple Lie algebras

Yu. A. Brailov

M. V. Lomonosov Moscow State University
References:
Abstract: Each orbit of the coadjoint representation of a semisimple Lie algebra can be equipped with a complete commutative family of polynomials; this family was obtained by the argument-translation method in papers of Mishchenko and Fomenko. This commutative family and the corresponding Euler's equations play an important role in the theory of finite-dimensional integrable systems. These Euler's equations admit a natural Lax representation with spectral parameter.
It is proved in the paper that the discriminant of the spectral curve coincides completely with the bifurcation diagram of the moment map for the algebra sl(n,C). The maximal degeneracy points of the moment map are described for compact semisimple Lie algebras in terms of the root structure. It is also proved that the set of regular points of the moment map is connected, and the inverse image of each regular point consists of precisely one Liouville torus.
Received: 10.06.2003
Bibliographic databases:
UDC: 513.944
MSC: Primary 14L24; Secondary 37B05, 37J35
Language: English
Original paper language: Russian
Citation: Yu. A. Brailov, “Geometry of translations of invariants on semisimple Lie algebras”, Sb. Math., 194:11 (2003), 1585–1598
Citation in format AMSBIB
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\by Yu.~A.~Brailov
\paper Geometry of translations of invariants on semisimple Lie algebras
\jour Sb. Math.
\yr 2003
\vol 194
\issue 11
\pages 1585--1598
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Linking options:
  • https://www.mathnet.ru/eng/sm778
  • https://doi.org/10.1070/SM2003v194n11ABEH000778
  • https://www.mathnet.ru/eng/sm/v194/i11/p3
  • This publication is cited in the following 9 articles:
    1. PETER CROOKS, MARKUS RÖSER, “ON THE SINGULARITIES OF MISHCHENKO–FOMENKO SYSTEMS”, Transformation Groups, 28:4 (2023), 1477  crossref
    2. Izosimov A., “Singularities of Integrable Systems and Algebraic Curves”, Int. Math. Res. Notices, 2017, no. 18, 5475–5524  crossref  mathscinet  zmath  isi  scopus
    3. S. V. Sokolov, “Integriruemyi sluchai Adlera–van Mërbeke. Vizualizatsiya bifurkatsii torov Liuvillya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:4 (2017), 532–539  mathnet  crossref  elib
    4. Pavel E. Ryabov, Andrej A. Oshemkov, Sergei V. Sokolov, “The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram”, Regul. Chaotic Dyn., 21:5 (2016), 581–592  mathnet  crossref  mathscinet  zmath
    5. P. E. Ryabov, E. O. Biryucheva, “Diskriminantnoe mnozhestvo i bifurkatsionnaya diagramma integriruemogo sluchaya M. Adlera i P. van Merbeke”, Nelineinaya dinam., 12:4 (2016), 633–650  mathnet  crossref  elib
    6. Anton Izosimov, “Algebraic geometry and stability for integrable systems”, Physica D: Nonlinear Phenomena, 2014  crossref  mathscinet  scopus  scopus  scopus
    7. Fomenko A.T., Konyaev A.Yu., “New Approach to Symmetries and Singularities in Integrable Hamiltonian Systems”, Topology Appl., 159:7, SI (2012), 1964–1975  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    8. A. Yu. Konyaev, “Bifurcation diagram and the discriminant of a spectral curve of integrable systems on Lie algebras”, Sb. Math., 201:9 (2010), 1273–1305  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Bolsinov, AV, “Bi-Hamiltonian structures and singularities of integrable systems”, Regular & Chaotic Dynamics, 14:4–5 (2009), 431  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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