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Mat. Sb., 2003, Volume 194, Number 11, Pages 3–16 (Mi msb778)  

This article is cited in 8 scientific papers (total in 8 papers)

Geometry of translations of invariants on semisimple Lie algebras

Yu. A. Brailov

M. V. Lomonosov Moscow State University

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 $\mathrm{sl}(n,\mathbb C)$. 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.

DOI: https://doi.org/10.4213/sm778

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English version:
Sbornik: Mathematics, 2003, 194:11, 1585–1598

Bibliographic databases:

UDC: 513.944
MSC: Primary 14L24; Secondary 37B05, 37J35
Received: 10.06.2003

Citation: Yu. A. Brailov, “Geometry of translations of invariants on semisimple Lie algebras”, Mat. Sb., 194:11 (2003), 3–16; Sb. Math., 194:11 (2003), 1585–1598

Citation in format AMSBIB
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\paper Geometry of translations of invariants on semisimple Lie algebras
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\pages 3--16
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\pages 1585--1598
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Bolsinov, AV, “Bi-Hamiltonian structures and singularities of integrable systems”, Regular & Chaotic Dynamics, 14:4–5 (2009), 431  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    2. 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
    3. 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
    4. Anton Izosimov, “Algebraic geometry and stability for integrable systems”, Physica D: Nonlinear Phenomena, 2014  crossref  mathscinet  scopus  scopus  scopus
    5. 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
    6. 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
    7. Izosimov A., “Singularities of Integrable Systems and Algebraic Curves”, Int. Math. Res. Notices, 2017, no. 18, 5475–5524  crossref  mathscinet  isi  scopus
    8. 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
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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