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Mat. Zametki, 2011, Volume 90, Issue 5, Pages 689–702 (Mi mz8737)  

This article is cited in 3 scientific papers (total in 3 papers)

Алгебраические и геометрические свойства квадратичных гамильтонианов, задаваемых секционными операторами

A. V. Bolsinov, A. Yu. Konyaev

M. V. Lomonosov Moscow State University

Abstract: Following the terminology introduced by V. V. Trofimov and A. T. Fomenko, we say that a self-adjoint operator $\phi\colon \mathfrak{g}^* \to \mathfrak{g}$ is sectional if it satisfies the identity $\operatorname{ad}^{*}_{\phi x}a=\operatorname{ad}^{*}_{\beta}x$, $x\in \mathfrak{g}^*$, where $\mathfrak{g}$ is a finite-dimensional Lie algebra and $a\in \mathfrak{g}^*$ and $\beta \in \mathfrak{g}$ are fixed elements. In the case of a semisimple Lie algebra $\mathfrak{g}$, the above identity takes the form $[\phi x,a]=[\beta,x]$ and naturally arises in the theory of integrable systems and differential geometry (namely, in the dynamics of $n$-dimensional rigid bodies, the argument shift method, and the classification of projectively equivalent Riemannian metrics). This paper studies general properties of sectional operators, in particular, integrability and the bi-Hamiltonian property for the corresponding Euler equation $\dot x=\operatorname{ad}^*_{\phi x} x$.

Keywords: sectional operator, integrable Euler equation, bi-Hamiltonian Euler equation, finite-dimensional Lie algebra, coadjoint representation, Poisson bracket, Frobenius Lie algebra, semi-simple Lie algebra

DOI: https://doi.org/10.4213/mzm8737

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English version:
Mathematical Notes, 2011, 90:5, 666–677

Bibliographic databases:

UDC: 517.944
Received: 13.10.2010

Citation: A. V. Bolsinov, A. Yu. Konyaev, “Алгебраические и геометрические свойства квадратичных гамильтонианов, задаваемых секционными операторами”, Mat. Zametki, 90:5 (2011), 689–702; Math. Notes, 90:5 (2011), 666–677

Citation in format AMSBIB
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\by A.~V.~Bolsinov, A.~Yu.~Konyaev
\paper Алгебраические и геометрические свойства квадратичных гамильтонианов, задаваемых секционными операторами
\jour Mat. Zametki
\yr 2011
\vol 90
\issue 5
\pages 689--702
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\crossref{https://doi.org/10.4213/mzm8737}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2962559}
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\jour Math. Notes
\yr 2011
\vol 90
\issue 5
\pages 666--677
\crossref{https://doi.org/10.1134/S0001434611110058}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Izosimov A., “Algebraic Geometry and Stability For Integrable Systems”, Physica D, 291 (2015), 74–82  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. A. V. Bolsinov, “Argument shift method and sectional operators: applications to differential geometry”, J. Math. Sci., 225:4 (2017), 536–554  mathnet  crossref  mathscinet  elib
    3. Bolsinov A.V., Izosimov A.M., Tsonev D.M., “Finite-dimensional integrable systems: A collection of research problems”, J. Geom. Phys., 115 (2017), 2–15  crossref  mathscinet  zmath  isi  scopus
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