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Mat. Zametki, 2011, Volume 90, Issue 3, Pages 384–393 (Mi mz6376)  

This article is cited in 2 scientific papers (total in 2 papers)

Uniqueness of Recovering the Parameters of Sectional Operators on Simple Complex Lie Algebras

A. Yu. Konyaev

M. V. Lomonosov Moscow State University

Abstract: By a sectional operator on a simple complex Lie algebra $\mathfrak g$ we mean a self-adjoint operator $\phi\colon\mathfrak g\to\mathfrak g$ satisfying the identity $[\phi x,a]=[x,b]$ for some chosen elements $a,b\in\mathfrak g$, $a\ne0$. The problem concerning the uniqueness of recovering the parameters of a given specific operator arises in many areas of geometry. The main result of the paper is as follows: if $a$ and $b$ are not proportional and $a$ is regular and semisimple, then every pair of parameters $p$, $q$ of the sectional operator is obtained from the pair $a$, $b$ by multiplying the pair by a nonzero scalar, i.e., the parameters are recovered uniquely in a sense. It follows that the Mishchenko–Fomenko subalgebras for regular semisimple elements of the Poisson–Lie algebra coincide for proportional values of the parameters only.

Keywords: simple complex Lie algebra, sectional operator, caustic, semi-simple element of a Poisson–Lie algebra, Mishchenko–Fomenko algebra, Killing form, Cartan subalgebra, root system, Weyl basis, Jacobi identity

DOI: https://doi.org/10.4213/mzm6376

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English version:
Mathematical Notes, 2011, 90:3, 365–372

Bibliographic databases:

UDC: 517.944
Received: 02.11.2007
Revised: 24.08.2010

Citation: A. Yu. Konyaev, “Uniqueness of Recovering the Parameters of Sectional Operators on Simple Complex Lie Algebras”, Mat. Zametki, 90:3 (2011), 384–393; Math. Notes, 90:3 (2011), 365–372

Citation in format AMSBIB
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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. 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
    2. 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
  • Математические заметки Mathematical Notes
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