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

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
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
\Bibitem{Kon11} \by A.~Yu.~Konyaev \paper Uniqueness of Recovering the Parameters of Sectional Operators on Simple Complex Lie Algebras \jour Mat. Zametki \yr 2011 \vol 90 \issue 3 \pages 384--393 \mathnet{http://mi.mathnet.ru/mz6376} \crossref{https://doi.org/10.4213/mzm6376} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2868367} \transl \jour Math. Notes \yr 2011 \vol 90 \issue 3 \pages 365--372 \crossref{https://doi.org/10.1134/S0001434611090057} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000296476500005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-80155213601} 

• http://mi.mathnet.ru/eng/mz6376
• https://doi.org/10.4213/mzm6376
• http://mi.mathnet.ru/eng/mz/v90/i3/p384

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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
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
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