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Uspekhi Mat. Nauk, 2016, Volume 71, Issue 1(427), Pages 117–168 (Mi umn9703)  

This article is cited in 5 scientific papers (total in 5 papers)

Lax operator algebras and integrable systems

O. K. Sheinman

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero–Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac–Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann–Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems.
Bibliography: 51 titles.

Keywords: gradings of semisimple Lie algebras, Lax operator algebras, integrable systems, spectral parameter on a Riemann surface, Tyurin parameters, Hamiltonian theory, prequantization.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


DOI: https://doi.org/10.4213/rm9703

Full text: PDF file (1056 kB)
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English version:
Russian Mathematical Surveys, 2016, 71:1, 109–156

Bibliographic databases:

ArXiv: 1602.04320
Document Type: Article
UDC: 512.554.3
MSC: 17B66, 17B67, 14H10, 14H15, 14H55, 30F30, 81R10, 81T40
Received: 14.01.2016

Citation: O. K. Sheinman, “Lax operator algebras and integrable systems”, Uspekhi Mat. Nauk, 71:1(427) (2016), 117–168; Russian Math. Surveys, 71:1 (2016), 109–156

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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. O. K. Sheinman, “Certain reductions of Hitchin systems of rank 2 and genera 2 and 3”, Dokl. Math., 97:2 (2018), 144–146  mathnet  mathnet  crossref  crossref  zmath  isi  elib  scopus
    2. Oleg K. Sheinman, “Global current algebras and localization on Riemann surfaces”, Mosc. Math. J., 15:4 (2015), 833–846  mathnet  mathscinet  zmath  elib
    3. V. M. Buchstaber, “Polynomial dynamical systems and the Korteweg–de Vries equation”, Proc. Steklov Inst. Math., 294 (2016), 176–200  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. O. K. Sheinman, “Matrix divisors on Riemann surfaces and Lax operator algebras”, Trans. Moscow Math. Soc., 78 (2017), 109–121  mathnet  crossref  elib
    5. Elena Yu. Bunkova, “Hirzebruch functional equation: classification of solutions”, Proc. Steklov Inst. Math., 302 (2018), 33–47  mathnet  crossref  crossref  isi  elib
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