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Uspekhi Mat. Nauk, 2011, Volume 66, Issue 1(397), Pages 151–178 (Mi umn9406)  

This article is cited in 1 scientific paper (total in 1 paper)

Lax operator algebras and Hamiltonian integrable hierarchies

O. K. Sheinman

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This paper considers the theory of Lax equations with a spectral parameter on a Riemann surface, proposed by Krichever in 2001. The approach here is based on new objects, the Lax operator algebras, taking into consideration an arbitrary complex simple or reductive classical Lie algebra. For every Lax operator, regarded as a map sending a point of the cotangent bundle on the space of extended Tyurin data to an element of the corresponding Lax operator algebra, a hierarchy of mutually commuting flows given by the Lax equations is constructed, and it is proved that they are Hamiltonian with respect to the Krichever–Phong symplectic structure. The corresponding Hamiltonians give integrable finite-dimensional Hitchin-type systems. For example, elliptic $A_n$, $C_n$, and $D_n$ Calogero–Moser systems are derived in the framework of our approach.
Bibliography: 13 titles.

Keywords: infinite-dimensional Lie algebras, current algebras, Lax integrable systems, Hamiltonian theory.

DOI: https://doi.org/10.4213/rm9406

Full text: PDF file (697 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2011, 66:1, 145–171

Bibliographic databases:

Document Type: Article
UDC: 517.9
MSC: 17B66, 17B67, 14H10, 14H15, 14H55, 30F30, 81R10, 81T40
Received: 09.12.2010

Citation: O. K. Sheinman, “Lax operator algebras and Hamiltonian integrable hierarchies”, Uspekhi Mat. Nauk, 66:1(397) (2011), 151–178; Russian Math. Surveys, 66:1 (2011), 145–171

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. M. Schlichenmaier, “Multipoint Lax operator algebras: almost-graded structure and central extensions”, Sb. Math., 205:5 (2014), 722–762  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Успехи математических наук Russian Mathematical Surveys
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