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Mat. Sb., 2014, Volume 205, Number 5, Pages 117–160 (Mi msb8258)  

This article is cited in 6 scientific papers (total in 6 papers)

Multipoint Lax operator algebras: almost-graded structure and central extensions

M. Schlichenmaier

University of Luxembourg

Abstract: Recently, Lax operator algebras appeared as a new class of higher genus current-type algebras. Introduced by Krichever and Sheinman, they were based on Krichever's theory of Lax operators on algebraic curves. These algebras are almost-graded Lie algebras of currents on Riemann surfaces with marked points (in-points, out-points and Tyurin points). In a previous joint article with Sheinman, the author classified the local cocycles and associated almost-graded central extensions in the case of one in-point and one out-point. It was shown that the almost-graded extension is essentially unique. In this article the general case of Lax operator algebras corresponding to several in- and out-points is considered. As a first step they are shown to be almost-graded. The grading is given by splitting the marked points which are non-Tyurin points into in- and out-points. Next, classification results both for local and bounded cocycles are proved. The uniqueness theorem for almost-graded central extensions follows. To obtain this generalization additional techniques are needed which are presented in this article.
Bibliography: 30 titles.

Keywords: infinite-dimensional Lie algebras, current algebras, Krichever-Novikov type algebras, central extensions, Lie algebra cohomology, integrable systems.


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English version:
Sbornik: Mathematics, 2014, 205:5, 722–762

Bibliographic databases:

UDC: 512.554.32
MSC: 17B65, 17B67, 17B80, 14H55, 14H70, 30F30, 81R10, 81T40
Received: 11.06.2013

Citation: M. Schlichenmaier, “Multipoint Lax operator algebras: almost-graded structure and central extensions”, Mat. Sb., 205:5 (2014), 117–160; Sb. Math., 205:5 (2014), 722–762

Citation in format AMSBIB
\by M.~Schlichenmaier
\paper Multipoint Lax operator algebras: almost-graded structure and central extensions
\jour Mat. Sb.
\yr 2014
\vol 205
\issue 5
\pages 117--160
\jour Sb. Math.
\yr 2014
\vol 205
\issue 5
\pages 722--762

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    This publication is cited in the following articles:
    1. O. K. Sheinman, “Lax operators algebras and gradings on semisimple Lie algebras”, Dokl. Math., 91:2 (2015), 160–162  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. O. K. Sheinman, “Semisimple Lie algebras and Hamiltonian theory of finite-dimensional Lax equations with spectral parameter on a Riemann surface”, Proc. Steklov Inst. Math., 290:1 (2015), 178–188  mathnet  crossref  crossref  isi  elib  elib
    3. O. K. Sheinman, “Hierarchies of finite-dimensional Lax equations with a spectral parameter on a Riemann surface and semisimple Lie algebras”, Theoret. and Math. Phys., 185:3 (2015), 1816–1831  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. Oleg K. Sheinman, “Global current algebras and localization on Riemann surfaces”, Mosc. Math. J., 15:4 (2015), 833–846  mathnet  crossref  mathscinet  zmath  elib
    5. O. K. Sheinman, “Lax operator algebras and integrable systems”, Russian Math. Surveys, 71:1 (2016), 109–156  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. O. K. Sheinman, “Lax operator algebras and gradings on semisimple Lie algebras”, Transform. Groups, 21:1 (2016), 181–196  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
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