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Semisimple Lie algebras and Hamiltonian theory of finite-dimensional Lax equations with spectral parameter on a Riemann surface

  • O. K. SheinmanEmail author
Article

Abstract

The Hamiltonian property and integrability of the Lax equations with spectral parameter on a Riemann surface are considered. The operators of Lax pairs are meromorphic functions of special form on a Riemann surface of arbitrary positive genus with values in an arbitrary semisimple Lie algebra. The study combines the theory of Lax equations with spectral parameter on a Riemann surface, as proposed by I.M. Krichever in 2001, with a “group-theoretic approach.”

Keywords

Riemann Surface STEKLOV Institute Meromorphic Function Spectral Parameter Central Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov, “Integrability and Seiberg–Witten exact solution,” Phys. Lett. B 355, 466–474 (1995); arXiv: hep-th/9505035.zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    N. Hitchin, “Stable bundles and integrable systems,” Duke Math. J. 54, 91–114 (1987).MathSciNetCrossRefGoogle Scholar
  3. 3.
    I. Krichever, “Vector bundles and Lax equations on algebraic curves,” Commun. Math. Phys. 229, 229–269 (2002).zbMATHCrossRefGoogle Scholar
  4. 4.
    I. M. Krichever and S. P. Novikov, “Holomorphic bundles over algebraic curves and non-linear equations,” Usp. Mat. Nauk 35 (6), 47–68 (1980)zbMATHGoogle Scholar
  5. 4.
    I. M. Krichever and S. P. Novikov, Russ. Math. Surv. 35 (6), 53–79 (1980)].zbMATHCrossRefGoogle Scholar
  6. 5.
    I. M. Krichever and O. K. Sheinman, “Lax operator algebras,” Funkts. Anal. Prilozh. 41 (4), 46–59 (2007)CrossRefGoogle Scholar
  7. 5.
    I. M. Krichever and O. K. Sheinman, Funct. Anal. Appl. 41, 284–294 (2007)]; arXiv:math/0701648 [math.RT].zbMATHMathSciNetCrossRefGoogle Scholar
  8. 6.
    S. P. Novikov and I. A. Taimanov, Modern Geometric Structures and Fields (MTsNMO, Moscow, 2005; Am. Math. Soc., Providence, RI, 2006).Google Scholar
  9. 7.
    A. G. Reiman and M. A. Semenov-Tyan-Shanskii, Integrable Systems: A Group Theoretical Approach (Inst. Kompyut. Issled., Moscow, 2003).Google Scholar
  10. 8.
    M. Schlichenmaier, “Multipoint Lax operator algebras: Almost-graded structure and central extensions,” Mat. Sb. 205 (5), 117–160 (2014)MathSciNetCrossRefGoogle Scholar
  11. 8.
    M. Schlichenmaier, Sb. Math. 205, 722–762 (2014)]; arXiv: 1304.3902 [math.QA].zbMATHMathSciNetCrossRefGoogle Scholar
  12. 9.
    M. Schlichenmaier and O. K. Sheinman, “Central extensions of Lax operator algebras,” Usp. Mat. Nauk 63 (4), 131–172 (2008)CrossRefGoogle Scholar
  13. 9.
    M. Schlichenmaier and O. K. Sheinman, Russ. Math. Surv. 63, 727–766 (2008)]; arXiv: 0711.4688 [math.QA].zbMATHCrossRefGoogle Scholar
  14. 10.
    O. K. Sheinman, Current Algebras on Riemann Surfaces: New Results and Applications (W. de Gruyter, Berlin, 2012), De Gruyter Exp. Math. 58.Google Scholar
  15. 11.
    O. K. Sheinman, “Lax operator algebras of type G2,” Dokl. Akad. Nauk 455 (1), 23–25 (2014)Google Scholar
  16. 11.
    O. K. Sheinman, Dokl. Math. 89 (2), 151–153 (2014)]; arXiv: 1304.2510 [math.RT].zbMATHMathSciNetCrossRefGoogle Scholar
  17. 12.
    O. K. Sheinman, “Lax operator algebras of type G2,” in Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov’s Seminar 2012–2014, Ed. by V. M. Buchstaber, B. A. Dubrovin, and I. M. Krichever (Am. Math. Soc., Providence, RI, 2014)Google Scholar
  18. 12.
    O. K. Sheinman, AMS Transl., Ser. 2, 234Google Scholar
  19. 12.
    O. K. Sheinman, Adv. Math. Sci. 67, pp. 373–392.Google Scholar
  20. 13.
    O. K. Sheinman, “Lax operators algebras and gradings on semisimple Lie algebras,” Dokl. Akad. Nauk 461 (2), 143–145 (2015)Google Scholar
  21. 13.
    O. K. Sheinman, Dokl. Math. 91 (2), 160–162 (2015)].CrossRefGoogle Scholar
  22. 14.
    O. K. Sheinman, “Lax operator algebras and gradings on semi-simple Lie algebras,” Transform. Groups, doi: 10.1007/s00031-015-9340-y (2015); arXiv: 1406.5017 [math.RA].Google Scholar
  23. 15.
    O. K. Sheinman, “Hierarchies of finite-dimensional Lax equations with spectral parameter on a Riemann surface and semisimple Lie algebras,” Teor. Mat. Fiz. (in press).Google Scholar
  24. 16.
    A. N. Tyurin, “Classification of vector bundles over an algebraic curve of arbitrary genus,” Izv. Akad. Nauk SSSR, Ser. Mat. 29 (3), 657–688 (1965)MathSciNetGoogle Scholar
  25. 16.
    A. N. Tyurin, Am. Math. Soc. Transl., Ser. 2, 63, 245–279 (1967)].Google Scholar
  26. 17.
    È. B. Vinberg, V. V. Gorbatsevich, and A. L. Onishchik, Structure of Lie Groups and Lie Algebras (VINITI, Moscow, 1990), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 41; Engl. transl. in Lie Groups and Lie Algebras III (Springer, Berlin, 1994), Encycl. Math. Sci. 41.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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