Proceedings of the Steklov Institute of Mathematics

, Volume 263, Issue 1, pp 204–213 | Cite as

Lax operator algebras and integrable hierarchies

  • O. K. SheinmanEmail author


We study applications of a new class of infinite-dimensional Lie algebras, called Lax operator algebras, which goes back to the works by I. Krichever and S. Novikov on finite-zone integration related to holomorphic vector bundles and on Lie algebras on Riemann surfaces. Lax operator algebras are almost graded Lie algebras of current type. They were introduced by I. Krichever and the author as a development of the theory of Lax operators on Riemann surfaces due to I. Krichever, and further investigated in a joint paper by M. Schlichenmaier and the author. In this article we construct integrable hierarchies of Lax equations of that type.


Riemann Surface STEKLOV Institute Central Extension Algebraic Curf Holomorphic Vector Bundle 
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  1. 1.
    H. Garland, “The Arithmetic Theory of Loop Groups,” Publ. Math., Inst. Hautes Étud. Sci. 52, 5–136 (1980).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    V. G. Kac, “Simple Irreducible Graded Lie Algebras of Finite Growth,” Izv. Akad. Nauk SSSR, Ser. Mat. 32(6), 1323–1367 (1968) [Math. USSR, Izv. 2, 1271–1311 (1968)].MathSciNetGoogle Scholar
  3. 3.
    V. G. Kac, Infinite Dimensional Lie Algebras (Cambridge Univ. Press, Cambridge, 1990).zbMATHGoogle Scholar
  4. 4.
    I. M. Krichever, “Vector Bundles and Lax Equations on Algebraic Curves,” Commun. Math. Phys. 229(2), 229–269 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    I. M. Krichever, “Isomonodromy Equations on Algebraic Curves, Canonical Transformations and Whitham Equations,” Moscow Math. J. 2, 717–752 (2002); arXiv: hep-th/0112096.zbMATHMathSciNetGoogle Scholar
  6. 6.
    I. M. Krichever and S. P. Novikov, “Holomorphic Bundles over Algebraic Curves and Non-linear Equations,” Usp. Mat. Nauk 35(6), 47–68 (1980) [Russ. Math. Surv. 35 (6), 53–79 (1980)].zbMATHMathSciNetGoogle Scholar
  7. 7.
    I.M. Krichever and S. P. Novikov, “Holomorphic Bundles over Riemann Surfaces and the Kadomtsev-Petviashvili Equation. I,” Funkts. Anal. Prilozh. 12(4), 41–52 (1978) [Funct. Anal. Appl. 12, 276–286 (1978)].zbMATHMathSciNetGoogle Scholar
  8. 8.
    I. M. Krichever, “Commutative Rings of Ordinary Linear Differential Operators,” Funkts. Anal. Prilozh. 12(3), 20–31 (1978) [Funct. Anal. Appl. 12, 175–185 (1978)].zbMATHMathSciNetGoogle Scholar
  9. 9.
    I. M. Krichever and S. P. Novikov, “Algebras of Virasoro Type, Riemann Surfaces and Structures of the Theory of Solitons,” Funkts. Anal. Prilozh. 21(2), 46–63 (1987) [Funct. Anal. Appl. 21, 126–142 (1987)].MathSciNetGoogle Scholar
  10. 10.
    I. M. Krichever and O. K. Sheinman, “Lax Operator Algebras,” Funkts. Anal. Prilozh. 41(4), 46–59 (2007) [Funct. Anal. Appl. 41, 284–294 (2007)]; arXiv:math.RT/0701648.MathSciNetGoogle Scholar
  11. 11.
    R. V. Moody, “Euclidean Lie Algebras,” Can. J. Math. 21, 1432–1454 (1969).zbMATHMathSciNetGoogle Scholar
  12. 12.
    M. Schlichenmaier, “Local Cocycles and Central Extensions for Multipoint Algebras of Krichever-Novikov Type,” J. Reine Angew. Math. 559, 53–94 (2003).zbMATHMathSciNetGoogle Scholar
  13. 13.
    M. Schlichenmaier, “Higher Genus Affine Algebras of Krichever-Novikov Type,” Moscow Math. J. 3, 1395–1427(2003).zbMATHMathSciNetGoogle Scholar
  14. 14.
    M. Schlichenmaier and O. K. Sheinman, “Central Extensions of Lax Operator Algebras,” arXiv: 0711.4688.Google Scholar
  15. 15.
    O. K. Sheinman, “Affine Krichever-Novikov Algebras, Their Representations and Applications,” in Geometry, Topology, and Mathematical Physics: S.P. Novikov’s Seminar 2002–2003, Ed. by V. M. Buchstaber and I. M. Krichever (Am. Math. Soc., Providence, RI, 2004), AMS Transl., Ser. 2, 212, pp. 297–316; arXiv:math.RT/0304020.Google Scholar
  16. 16.
    O. K. Sheinman, “On Certain Current Algebras Related to Finite-Zone Integration,” in Geometry, Topology, and Mathematical Physics: S.P. Novikov’s Seminar 2006–2007, Ed. by V. M. Buchstaber and I. M. Krichever (Am. Math. Soc., Providence, RI, 2008), AMS Transl., Ser. 2, 224, pp. 271–284.Google Scholar
  17. 17.
    A. N. Tyurin, “Classification of Vector Bundles on an Algebraic Curve of an Arbitrary Genus,” Izv. Akad. Nauk SSSR, Ser. Mat. 29(3), 657–688 (1965).MathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2008

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.Independent University of MoscowMoscowRussia

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