Advertisement

Functional Analysis and Its Applications

, Volume 41, Issue 4, pp 284–294 | Cite as

Lax operator algebras

  • I. M. Krichever
  • O. K. Sheinman
Article

Abstract

In this paper, we develop the general approach, introduced in [l], to Lax operators on algebraic curves. We observe that the space of Lax operators is closed with respect to their usual multiplication as matrix-valued functions. We construct orthogonal and symplectic analogs of Lax operators, prove that they form almost graded Lie algebras, and construct local central extensions of these Lie algebras.

Key words

Lax operator current algebra Tyurin data almost graded structure local central extension 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    I. M. Krichever, “Vector bundles and Lax equations on algebraic curves,” Comm. Math. Phys., 229:2 (2002), 229–269; http://arxiv.org/abs/hep-th/0108110.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    I. M. Krichever, “Isomonodromy equations on algebraic curves, canonical transformations and Witham equations,” Mosc. Math. J., 2:4 (2002), 717–752, 806; http://arxiv.org/abs/hep-th/0112096.MATHMathSciNetGoogle Scholar
  3. [3]
    I. M. Krichever and S. P. Novikov, “Algebras ofVirasoro type, Riemann surfaces and structures of the theory of solitons,” Funkts. Anal. Prilozhen., 21:2 (1987), 46–63.MathSciNetGoogle Scholar
  4. [4]
    I. M. Krichever and S. P. Novikov, “Virasoro type algebras, Riemann surfaces and strings in Minkowski space,” Funkts. Anal. Prilozhen., 21:4 (1987), 47–61.MathSciNetGoogle Scholar
  5. [5]
    I. M. Krichever and S. P. Novikov, “Algebras of Virasoro type, energy-momentum tensors and decompositions ofoperators on Riemann surfaces,” Funkts. Anal. Prilozhen., 23:1 (1989), 46–63.MathSciNetCrossRefGoogle Scholar
  6. [6]
    I. M. Krichever and S. P. Novikov, “Holomorphic bundles and commuting difference operators. Two-point constructions,” Uspekhi Mat. Nauk, 55:4 (2000), 181–182.MathSciNetGoogle Scholar
  7. [7]
    I. M. Krichever and S. P. Novikov, “Holomorphic bundles on Riemann surfaces and Kadomtsev-Petviashvili equation (KP). I,” Funkts. Anal. Prilozhen., 12:4 (1978), 41–52.MATHMathSciNetGoogle Scholar
  8. [8]
    I. M. Krichever and S. P. Novikov, “Holomorphic bundles on algebraic curves and nonlinear equations,” Uspekhi Math. Nauk, 35:6 (1980), 47–68.MATHMathSciNetGoogle Scholar
  9. [9]
    M. Schlichenmaier, “Local cocycles and central extensions for multi-point algebras of Krichever-Novikov type,” J. Reine Angew. Math., 559 (2003), 53–94.MATHMathSciNetGoogle Scholar
  10. [10]
    M. Schlichenmaier, “Higher genus affine algebras of Krichever-Novikov type,” Moscow Math. J., 3:4 (2003), 1395–1427; http://arxiv.org/abs/math/0210360.MATHMathSciNetGoogle Scholar
  11. [11]
    O. K. Sheinman, “Affine Krichever-Novikov algebras, their representations and applications,” in: Geometry, Topology and Mathematical Physics. S. P. Novikov’s Seminar 2002–2003, Amer. Soc. Transl. (2), vol. 212 (eds. V. M. Buchstaber, I. M. Krichever), Amer. Math. Soc., Providence, R.I., 2004, 297–316; http://arxiv.org/abs/Math.RT/0304020.Google Scholar
  12. [12]
    A. N. Tyurin, “The classification of vector bundles over an algebraic curve of arbitrary genus,” Izv. Akad. Nauk SSSR Ser. Mat., 29 (1965), 657–688.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsUSA
  2. 2.Columbia University
  3. 3.Steklov Mathematical InstituteRussia
  4. 4.Independent University of MoscowMoscow

Personalised recommendations