Lax Equations and the Knizhnik–Zamolodchikov Connection

  • Oleg K. SheinmanEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)


Given a Lax system of equations with the spectral parameter on a Riemann surface we construct a projective unitary representation of the Lie algebra of Hamiltonian vector fields by Knizhnik–Zamolodchikov operators. This provides a prequantization of the Lax system. The representation operators of Poisson commutingH amiltonians of the Lax system projectively commute. If Hamiltonians depend only on the action variables then the correspondingop erators commute


Current algebra Lax operator algebra Lax integrable system Knizhnik-Zamolodchikov connection 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Krichever, I.M. Vector bundles and Lax equations on algebraic curves. Comm. Math. Phys. 229, 229–269 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Krichever, I.M., Sheinman, O.K. Lax operator algebras. Funct. Anal. i Prilozhen., 41 (2007), no. 4, pp. 46–59. math.RT/0701648.Google Scholar
  3. 3.
    Sheinman, O.K. Lax operator algebras and Hamiltonian integrable hierarchies. Arxiv.math.0910.4173. Russ. Math. Surv., 2011, no. 1, 151–178. Lax operator algebras and integrable hierarchies. In: Proc. of the Steklov Mathematical Institute, 2008, v. 263.Google Scholar
  4. 4.
    Hitchin, N.J. Flat connections and geometric quantization. Comm. Math. Phys. 131 (1990), 347–380.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Sheinman, O.K. Lax equations and Knizhnik–Zamolodchikov connection. Arxiv. math.1009.4706.Google Scholar
  6. 6.
    Krichever I.M., Novikov S.P. Holomorphic bundles on algebraic curves and nonlinear equations. Uspekhi Math. Nauk (Russ. Math. Surv), 35 (1980), 6, 47–68.Google Scholar
  7. 7.
    Schlichenmaier, M., Sheinman, O.K. Central extensions of Lax operator algebras. Russ.Math.Surv., 63, no. 4, pp. 131–172. ArXiv:0711.4688.Google Scholar
  8. 8.
    Schlichenmaier, M., Sheinman, O.K. Central extensions of Lax operator algebras. The multi-point case. (In progress.)Google Scholar
  9. 9.
    Schlichenmaier, M., Sheinman, O.K. Wess-Zumino-Witten-Novikov Theory, Knizhnik-Zamolodchikov equations and Krichever-Novikov algebras. Russ. Math. Surv., 1999, v. 54, no. 1, pp. 213–249. Knizhnik-Zamolodchikov equations for positive genus and Krichever-Novikov algebras. Russ. Math. Surv., 2004, v. 59, no. 4, pp. 737–770.Google Scholar
  10. 10.
    Sheinman, O.K. Krichever-Novikov algebras, their representations and applications in geometry and mathematical physics. In: Contemporary mathematical problems, Steklov Mathematical Institute publications, v. 10 (2007), 142 p. (in Russian). Krichever-Novikov algebras, their representations and applications. In: Geometry, Topology and Mathematical Physics. S.P. Novikov’s Seminar 2002–2003, V.M. Buchstaber, I.M. Krichever, eds., AMS Translations, Ser. 2, v. 212 (2004), 297–316, math.RT/0304020.Google Scholar
  11. 11.
    Kac V.G., Raina A.K. Highest Weight Representations of Infinite Dimensional Lie Algebras. Adv. Ser. in Math. Physics Vol. 2, World Scientific, 1987.Google Scholar
  12. 12.
    Krichever, I.M. Novikov, S.P. Algebras of Virasoro type, Riemann surfaces and strings in Minkowski space. Funct. Anal. Appl. 21 (1987), 4, 294–307.Google Scholar
  13. 13.
    Schlichenmaier, M., Sheinman, O.K. Sugawara construction and Casimir operators for Krichever-Novikov algebras. Jour.of Math. Science 92 (1998), 3807–3834.Google Scholar
  14. 14.
    Sheinman, O.K. The fermion model of representations of affine Krichever-Novikov algebras. Func. Anal. Appl. 35 (2001), 3, pp. 209–219.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

Personalised recommendations