Abstract
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
Mathematics Subject Classification (2010). 17B66, 17B67, 14H10, 14H15, 14H55, 30F30, 81R10, 81T40.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Krichever, I.M. Vector bundles and Lax equations on algebraic curves. Comm. Math. Phys. 229, 229–269 (2002).
Krichever, I.M., Sheinman, O.K. Lax operator algebras. Funct. Anal. i Prilozhen., 41 (2007), no. 4, pp. 46–59. math.RT/0701648.
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.
Hitchin, N.J. Flat connections and geometric quantization. Comm. Math. Phys. 131 (1990), 347–380.
Sheinman, O.K. Lax equations and Knizhnik–Zamolodchikov connection. Arxiv. math.1009.4706.
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.
Schlichenmaier, M., Sheinman, O.K. Central extensions of Lax operator algebras. Russ.Math.Surv., 63, no. 4, pp. 131–172. ArXiv:0711.4688.
Schlichenmaier, M., Sheinman, O.K. Central extensions of Lax operator algebras. The multi-point case. (In progress.)
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.
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.
Kac V.G., Raina A.K. Highest Weight Representations of Infinite Dimensional Lie Algebras. Adv. Ser. in Math. Physics Vol. 2, World Scientific, 1987.
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.
Schlichenmaier, M., Sheinman, O.K. Sugawara construction and Casimir operators for Krichever-Novikov algebras. Jour.of Math. Science 92 (1998), 3807–3834.
Sheinman, O.K. The fermion model of representations of affine Krichever-Novikov algebras. Func. Anal. Appl. 35 (2001), 3, pp. 209–219.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this paper
Cite this paper
Sheinman, O.K. (2013). Lax Equations and the Knizhnik–Zamolodchikov Connection. In: Kielanowski, P., Ali, S., Odzijewicz, A., Schlichenmaier, M., Voronov, T. (eds) Geometric Methods in Physics. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0448-6_37
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0448-6_37
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0447-9
Online ISBN: 978-3-0348-0448-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)