Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 043, 14 pages      nlin.SI/0604032      http://dx.doi.org/10.3842/SIGMA.2006.043

Quasigraded Lie Algebras and Modified Toda Field Equations

Taras V. Skrypnyk a, b
a) Bogolyubov Institute for Theoretical Physics, 14-b Metrologichna Str., Kyiv, 03143 Ukraine
b) Institute of Mathematics, 3 Tereshchenkivs'ka Str., Kyiv-4, 01601 Ukraine

Received October 31, 2005, in final form March 03, 2006; Published online April 16, 2006

Abstract
We construct a family of quasigraded Lie algebras that coincide with the deformations of the loop algebras in "principal" gradation and admit Kostant-Adler-Symes scheme. Using them we obtain new Volterra coupled systems and modified Toda field equations for all series of classical matrix Lie algebras g.

Key words: infinite-dimensional Lie algebras; soliton equations.

pdf (284 kb)   ps (202 kb)   tex (17 kb)

References

  1. Zaharov V., Shabat A., Integration of nonlinear equations of mathematical physics using the method of inverse scattering problem. II, Funktsional. Anal. i Prilozhen., 1979, V.13, N 3, 13-21 (in Russian).
  2. Takhtadzhyan L.A., Faddeev L.D., The Hamiltonian approach in soliton theory, Moscow, Nauka, 1986 (in Russian).
  3. Newell A., Solitons in mathematics and physics, University of Arizona, Society for Industrial and Applied Mathematics, 1985.
  4. Flaschka H., Newell A., Ratiu T., Kac-Moody Lie algebras and soliton equations. II. Lax equations associated with A1(1), Phys. D, 1983, V.9, 303-323.
  5. Flaschka H., Newell A., Ratiu T., Kac-Moody Lie algebras and soliton equations. III. Stationary equations associated with A1(1), Phys. D, 1983, V.9, 324-332.
  6. Holod P., Integrable Hamiltonian systems on the orbits of affine Lie groups and periodical problem for mKdV equation, Preprint ITF-82-144R, Kyiv, Institute for Theoretical Physics, 1982 (in Russian).
  7. Reyman A., Semenov-Tian-Shansky M., Group theoretical methods in the theory of finite-dimensional integrable systems, VINITI, Current Problems in Mathematics. Fundamental Directions, 1989, V.6, 145-147 (in Russian).
  8. Holod P., Skrypnyk T., Anisotropic quasigraded Lie algebras on the algebraic curves and integrable Hamiltonian systems, Naukovi Zapysky NAUKMA, Ser. Phys.-Math. Sciences, 2000, V.18, 20-25 (in Ukrainian).
  9. Skrypnyk T., Lie algebras on algebraic curves and finite-dimensional integrable systems, nlin.SI-0010005 (talk on the XXIII International Colloquium on Group Theoretical Methods in Physics, July 31 - August 5, 2000, Dubna, Russia).
  10. Skrypnyk T., Quasigraded Lie algebras on hyperelliptic curves and classical integrable systems, J. Math. Phys., 2001, V.42, 4570-4581.
  11. Skrypnyk T., Quasigraded deformations of Lie algebras and finite-dimensional integrable systems, Czechoslovak J. Phys., 2002, V.52, 1283-1288.
  12. Skrypnyk T., Quasigraded Lie algebras and hierarchies of integrable equations, Czechoslovak J. Phys., 2003, V.53, 1119-1124.
  13. Golubchik I., Sokolov V., Compatible Lie brackets and integrable equations of the principle chiral model type, Funktsional. Anal. i Prilozhen., 2002, V.36, N 3, 9-19 (in Russian).
  14. Holod P., Hamiltonian systems connected with anisotropic affine Lie algebras and higher Landau-Lifshits equations, Dokl. Akad. Nauk Ukrain. SSR Ser. A, 1984, N 5, 5-8 (in Russian).
  15. Holod P., The hidden symmetry of the Landau-Lifshits equation, the hierarchy of higher equations and a dual equation for an asymmetric chiral field, Teoret. Mat. Fiz., 1987, V.70, N 1, 18-29 (in Russian).
  16. Skrypnyk T., Doubled generalized Landau-Lifshitz hierarchies and special quasigraded Lie algebras, J. Phys. A: Math. Gen., 2004, V.37, 7755-7768, nlin.SI/0403046.
  17. Skrypnyk T., Deformations of loop algebras and integrable systems: hierarchies of integrable equations, J. Math. Phys., 2004, V.45, 4578-4595.
  18. Skrypnyk T., Quasigraded Lie algebras, the Kostant-Adler scheme, and integrable hierarchies, Teoret. Mat. Fiz., 2005, V.142, N 2, 275-288 (in Russian).
  19. Mikhailov A., The reduction in integrable systems. Groups of reduction, Pis'ma Zh. Eksper. Teoret. Fiz., 1980, V.32, N 1, 187-192 (in Russian).
  20. Kac V., Infinite-dimentional Lie algebras, Moscow, Mir, 1993 (in Russian).
  21. Drinfel'd V., Sokolov V., Lie algebras and KdV-type equations, VINITI, Current Problems in Mathematics. Fundamental Directions, 1984, V.24, 81-180 (in Russian).
  22. Shabat A., Yamilov R., To a transformation theory of two-dimensional integrable systems, Phys. Lett. A, 1997, V.277, 15-23.
  23. Leznov A., Saveliev M., Group methods for the integration of nonlinear dynamical systems, Moscow, Nauka, 1985 (in Russian).
  24. Krichiver I., Novikov S., Virasoro-type algebras, Riemannian surfaces and structures of the soliton theory, Funktsional. Anal. i Prilozhen., 1987, V.21, N 2, 46-64 (in Russian).
  25. Cantor I., Persits D., Closed stacks of Poisson brackets, in Proceedinds of the IX USSR Conference in Geometry, Kishinev, Shtinitsa, 1988, 141 (in Russian).
  26. Bolsinov A., Completeness of families of functions in involution that are connected with compatible Poisson brackets, Trudy Sem. Vektor. Tenzor. Anal., 1988, N 23, 18-38 (in Russian).
  27. Borisov A., Zykov S., The chain of the dressing discrete symmetries and generation of the non-linear equations, Teoret. Mat. Fiz., 1998, V.115, 199-214 (in Russian).
  28. de Groot M., Hollowood T., Miramontes J., Generalized Drienfield-Sokolov hierarchies, Comm. Math. Phys., 1992, V.145, 57-78.
  29. Skrypnyk T., Integrable deformations of the mKdV and SG hierarchies and quasigraded Lie algebras, Phys. D, to appear.

Previous article   Next article   Contents of Volume 2 (2006)