
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, 14b Metrologichna Str., Kyiv, 03143 Ukraine
^{b)} Institute of Mathematics, 3 Tereshchenkivs'ka Str., Kyiv4, 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 KostantAdlerSymes 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:
infinitedimensional Lie algebras; soliton equations.
pdf (284 kb)
ps (202 kb)
tex (17 kb)
References
 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, 1321 (in Russian).
 Takhtadzhyan L.A., Faddeev L.D., The Hamiltonian approach in soliton theory, Moscow,
Nauka, 1986 (in Russian).
 Newell A., Solitons in mathematics and physics,
University of Arizona, Society for Industrial and Applied
Mathematics, 1985.
 Flaschka H., Newell A., Ratiu T.,
KacMoody Lie algebras and soliton equations. II. Lax equations
associated with A_{1}^{(1)},
Phys. D, 1983, V.9, 303323.
 Flaschka H., Newell A., Ratiu T., KacMoody Lie algebras and
soliton equations. III. Stationary equations associated with
A_{1}^{(1)}, Phys. D, 1983, V.9, 324332.
 Holod P., Integrable Hamiltonian systems on the orbits of
affine Lie groups and periodical problem for mKdV equation,
Preprint ITF82144R, Kyiv, Institute for Theoretical Physics,
1982 (in Russian).
 Reyman A., SemenovTianShansky M.,
Group theoretical methods in the theory of finitedimensional
integrable systems, VINITI, Current Problems in Mathematics.
Fundamental Directions,
1989, V.6, 145147 (in Russian).
 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, 2025 (in Ukrainian).
 Skrypnyk T., Lie algebras on algebraic curves and
finitedimensional integrable systems,
nlin.SI0010005 (talk on the XXIII
International Colloquium on Group Theoretical Methods in Physics, July 31  August 5, 2000, Dubna, Russia).
 Skrypnyk T., Quasigraded Lie algebras on
hyperelliptic curves and classical integrable systems, J.
Math. Phys., 2001, V.42, 45704581.
 Skrypnyk T., Quasigraded deformations of Lie algebras
and finitedimensional integrable systems, Czechoslovak J.
Phys., 2002, V.52, 12831288.
 Skrypnyk T., Quasigraded Lie algebras and
hierarchies of integrable equations, Czechoslovak J. Phys.,
2003, V.53, 11191124.
 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, 919 (in Russian).
 Holod P., Hamiltonian systems connected with anisotropic affine Lie algebras and higher
LandauLifshits equations, Dokl. Akad. Nauk Ukrain. SSR Ser.
A, 1984, N 5, 58 (in Russian).
 Holod P., The hidden symmetry of the LandauLifshits equation, the hierarchy
of higher equations and a dual equation for an asymmetric chiral
field, Teoret. Mat. Fiz., 1987, V.70, N 1, 1829 (in
Russian).
 Skrypnyk T.,
Doubled generalized LandauLifshitz hierarchies and special
quasigraded Lie algebras, J. Phys. A: Math. Gen., 2004,
V.37, 77557768, nlin.SI/0403046.
 Skrypnyk T., Deformations of loop algebras and integrable systems: hierarchies of
integrable equations, J. Math. Phys., 2004, V.45,
45784595.
 Skrypnyk T., Quasigraded Lie algebras, the
KostantAdler scheme, and integrable hierarchies, Teoret.
Mat. Fiz., 2005, V.142, N 2, 275288 (in Russian).
 Mikhailov A., The reduction in integrable systems.
Groups of reduction, Pis'ma Zh. Eksper. Teoret. Fiz., 1980, V.32, N 1,
187192 (in Russian).
 Kac V., Infinitedimentional Lie algebras, Moscow,
Mir, 1993 (in Russian).
 Drinfel'd V., Sokolov V., Lie algebras and KdVtype
equations, VINITI, Current Problems in Mathematics. Fundamental Directions, 1984, V.24, 81180 (in Russian).
 Shabat A., Yamilov R., To a transformation theory of twodimensional integrable
systems, Phys. Lett. A, 1997, V.277, 1523.
 Leznov A., Saveliev M., Group methods for the integration of nonlinear dynamical systems,
Moscow, Nauka, 1985 (in Russian).
 Krichiver I., Novikov S., Virasorotype algebras, Riemannian surfaces and structures of the
soliton theory, Funktsional. Anal. i Prilozhen., 1987, V.21,
N 2, 4664 (in Russian).
 Cantor I., Persits D., Closed stacks of Poisson
brackets, in Proceedinds of the IX USSR Conference in Geometry,
Kishinev, Shtinitsa, 1988, 141 (in Russian).
 Bolsinov A., Completeness of families
of functions in involution that are connected with compatible
Poisson brackets, Trudy Sem. Vektor. Tenzor. Anal., 1988,
N 23, 1838 (in Russian).
 Borisov A., Zykov S., The chain of the dressing
discrete symmetries and generation of the nonlinear equations,
Teoret. Mat. Fiz., 1998, V.115, 199214 (in Russian).
 de Groot M., Hollowood T.,
Miramontes J., Generalized DrienfieldSokolov hierarchies,
Comm. Math. Phys., 1992, V.145, 5778.
 Skrypnyk T., Integrable deformations of the mKdV and SG
hierarchies and quasigraded Lie algebras, Phys. D, to
appear.

