Symmetry algebras of Lagrangian Liouville-type systems

@article{Kiselev2009SymmetryAO,
  title={Symmetry algebras of Lagrangian Liouville-type systems},
  author={A. V. Kiselev and Johan van de Leur},
  journal={Theoretical and Mathematical Physics},
  year={2009},
  volume={162},
  pages={149-162}
}
  • A. KiselevJ. Leur
  • Published 20 February 2009
  • Mathematics
  • Theoretical and Mathematical Physics
We calculate the generators and commutation relations explicitly for higher symmetry algebras of a class of hyperbolic Lagrangian systems of Liouville type, in particular, for two-dimensional Toda chains associated with semisimple complex Lie algebras. 

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References

SHOWING 1-10 OF 29 REFERENCES

Symmetries of nonlinear hyperbolic systems of the Toda chain type

We consider hyperbolic systems of equations that have full sets of integrals along both characteristics. The best known example of models of this type is given by two-dimensional open Toda chains.

On construction of symmetries from integrals of hyperbolic partial differential systems

An algorithm is proposed which allows one to construct higher symmetries of arbitrary order for some special classes of hyperbolic systems possessing integrals. The Pohlmeyer-Lund-Regge system and

Lie algebras and equations of Korteweg-de Vries type

The survey contains a description of the connection between the infinite-dimensional Lie algebras of Kats-Moody and systems of differential equations generalizing the Korteweg-de Vries and

Laplace Invariants of Two-Dimensional Open Toda Lattices

We show that Toda lattices with the Cartan matrices An, Bn, Cn, and Dn are Liouville-type systems. For these systems of equations, we obtain explicit formulas for the invariants and generalized

Exactly integrable hyperbolic equations of Liouville type

This is a survey of the authors' results concerning non-linear hyperbolic equations of Liouville type. The definition is based on the condition that the chain of Laplace invariants of the linearized

Higher symmetries of two-dimensional lattices

Algebraic properties of Gardner’s deformations for integrable systems

We formulate an algebraic definition of Gardner’s deformations for completely integrable bi-Hamiltonian evolutionary systems. The proposed approach extends the class of deformable equations and

Integrable Systems and Metrics of Constant Curvature

. In this article we present a Lagrangian representation for evolutionary systems with a Hamiltonian structure determined by a differential-geometric Poisson bracket of the first order associated with

On the symmetries of evolution equations

GONTENTS § 0. Introduction § 1. Finite dimensionality of the algebra of classical symmetries § 2. Finite-dimensional subalgebras of K3 § 3. Subalgebras of K3 and differential substitutions § 4. Group

Homological Methods in Equations of Mathematical Physics

These lecture notes are a systematic and self-contained exposition of the cohomological theories naturally related to partial differential equations: the Vinogradov C-spectral sequence and the