Lax operator algebras and integrable systems
TLDR
In this article, a new class of infinite-dimensional Lie algebras, called Lax operator algesas, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface.Abstract:
A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero-Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac-Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann-Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. Bibliography: 51 titles.read more
Citations
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Polynomial Dynamical Systems and the Korteweg de Vries Equation
TL;DR: In this paper, a polynomial Lie algebra of the fields L>>\s k>>\s and the structure of the ring C[X,Z] as a graded module with two generators x>>\s 2 and z>>\s 4 over this algebra is presented.
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Polynomial dynamical systems and Korteweg--de Vries equation
TL;DR: In this paper, a polynomial Lie algebra of vector fields was constructed for the complex linear space with coordinates of the fields and the structure of polynomials in the ring was described.
Krichever{Novikov Type Algebras. An Introduction
TL;DR: The Krichever-Novikov type algebras as discussed by the authors are generalizations of the Witt, Virasoro, affine and affine Lie algesbras and their relatives to Riemann surfaces of ar- bitrary genus.
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Spectral Curves of the Hyperelliptic Hitchin Systems
TL;DR: In this article, the Hitchin systems of types Al, Bl, and Cl on hyperelliptic curves are described and the current state of the problem in the case of type Dl is described.
References
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Book
Mathematical Methods of Classical Mechanics
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
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Lie algebras and equations of Korteweg-de Vries type
TL;DR: A survey of the theory of Kats-Moody algebras is given in this paper, which contains a description of the connection between the infinite-dimensional Lie algebra of Kats and systems of differential equations generalizing the Korteweg-de Vries and sine-Gordon equations and integrable by the inverse scattering problem.
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Stable bundles and integrable systems
TL;DR: In this article, a geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur a surface de Riemann is considered.
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Classical integrable finite-dimensional systems related to Lie algebras
TL;DR: In this article, a review of the results obtained in this field during the last few years is presented, and some new results both of physical and mathematical interest are also presented, as well as some generalizations of these models, naturally suggested by their association with Lie algebras.