Poisson vertex algebras in the theory of Hamiltonian equations

@article{Barakat2009PoissonVA,
  title={Poisson vertex algebras in the theory of Hamiltonian equations},
  author={Ali Barakat and Alberto De Sole and Victor G. Kac},
  journal={Japanese Journal of Mathematics},
  year={2009},
  volume={4},
  pages={141-252}
}
Abstract.We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme… 

The variational Poisson cohomology

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st

Computing with Hamiltonian operators

  • R. Vitolo
  • Mathematics
    Comput. Phys. Commun.
  • 2019

Representations of Lie Algebras and Partial Differential Equations

This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial

On Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

The theory of Poisson vertex algebras (PVAs) (Barakat et al. in Jpn J Math 4(2):141–252, 2009) is a good framework to treat Hamiltonian partial differential equations. A PVA consists of a pair

Integrable Hamiltonian Hierarchies for Classical Lie Algebras

We prove that any classical affineW -algebraW(g, f ), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrableHamiltonian hierarchy of Lax type

Cohomology of the Schrödinger–Virasoro conformal algebra

Lie conformal algebra, which was introduced by Kac in [6, 7], gives an axiomatic description of the operator product expansion (or rather its Fourier transform) of chiral fields in conformal field

Classical Affine W -Algebras and the Associated Integrable Hamiltonian Hierarchies for Classical Lie Algebras

We prove that any classical affine W -algebra W(g, f), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrable Hamiltonian hierarchy of Lax type

Classical $${\mathcal{W}}$$W -Algebras and Generalized Drinfeld-Sokolov Bi-Hamiltonian Systems Within the Theory of Poisson Vertex Algebras

We describe of the generalized Drinfeld-Sokolov Hamiltonian reduction for the construction of classical $${\mathcal{W}}$$W -algebras within the framework of Poisson vertex algebras. In this context,

Lie Conformal Algebra Cohomology and the Variational Complex

We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give

Higher-Order Dispersive Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

  • M. Casati
  • Mathematics
    Theoretical and Mathematical Physics
  • 2018
The theory of multidimensional Poisson vertex algebras provides a completely algebraic formalism for studying the Hamiltonian structure of partial differential equations for any number of dependent
...

References

SHOWING 1-10 OF 23 REFERENCES

Cohomology of Conformal Algebras

Abstract:The notion of a conformal algebra encodes an axiomatic description of the operator product expansion of chiral fields in conformal field theory. On the other hand, it is an adequate tool for

Finite vs affine W-algebras

Abstract.In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we

Lie Conformal Algebra Cohomology and the Variational Complex

We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give

A Simple model of the integrable Hamiltonian equation

A method of analysis of the infinite‐dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested. This analysis is based

Soliton Equations and Hamiltonian Systems

Integrable Systems Generated by Linear Differential nth Order Operators Hamiltonian Structures Hamiltonian Structure of the GD Hierarchies Modified KdV and GD. The Kupershmidt-Wilson Theorem The KP

Applications of lie groups to differential equations

1 Introduction to Lie Groups.- 1.1. Manifolds.- Change of Coordinates.- Maps Between Manifolds.- The Maximal Rank Condition.- Submanifolds.- Regular Submanifolds.- Implicit Submanifolds.- Curves and

Vertex algebras for beginners

Preface. 1: Wightman axioms and vertex algebras. 1.1: Wightman axioms of a QFT. 1.2: d = 2 QFT and chiral algebras. 1.3: Definition of a vertex algebra. 1.4: Holomorphic vertex algebras. 2: Calculus

Dirac Structures and Integrability of Nonlinear Evolution Equations

Algebraic theory of Dirac structures Nijenhuis operators and pairs of Dirac structures the complex of formal variational calculus local Hamiltonian operators local symplectic operators and evolution

From Poisson algebras to Gerstenhaber algebras

© Annales de l’institut Fourier, 1996, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions