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

@article{DeSole2012Classical,
  title={Classical \$\$\{\mathcal\{W\}\}\$\$W -Algebras and Generalized Drinfeld-Sokolov Bi-Hamiltonian Systems Within the Theory of Poisson Vertex Algebras},
  author={Alberto De Sole and Victor G. Kac and Daniele Valeri},
  journal={Communications in Mathematical Physics},
  year={2012},
  volume={323},
  pages={663-711}
}
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, the gauge group action on the phase space is translated in terms of (the exponential of) a Lie conformal algebra action on the space of functions. Following the ideas of Drinfeld and Sokolov, we then establish under certain sufficient conditions the applicability of the Lenard-Magri scheme of… 
View on Springer
arxiv.org
45 Citations
Highly Influential Citations
5
Background Citations
13
Methods Citations
11
Results Citations
1

45 Citations

Classical $${\mathcal{W}}$$W-Algebras in Types A, B, C, D and G

We produce explicit generators of the classical $${\mathcal{W}}$$W-algebras associated with the principal nilpotents in the simple Lie algebras of all classical types and in the exceptional Lie…

Classical Affine $${\mathcal{W}}$$W-Algebras for $${\mathfrak{gl}_N}$$glN and Associated Integrable Hamiltonian Hierarchies

We apply the new method for constructing integrable Hamiltonian hierarchies of Lax type equations developed in our previous paper to show that all $${\mathcal{W}}$$W-algebras…

Classical and Quantum $${\mathcal {W}}$$-Algebras and Applications to Hamiltonian Equations

  • A. Sole
  • Mathematics
    Springer INdAM Series
  • 2019
We start by giving an overview of the four fundamental physical theories, namely classical mechanics, quantum mechanics, classical field theory and quantum field theory, and the corresponding…

Classical Affine W-Superalgebras via Generalized Drinfeld–Sokolov Reductions and Related Integrable Systems

The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel’d–Sokolov (D–S) reduction…

Classical $${\mathcal{W}}$$W-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

We derive explicit formulas for λ-brackets of the affine classical $${\mathcal{W}}$$W -algebras attached to the minimal and short nilpotent elements of any simple Lie algebra $${\mathfrak{g}}$$g .…

Structures of (supersymmetric) classical W-algebras

In the first part of this paper, we discuss the classical W-algebra $\mathcal{W}(\mathfrak{g}, F)$ associated with a Lie superalgebra $\mathfrak{g}$ and the nilpotent element $F$ in an…

Classical W-algebras for centralizers

We introduce a new family of Poisson vertex algebras $\mathcal{W}(\mathfrak{a})$ analogous to the classical $\mathcal{W}$-algebras. The algebra $\mathcal{W}(\mathfrak{a})$ is associated with the…

Supersymmetric Bi-Hamiltonian Systems

We construct super Hamiltonian integrable systems within the theory of supersymmetric Poisson vertex algebras (SUSY PVAs). We provide a powerful tool for the understanding of SUSY PVAs called the…

MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $${\mathcal {W}}$$W-algebras

We give an introduction to the Mathematica packages MasterPVA and MasterPVAmulti used to compute $$\lambda $$λ-brackets in Poisson vertex algebras, which play an important role in the theory of…

Classical $$\mathcal {W}$$-algebras for Centralizers

We introduce a new family of Poisson vertex algebras $\mathcal{W}(\mathfrak{a})$ analogous to the classical $\mathcal{W}$-algebras. The algebra $\mathcal{W}(\mathfrak{a})$ is associated with the…

References

SHOWING 1-10 OF 34 REFERENCES

Poisson vertex algebras in the theory of Hamiltonian equations

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…

Generalized Drinfeld-Sokolov Hierarchies II: The Hamiltonian Structures

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and…

Generalized Drinfel'd-Sokolov hierarchies

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and…

A FRACTIONAL KdV HIERARCHY

We construct a new system of integrable nonlinear differential equations associated with. the operator algebra of Polyakov. Its members are fractional generalizations of KdV type flows corresponding…

Generalized Drinfeld-Sokolov reductions and KdV type hierarchies

Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of…

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…

Generalized Drinfel'd-Sokolov hierarchies

A general approach is adopted to the construction of integrable hierarchies of partial differential equations. A series of hierarchies associated to untwisted Kac-Moody algebras, and conjugacy…

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…

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…

Lie Algebras and Lie Groups

In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is…