Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras

TL;DR

This paper explores the construction of classical W-algebras through Drinfeld-Sokolov Hamiltonian reduction within Poisson vertex algebras, establishing conditions for integrability and bi-Hamiltonian hierarchies.

Contribution

It provides a new framework for understanding classical W-algebras using Poisson vertex algebras and extends the applicability of the Lenard-Magri scheme for integrability.

Findings

Established conditions for the applicability of the Lenard-Magri scheme.

Constructed integrable hierarchies of bi-Hamiltonian equations.

Linked gauge group actions to Lie conformal algebra actions.

Abstract

We provide a description of the Drinfeld-Sokolov Hamiltonian reduction for the construction of classical 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 integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations.