The Hamiltonian Structure of Soliton Equations and Deformed W-Algebras

TL;DR

This paper unifies the understanding of the Hamiltonian structures of a broad class of integrable soliton equations and their relation to deformed W-algebras, revealing that these structures are deformations of W-algebra Dirac brackets.

Contribution

It provides a comprehensive framework linking the Hamiltonian structures of generalized KdV/mKdV hierarchies to deformed W-algebras, extending previous specific case studies.

Findings

The second Poisson bracket algebra is a deformation of the W-algebra Dirac bracket.

Includes almost all known generalizations of Drinfel'd-Sokolov hierarchies.

Offers a unified description of soliton equations' Hamiltonian structures and W-algebras.

Abstract

The Poisson bracket algebra corresponding to the second Hamiltonian structure of a large class of generalized KdV and mKdV integrable hierarchies is carefully analysed. These algebras are known to have conformal properties, and their relation to $\cal W$-algebras has been previously investigated in some particular cases. The class of equations that is considered includes practically all the generalizations of the Drinfel'd-Sokolov hierarchies constructed in the literature. In particular, it has been recently shown that it includes matrix generalizations of the Gelfand-Dickey and the constrained KP hierarchies. Therefore, our results provide a unified description of the relation between the Hamiltonian structure of soliton equations and $\cal W$-algebras, and it comprises almost all the results formerly obtained by other authors. The main result of this paper is an explicit general equation showing that the second Poisson bracket algebra is a deformation of the Dirac bracket algebra corresponding to the $\cal W$-algebras obtained through Hamiltonian reduction.