THE HAMILTONIAN-STRUCTURE OF SOLITON-EQUATIONS AND DEFORMED W-ALGEBRAS

The Poisson bracket algebra corresponding to the second Hamiltonian st ructure of a large class of generalized KdV and mKdV integrable hierar chies is carefully analysed. These algebras are known to have conforma l properties and their relation to W-algebras has been previously inve stigated in some particular cases. The class of equations that is cons idered includes practically all the generalizations of the Drinfel'd-S okolov hierarchies constructed in the literature. In particular, it ha s been recently shown that it includes matrix generalizations of the G elfand-Dickey and the constrained KP hierarchies. Therefore, our resul ts provide a unified description of the relation between the Hamiltoni an structure of soliton equations and W-algebras, and it comprises alm ost all the results formerly obtained by other authors. The main resul t of this paper is an explicit general equation showing that the secon d Poisson bracket algebra is a deformation of the Dirac bracket algebr a corresponding to the W-algebras obtained through Hamiltonian reducti on. (C) 1997 Academic Press