AFFINE LIE ALGEBRAS AND PSEUDO-DIFFERENTIAL OPERATORS OF INFINITE ORDER

In this paper we make a careful study of some connection between pseudo-differential operators, Kadomtsev-Petviashvili (KP) hierarchy and tau functions based on the Sato-Date-Jimbo-Miwa-Kashiwara theory. A few other connections and ideas concerning the Korteweg-de Vries (KdV) and Boussinesq equations, the Gelfand-Dickey flows, the Heisenberg and Virasoro algebras are given. The study of the KP and KdV hierarchies, the use of tau functions related to infinite dimensional Grassmannians, vertex operators and the Hirota's bilinear formalism led to obtaining remarkable properties concerning these algebras as for example the existence of an infinite family of first integrals functionally independent and in involution. The paper is supported by an appendix which contains some information about coadjoint orbits in Kac-Moody algebras and a proof of the Adler-Kostant-Symes theorem.