THE ALGEBRA OF Q-PSEUDODIFFERENTIAL SYMBOLS AND THE Q-W-KP((N)) ALGEBRA

In this paper we continue with the program to explore the topography o f the space of W-type algebras. In the present case, the starting poin t is the work of Khesin, Lyubashenko, and Roger on the algebra of q-de formed pseudodifferential symbols and their associated integrable hier archies. The analysis goes on by studying the associated Hamiltonian s tructures for which compact expressions are found. The fundamental Poi sson brackets yield q-deformations of W-KP and related W-type algebras which, in specific cases, coincide with the ones constructed by Frenk el and Reshetikhin. The construction underlies a continuous correspond ence between the Hamiltonian structures of the Toda lattice and the KP hierarchies. (C) 1996 American Institute of Physics.