Intertwining Operators and Soliton Equations

Abstract

In this paper we generalize the fermionic approach to the KP hierarchy sudgested in the papers of Kyoto school 1981-1984 (Sato,Jimbo, Miwa...). The main idea is that the components of the intertwiningoperators are in some sense a generalization of free fermions for $gl_{\infty}$. We formulate in terms of intertwining operators the integrable hierarchies related to Kac-Moody Lie algebra symmetries. We write down explicitly the bosonization of these operators for different choices of Heisenberg subalgebras. These different realizations lead to different hierarchies of soliton equations. For example, for $sl_N$-symmetries we get hierarchies obtained as $(n_1,..., n_s)$-reduction from $s$-component KP $(n_1+...+n_s = N)$ introduced by V.Kac and J.Van de Leur.