Intertwining Operators and Soliton Equations

TL;DR

This paper extends the fermionic approach to the KP hierarchy using intertwining operators, linking integrable hierarchies to Kac-Moody symmetries and deriving various soliton equations through bosonization.

Contribution

It introduces a generalized fermionic framework with intertwining operators for KP hierarchies, connecting them to Kac-Moody symmetries and explicit bosonization methods.

Findings

Derived hierarchies for $sl_N$-symmetries as reductions of multi-component KP.

Explicit bosonization formulas for intertwining operators.

Demonstrated connections between Lie algebra symmetries 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.