Intertwining Operators and Soliton Equations

M. I. Golenishcheva-Kutuzova ; D. R. Lebedev

Funkcionalʹnyj analiz i ego priloženiâ, Tome 33 (1999) no. 4, pp. 1-24

Résumé

The fermionic approach to the Kadomtsev–Petviashvili hierarchy, suggested by the Kyoto school (Sato, Date, Jimbo, Kashiwara, and Miwa) in 1981–4, is generalized on the basis of the idea that, in a sense, the components of intertwining operators are a generalization of free fermions for $gl_\infty$. Integrable hierarchies related to symmetries of Kac–Moody algebras are described in terms of intertwining operators. The bosonization of these operators for various choices of the Heisenberg subalgebra is explicitly written out. These various realizations result in distinct hierarchies of soliton equations. For example, for $sl_N$-symmetries this gives the hierarchies obtained by the $(n_1,\dots,n_s)$-reduction from the $s$-component KP hierarchy introduced by Kac and van de Leur.