An Extension of the KdV Hierarchy Arising from a Representation of a Toroidal Lie Algebra

0. Introduction. In this article we show how to construct hierarchies of partial differential equations and their soliton-type solutions from the vertex operator representations of toroidal Lie algebras. Soliton theory was given a new impetus when it was linked with the representation theory of infinite-dimensional Lie algebras in the works of Sato [S], Date-Jimbo-Kashiwara-Miwa [DJKM] and Drinfeld-Sokolov [DS]. It was discovered that for various partial differential equations the space of soliton solutions has a large group of hidden symmetries. Moreover, for every KacMoody algebra one can construct a hierarchy of PDEs whose symmetries form the corresponding Kac-Moody group [KW]. The most famous example occurs in the context of the affine Kac-Moody algebra A (1) 1 which is a central extension of the loop algebra sl2(C[t, t ]). In this case the hierarchy contains the Korteweg-de Vries equation ft = ffx + fxxx