On the relation between integrability and infinite - dimensional algebras ∗

We review our work on the relation between integrability and infinite-dimensional algebras. We first consider the question of what sets of commuting charges can be constructed from the current of a U(1) Kac-Moody algebra. It emerges that there exists a set S n of such charges for each positive integer n > 1; the corresponding value of the central charge in the Feigin-Fuchs realization of the stress tensor is c = 13−6n−6/n. The charges in each series can be written in terms of the generators of an exceptional W-algebra. We show that the W-algebras that arise in this way are symmetries of Liouville theory for special values of the coupling. We then exhibit a relationship between the non-linear Schrödinger equation and the KP hierarchy. From this it follows that there is a relationship between the non-linear Schrödinger equation and the algebra W 1+∞. These examples provide evidence for our conjecture that the phenomenon of integrability is intimately linked with properties of infinite dimensional algebras.