  We investigate the algebraic structure of integrable hierarchies that, we
propose, underlie models of $W$-gravity coupled to matter. More precisely, we
concentrate on the dispersionless limit of the topological subclass of such
theories, by making use of a correspondence between Drinfeld-Sokolov systems,
principal $s\ell(2)$ embeddings and certain chiral rings. We find that the
integrable hierarchies can be viewed as generalizations of the usual matrix
Drinfeld-Sokolov systems to higher fundamental representations of $s\ell(n)$.
The underlying Heisenberg algebras have an intimate connection with the quantum
cohomology of grassmannians. The Lax operators are directly given in terms of
multi-field superpotentials of the associated topological LG theories. We view
our construction as a prototype for a multi-variable system and suspect that it
might be useful also for a class of related problems.
