The second Gelfand-Dickey bracket as a bracket on a Poisson-Lie Grassmannian

We introduce a Poisson structure on a Grassmannian Gr _k (V) on which the Poisson-Lie group GL(V) acts in a Poisson-Lie way. We discuss the analytic complications connected with the infinite-dimensional caseV=C () and show that an open subset of Gr _k (V) with this Poisson structure is isomorphic to the Gelfand-Dickey manifold of differential operators of orderk with the second Gelfand-Dickey bracket. In fact we introduce as a consequence a Poisson-Lie action of an enormous group on the Gelfand-Dickey manifold generalizing (on the semiclassical level) the Sugavara inclusion.Dedicated to Israel M. Gelfand at his80 ^th birthday