Seminarium KMMF "Teoria Dwoistości"

We will consider particular examples of Poisson manifolds, namely Banach Poisson-Lie groups related to the Korteweg-de-Vries hierarchy.We construct a Banach Poisson-Lie group structure on the unitary restricted group, as well as on a Banach Lie group consisting of (a class of) upper triangular bounded operators. We show that the restricted Grassmannian inherite a Bruhat-Poisson structure from the unitary restricted group, and that the action of the triangular Banach Lie group on it by ``dressing transformations'' is a Poisson map. This action generates the KdV hierarchy, and its orbits are the Schubert cells of the restricted Grassmannian.