Banach Poisson–Lie Groups and Bruhat–Poisson Structure of the Restricted Grassmannian

Abstract

In this paper, we construct a (generalized) Banach Poisson-Lie group structure on the unitary restricted Banach Lie group acting transitively on the restricted Grassmannian. A "dual" Banach Lie group consisting of (a class of) upper triangular bounded operators admits also a Poisson-Lie group structure. We show that the restricted Grassmannian inherits a Bruhat-Poisson structure from the unitary Banach Lie group, and that the action of the dual Banach Lie group on it (by "dressing transformations") is a Poisson map. This action generates the KdV hierarchy as explained in [SW85], and its orbits are the Schubert cells of the restricted Grassmannian as described in [PS88].