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

Abstract

The first part of this paper is devoted to the theory of Poisson–Lie groups in the Banach setting. Our starting point is the straightforward adaptation of the notion of Manin triples to the Banach context. The difference with the finite-dimensional case lies in the fact that a duality pairing between two non-reflexive Banach spaces is necessary weak (as opposed to a strong pairing where one Banach space can be identified with the dual space of the other). The notion of generalized Banach Poisson manifolds introduced in this paper is compatible with weak duality pairings between the tangent space and a subspace of the dual. We investigate related notion like Banach Lie bialgebras and Banach Poisson–Lie groups, suitably generalized to the non-reflexive Banach context. The second part of the paper is devoted to the treatment of particular examples of Banach Poisson–Lie groups related to the restricted Grassmannian and the KdV hierarchy. More precisely, we construct a Banach Poisson–Lie group structure on the unitary restricted Banach Lie group which acts transitively on the restricted Grassmannian. A“dual” Banach Lie group consisting of (a class of) upper triangular bounded operators admits also a Banach Poisson–Lie group structure of the same kind. We show that the restricted Grassmannian inherits a generalized Banach Poisson structure from the unitary Banach Lie group, called Bruhat–Poisson structure. Moreover the action of the triangular Banach Poisson–Lie group on it is a Poisson map. This action generates the KdV hierarchy, and its orbits are the Schubert cells of the restricted Grassmannian.