17 research outputs found
Weak Poisson structures on infinite dimensional manifolds and hamiltonian actions
We introduce a notion of a weak Poisson structure on a manifold modeled
on a locally convex space. This is done by specifying a Poisson bracket on a
subalgebra \cA \subeq C^\infty(M) which has to satisfy a non-degeneracy
condition (the differentials of elements of \cA separate tangent vectors) and
we postulate the existence of smooth Hamiltonian vector fields. Motivated by
applications to Hamiltonian actions, we focus on affine Poisson spaces which
include in particular the linear and affine Poisson structures on duals of
locally convex Lie algebras. As an interesting byproduct of our approach, we
can associate to an invariant symmetric bilinear form on a Lie algebra
\g and a -skew-symmetric derivation a weak affine Poisson
structure on \g itself. This leads naturally to a concept of a Hamiltonian
-action on a weak Poisson manifold with a \g-valued momentum map and hence
to a generalization of quasi-hamiltonian group actions
Geometric representation theory for unitary groups of operator algebras
AbstractGeometric realizations for the restrictions of GNS representations to unitary groups of C∗-algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach–Lie groups is described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed
The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits
AbstractWe investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie–Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach Lie–Poisson spaces, and the underlying Banach space can be chosen to be even a Hilbert space. Smoothness of numerous adjoint and coadjoint orbits of the restricted unitary group is also established. Several pathological properties of the restricted algebra are pointed out