Weak Poisson structures on infinite dimensional manifolds and hamiltonian actions
Article in Edited Volumes
(Book chapter)
Publication Details
Author(s): Neeb KH, Thiemann T, Sahlmann H
Editor(s): V. Dobrev
Title edited volumes: Springer Proceedings in Mathematics & Statistics
Publisher: Springer Japan
Publication year: 2015
Volume: 111
Pages range: 105136
ISBN: 9784431552840
Abstract
We introduce a notion of a weak Poisson structure on a manifold M modeled on a locally convex space. This is done by specifying a Poisson bracket on a subalgebra A⊆C∞(M) which has to satisfy a nondegeneracy condition (the differentials of elements of A 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 κ on a Lie algebra g and a κskewsymmetric derivation D a weak affine Poisson structure on g itself. This leads naturally to a concept of a Hamiltonian Gaction on a weak Poisson manifold with a g
valued momentum map and hence to a generalization of quasihamiltonian group actions.
FAU Authors / FAU Editors
 Neeb, KarlHermann Prof. Dr. 
  
 Sahlmann, Hanno Prof. Dr. 
  Professur für Theoretische Physik 

 Thiemann, Thomas Prof. Dr. 
  Lehrstuhl für Theoretische Physik 

How to cite
APA:  Neeb, K.H., Thiemann, T., & Sahlmann, H. (2015). Weak Poisson structures on infinite dimensional manifolds and hamiltonian actions. In V. Dobrev (Eds.), Springer Proceedings in Mathematics & Statistics (pp. 105136). Springer Japan. 
MLA:  Neeb, KarlHermann, Thomas Thiemann, and Hanno Sahlmann. "Weak Poisson structures on infinite dimensional manifolds and hamiltonian actions." Springer Proceedings in Mathematics & Statistics Ed. V. Dobrev, Springer Japan, 2015. 105136. 