# 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: 105-136
ISBN: 978-4-431-55284-0

## 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)" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">A⊆C∞(M) which has to satisfy a non-degeneracy condition (the differentials of elements of A" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" tabindex="0">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" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">g and a κ-skew-symmetric derivation D a weak affine Poisson structure on g" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">g itself. This leads naturally to a concept of a Hamiltonian G-action on a weak Poisson manifold with a g" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0">g

-valued momentum map and hence to a generalization of quasi-hamiltonian group actions.

## FAU Authors / FAU Editors

 →
 Neeb, Karl-Hermann Prof. Dr.
 →
 Lehrstuhl für Mathematik (Lie-Gruppen und Darstellungstheorie)
 →
 Sahlmann, Hanno Prof. Dr.
 →
 Professur für Theoretische Physik
 →
 Thiemann, Thomas Prof. Dr.
 →
 Chair for Theoretical Physics III (Quantum Gravity)

## 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. 105-136). Springer Japan.

 MLA: Neeb, Karl-Hermann, 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. 105-136.

 BibTeX: http://cris.fau.de/bibtex/publication/116261244.bib

Last updated on 2019-13-05 at 13:15