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Erschienen in:
Revisiting Trace Anomalies in Chiral Theories Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

Weak Poisson Structures on Infinite Dimensional Manifolds and Hamiltonian Actions

verfasst von : K.-H. Neeb, H. Sahlmann, T. Thiemann

Erschienen in: Lie Theory and Its Applications in Physics

Verlag: Springer Japan

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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 \(\mathcal{A}\subseteq C^{\infty }(M)\) which has to satisfy a non-degeneracy condition (the differentials of elements of \(\mathcal{A}\) 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 \(\mathfrak{g}\) and a κ-skew-symmetric derivation D a weak affine Poisson structure on \(\mathfrak{g}\) itself. This leads naturally to a concept of a Hamiltonian G-action on a weak Poisson manifold with a \(\mathfrak{g}\)-valued momentum map and hence to a generalization of quasi-hamiltonian group actions.

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Vorheriges Kapitel Semi-classical Scalar Products in the Generalised SU(2) Model
Nächstes Kapitel Bethe Vectors of gl(3)-Invariant Integrable Models, Their Scalar Products and Form Factors
Fußnoten
1
A symplectic form ω on M is called strong if, for every p ∈ M, every continuous linear functional on T p (M) is of the form ω p (v, ⋅ ) for some v ∈ T p (M).
 
2
This condition is satisfied for finite dimensional symplectic manifolds, for strongly symplectic smoothly paracompact Banach manifolds (cf. [14]) and for symplectic vector spaces.
 
3
By definition of the weak-∗-topology on \(\mathfrak{g}^{{\prime}}\), which corresponds to the subspace topology with respect to the embedding \(\mathfrak{g}^{{\prime}}\hookrightarrow \mathbb{R}^{\mathfrak{g}}\), a map \(\varphi: M \rightarrow \mathfrak{g}^{{\prime}}\) is smooth with respect to this topology if and only if all functions \(\varphi _{X}(m):=\varphi (m)(X)\) are smooth on M.
 
4
One can ask more generally, for which locally convex spaces V and which topologies on V ′ the evaluation map \(V \times V ^{{\prime}}\rightarrow \mathbb{R}\) is continuous. This happens if and only if the topology on V can be defined by a norm, and then the operator norm turns V ′ into a Banach space for which the evaluation map is continuous.
 
5
This is the case for so-called regular Lie groups (cf. [21]). Banach–Lie groups and in particular finite dimensional Lie groups are regular.
 
6
This concept depends on the choice of the invariant symmetric bilinear form \(\langle \cdot,\cdot \rangle\) on the Lie algebra \(\mathfrak{k}\). Changing this form leads to a different Poisson structure on \(\mathcal{L}(\mathfrak{k})\).
 
7
In [1] one finds this concept for the special case where (M, ω) is a weak symplectic manifold. In this case one requires the action σ to be symplectic and the existence of a smooth \(\mathcal{L}(K)\)-equivariant map \(\varPhi: M \rightarrow \mathcal{L}(\mathfrak{k})\) such that the functions
$$\displaystyle{\varphi (\xi )(m):=\kappa (\varPhi (m),\xi )\quad \mbox{ satisfy }\quad i_{\xi _{\sigma }}\omega = \mathtt{d}(\varphi (\xi )).}$$
These conditions are easily verified to be equivalent to ours (cf. Proposition 3.1).
 
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Metadaten
Titel
Weak Poisson Structures on Infinite Dimensional Manifolds and Hamiltonian Actions
verfasst von
K.-H. Neeb
H. Sahlmann
T. Thiemann
Copyright-Jahr
2014
Verlag
Springer Japan
Buch
Lie Theory and Its Applications in Physics

Print ISBN: 978-4-431-55284-0

Electronic ISBN: 978-4-431-55285-7

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-4-431-55285-7_8