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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- 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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Acknowledgements
We thank Helge Glöckner, Stefand Waldmann and Anton Alekseev for discussions on the subject matter of this manuscript and for pointing out references.
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Neeb, KH., Sahlmann, H., Thiemann, T. (2014). Weak Poisson Structures on Infinite Dimensional Manifolds and Hamiltonian Actions. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics & Statistics, vol 111. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55285-7_8
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