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Keywords:
Poisson structure; Poisson bivector; infinite dimensional manifolds
Poisson structure; Poisson bivector; infinite dimensional manifolds
Summary:
We show that, on a smoothly paracompact convenient manifold $M$ modeled on a convenient space with the bornological approximation property, the dual map of a Poisson bracket factors as a smooth section of the vector bundle $L_{\text{skew}}^2(T^*M,\mathbb{R})$.
References:
[1] Beltiţă, Daniel, Goliński, Tomasz, Tumpach, Alice-Barbara: Queer Poisson brackets. J. Geom. Phys. 132 (2018), 358–362. DOI 10.1016/j.geomphys.2018.06.013
[2] Drinfeld, V.G.: HAMILTONIAN STRUCTURES ON LIE GROUPS. Yang-Baxter Equation in Integrable Systems 10 (1990), no. 2, 222.
[3] Glöckner, Helge: Applications of Hypocontinuous Bilinear Maps in Infinite-Dimensional Differential Calculus. Generalized Lie Theory in Mathematics, Physics and Beyond (Silvestrov, Sergei, Paal, Eugen, Abramov, Viktor, Stolin, Alexander, eds.), Springer, Berlin, Heidelberg, 2009, pp. 171–186.
[4] Goliński, Tomasz, Rahangdale, Praful, Tumpach, Alice Barbora: Poisson structures in the Banach setting: comparison of different approaches. Workshop on Geometric Methods in Physics, Springer, 2024, pp. 97–118.
[5] Hörmander, L.: The Analysis of linear partial differential operators I. Springer-Verlag, Berlin, 1983, Grundlehren 256.
[6] Jarchow, Hans: Locally convex spaces. Springer Science & Business Media, 2012.
[7] Kriegl, Andreas, Michor, Peter W.: The convenient setting of global analysis. Mathematical Surveys and Monographs, vol. 53, American Mathematical Society, Providence, RI, 1997. MR 1471480 DOI: http://dx.doi.org/10.1090/surv/053 DOI 10.1090/surv/053
[9] Michor, P: Manifolds of smooth maps IV: Theorem of de Rham. Cah. Topol. Géom. Différ. Catég. 24 (1983), no. 1, 57–86.
[10] Michor, Peter W.: The Schouten-Nijenhuis bracket in infinite dimensions. arXiv.2504.15010, 2025. DOI: http://dx.doi.org/10.48550/arXiv.2504.15010 DOI 10.48550/arXiv.2504.15010
[11] Michor, Peter W, Mumford, David: A zoo of diffeomorphism groups on $\mathbb{R}^n$. Ann. Global Anal. Geom. 44 (2013), no. 4, 529–540. DOI 10.1007/s10455-013-9380-2
[12] Neeb, K.-H., Sahlmann, H., Thiemann, T.: Weak Poisson structures on infinite dimensional manifolds and Hamiltonian actions. Lie Theory and Its Applications in Physics: Varna, Bulgaria, June 2013, Springer, 2014, pp. 105–135.
[13] Schaefer, H.H.: Topological vector spaces. Springer-Verlag, New York, 1971, GTM 3.
[14] Thomas, E.G.F.: Vector Fields as Derivations on Nuclear Manifolds. Math. Nachr. 176 (1995), no. 1, 277–286. DOI 10.1002/mana.19951760120
[15] Treves, François: Topological Vector Spaces, Distributions and Kernels: Pure and Applied Mathematics, Vol. 25. vol. 25, Elsevier, 2016.
[16] Tumpach, Alice Barbara: Banach Poisson–Lie groups and Bruhat–Poisson structure of the restricted Grassmannian. Comm. Math. Phys. 373 (2020), no. 3, 795–858. DOI 10.1007/s00220-019-03674-3
[17] Weinstein, Alan: The local structure of Poisson manifolds. J. Differential Geometry 18 (1983), no. 3, 523–557.
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