We present a general theorem describing the isomorphisms of the local Lie algebra structures on the spaces of smooth (real-analytic or holomorphic) functions on smooth (resp. real-analytic, Stein) manifolds, as, for example, those given by Poisson or contact structures. We admit degenerate structures as well, which seems to be new in the literature.
@article{bwmeta1.element.bwnjournal-article-bcpv51z1p79bwm,
author = {Grabowski, Janusz},
title = {Isomorphisms of Poisson and Jacobi brackets},
journal = {Banach Center Publications},
volume = {51},
year = {2000},
pages = {79-85},
zbl = {1017.53071},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv51z1p79bwm}
}
Grabowski, Janusz. Isomorphisms of Poisson and Jacobi brackets. Banach Center Publications, Tome 51 (2000) pp. 79-85. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv51z1p79bwm/
[000] [1] C. J. Atkin and J. Grabowski, Homomorphisms of the Lie algebras associated with a symplectic manifold, Compos. Math. 76 (1990), 315-349. | Zbl 0718.53025
[001] [2] P. Dazord, A. Lichnerowicz and Ch.-M. Marle, Structure locale des variétés de Jacobi, J. Math. pures et appl. 70 (1991), 101-152. | Zbl 0659.53033
[002] [3] J. Grabowski, Isomorphism and ideals of the Lie algebras of vector fields, Inv. Math. 50 (1978), 13-33. | Zbl 0378.57010
[003] [4] J. Grabowski, Abstract Jacobi and Poisson structures. Quantization and star-products, J. Geom. Phys. 9 (1992), 45-73. | Zbl 0761.16012
[004] [5] F. Guedira and A. Lichnerowicz, Géométrie des algèbres de Lie locales de Kirillov, J. Math. pures et appl. 63 (1984), 407-484. | Zbl 0562.53029
[005] [6] A. A. Kirillov, Local Lie algebras, Russ. Math. Surv. 31 (1976), 55-75. | Zbl 0357.58003
[006] [7] A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom. 12 (1977), 253-300. | Zbl 0405.53024
[007] [8] A. Lichnerowicz, Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. pures et appl. 57 (1978), 453-488. | Zbl 0407.53025
[008] [9] H. Omori, Infinite Dimensional Lie Transformation Groups, Lecture Notes in Math. 427, Springer, Berlin, 1974. | Zbl 0328.58005
[009] [10] S. M. Skriabin, Lie algebras of derivations of commutative rings: Generalizations of the Lie algebras of Cartan type, Preprint WINITI 4405-W87, Moscow University, 1987 (in Russian).
[010] [11] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983), 523-557. | Zbl 0524.58011