Isomorphisms of Poisson and Jacobi brackets

Janusz Grabowski

Banach Center Publications (2000)

  • Volume: 51, Issue: 1, page 79-85
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Grabowski, Janusz. "Isomorphisms of Poisson and Jacobi brackets." Banach Center Publications 51.1 (2000): 79-85. <http://eudml.org/doc/209046>.

@article{Grabowski2000,
abstract = {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.},
author = {Grabowski, Janusz},
journal = {Banach Center Publications},
keywords = {isomorphisms; local Lie algebra; Poisson; contact},
language = {eng},
number = {1},
pages = {79-85},
title = {Isomorphisms of Poisson and Jacobi brackets},
url = {http://eudml.org/doc/209046},
volume = {51},
year = {2000},
}

TY - JOUR
AU - Grabowski, Janusz
TI - Isomorphisms of Poisson and Jacobi brackets
JO - Banach Center Publications
PY - 2000
VL - 51
IS - 1
SP - 79
EP - 85
AB - 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.
LA - eng
KW - isomorphisms; local Lie algebra; Poisson; contact
UR - http://eudml.org/doc/209046
ER -

References

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  1. [1] C. J. Atkin and J. Grabowski, Homomorphisms of the Lie algebras associated with a symplectic manifold, Compos. Math. 76 (1990), 315-349. Zbl0718.53025
  2. [2] P. Dazord, A. Lichnerowicz and Ch.-M. Marle, Structure locale des variétés de Jacobi, J. Math. pures et appl. 70 (1991), 101-152. Zbl0659.53033
  3. [3] J. Grabowski, Isomorphism and ideals of the Lie algebras of vector fields, Inv. Math. 50 (1978), 13-33. Zbl0378.57010
  4. [4] J. Grabowski, Abstract Jacobi and Poisson structures. Quantization and star-products, J. Geom. Phys. 9 (1992), 45-73. Zbl0761.16012
  5. [5] F. Guedira and A. Lichnerowicz, Géométrie des algèbres de Lie locales de Kirillov, J. Math. pures et appl. 63 (1984), 407-484. Zbl0562.53029
  6. [6] A. A. Kirillov, Local Lie algebras, Russ. Math. Surv. 31 (1976), 55-75. Zbl0357.58003
  7. [7] A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom. 12 (1977), 253-300. Zbl0405.53024
  8. [8] A. Lichnerowicz, Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. pures et appl. 57 (1978), 453-488. Zbl0407.53025
  9. [9] H. Omori, Infinite Dimensional Lie Transformation Groups, Lecture Notes in Math. 427, Springer, Berlin, 1974. Zbl0328.58005
  10. [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). 
  11. [11] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983), 523-557. Zbl0524.58011

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