Stability of higher order singular points of Poisson manifolds and Lie algebroids

Jean-Paul Dufour[1]; Aïssa Wade[2]

  • [1] Université Montpellier 2 Département de Mathématiques Place Eugène Bataillon 34095 Montpellier Cedex 5 (France)
  • [2] Penn State University Department of Mathematics University Park PA 16802 (USA)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 3, page 545-559
  • ISSN: 0373-0956

Abstract

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We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first-order approximation (not necessarily linear) of a given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algebroids recently introduced by Crainic and Moerdijk. We also provide several examples of stable singular points of order k ≥ 1 for Poisson structures and Lie algebroids. Finally, we apply our results to pre-symplectic leaves of Dirac manifolds.

How to cite

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Dufour, Jean-Paul, and Wade, Aïssa. "Stability of higher order singular points of Poisson manifolds and Lie algebroids." Annales de l’institut Fourier 56.3 (2006): 545-559. <http://eudml.org/doc/10157>.

@article{Dufour2006,
abstract = {We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first-order approximation (not necessarily linear) of a given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algebroids recently introduced by Crainic and Moerdijk. We also provide several examples of stable singular points of order $k \ge 1$ for Poisson structures and Lie algebroids. Finally, we apply our results to pre-symplectic leaves of Dirac manifolds.},
affiliation = {Université Montpellier 2 Département de Mathématiques Place Eugène Bataillon 34095 Montpellier Cedex 5 (France); Penn State University Department of Mathematics University Park PA 16802 (USA)},
author = {Dufour, Jean-Paul, Wade, Aïssa},
journal = {Annales de l’institut Fourier},
keywords = {Poisson structure; Lie algebroid; Lichnerowicz-Poisson cohomology; stable point; stability of singular points; Poisson manifolds; Lie algebroids},
language = {eng},
number = {3},
pages = {545-559},
publisher = {Association des Annales de l’institut Fourier},
title = {Stability of higher order singular points of Poisson manifolds and Lie algebroids},
url = {http://eudml.org/doc/10157},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Dufour, Jean-Paul
AU - Wade, Aïssa
TI - Stability of higher order singular points of Poisson manifolds and Lie algebroids
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 3
SP - 545
EP - 559
AB - We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first-order approximation (not necessarily linear) of a given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algebroids recently introduced by Crainic and Moerdijk. We also provide several examples of stable singular points of order $k \ge 1$ for Poisson structures and Lie algebroids. Finally, we apply our results to pre-symplectic leaves of Dirac manifolds.
LA - eng
KW - Poisson structure; Lie algebroid; Lichnerowicz-Poisson cohomology; stable point; stability of singular points; Poisson manifolds; Lie algebroids
UR - http://eudml.org/doc/10157
ER -

References

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  14. P. Monnier, Une cohomologie associée à une fonction : applications aux cohomologies de Poisson et de Nambu-Poisson, (2001) 
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  16. O. Radko, A classification of topologically stable Poisson structures on a compact oriented surface, J. Symplectic Geom. 1 (2002), 523-542 Zbl1093.53087MR1959058
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