Abstract
We review the linearization of Poisson brackets and related problems, in the formal, analytic and smooth categories.
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References
V.I. Arnold (1988) Geometrical Methods in the Theory of Ordinary Differential Equations Spinger-Verlag New York
J. Basto-Gonçalves I. Cruz (1999) ArticleTitleAnalytic k-linearizability of some resonant Poisson structures Lett. Math. Phys. 49 59–66
S. Bochner (1945) ArticleTitleCompact groups of differentiable transformations Ann. of Math. 46 372–381
Bordemann M., Makhlouf A., Petit T. Déformation par quantification et rigidité des algèbres enveloppantes, preprint math.RA/0211416 (2002)
Brahic O. Normal Forms of Poisson structures near a symplectic leaf, Preprint math.SG/0403136 (2004).
G. Cairns E. Ghys (1997) ArticleTitleThe local linearization problem for smooth SL(n)-actions, Enseign Math. 43 IssueID2 133–171
Cannas da Silva A., Weinstein A. (1999). Geometric Models for Noncommutative Algebras. Berkeley Math. Lectures, 10, Amer. Math. Soc., Providence
Cartan H. (1935). Sur les groupes de transformation analytiques. Act. Sci. et Industrielles 198
J. Conn (1984) ArticleTitleNormal forms for analytic Poisson structures Ann. of Math. 119 576–601
J. Conn (1985) ArticleTitleNormal forms for smooth Poisson structures Ann. of Math. 121 565–593
Crainic M. Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes, to appear in Comment. Math. Helv. (preprint math.DG/0008064).
M. Crainic R.L. Fernandes (2003) ArticleTitleIntegrability of Lie brackets Ann. of Math. 157 IssueID2 575–620
Crainic M., Fernandes R.L. Rigidity of Poisson brackets, in preparation.
Crainic M., Fernandes R.L., Moerdijk I. Deformation Cohomology, in preparation
Desolneux-Moulis, N.: Linéarisation de certaines structures de Poisson de classe C∞. Séminaire de géométrie, 1985–1986, Publ. Dép. Math. Nouvelle Sér. B, 86–4, Univ. Claude-Bernard, Lyon (1986), 55-68.
J.P. Dufour (1990) ArticleTitleLinéarisation de certaines structures de Poisson J. Differential Geom. 32 IssueID2 415–428
J.P. Dufour (2001) ArticleTitleNormal forms of Lie algebroids Banach Center Publications. 54 35–41
J.P. Dufour N.T. Zung (2002) ArticleTitleNondegeneracy of the Lie algebra aff(n) C. R. Acad. Sci. Paris Sér. I Math. 335 1043–1046
Dufour J.P., Zung N.T. Poisson Structures and their Normal Forms, in preparation.
J. Duistermaat J. Kolk (2000) Lie Groups Springer-Verlag Berlin
R.L. Fernandes (2002) ArticleTitleLie algebroids, holonomy and characteristic class Adv. in Math. 170 119–179
M. Flato G. Pinczon J. Simon (1977) ArticleTitleNon linear representations of Lie groups Ann. Sci. Ecole Norm. Sup. 10 405–418
V. Guillemnin S. Sternberg (1968) ArticleTitleRemarks on a paper of Hermann Trans. Amer. Math. Soc. 130 110–116
R. Hamilton (1982) ArticleTitleThe inverse function theorem of Nash and Moser Bull. Amer. Math. Soc. (N.S.). 7 IssueID1 65–222
R. Hermann (1968) ArticleTitleThe formal linearization of a semisimple Lie algebra of vector fields about a singular point Trans. Amer. Math. Soc. 130 105–109
A.G. Kushnirenko (1967) ArticleTitleLinear-equivalent action of a semi-simple Lie group in the neighborhood of a stationary point Funkts. Anal. Prilozh. 1 103–104
I. Moerdijk J. Mrčun (2002) ArticleTitleOn integrability of infinitesimal actions Amer. J. Math. 124 567–593
Molinier J.C. (1993). Linéarisation de structures de Poisson, PhD Thesis, Montpellier
Monnier Ph., Zung N.T. Levi decomposition for smooth Poisson structures, Preprint math.DG/0209004.
J. Moser (1961) ArticleTitleA new technique for the construction of solutions of nonlinear differential equations. Proc Nat. Acad. Sci. USA. 47 1824–1831
J. Mrčun. (1996) ArticleTitleAn extension of the Reeb stability theorem Topology Appl. 70 25–55
A. Wade (1997) ArticleTitleNormalisation formelle de structures de Poisson C. R. Acad. Sci. Paris. Sér. I Math. 324 IssueID5 531–536
C. Weibel (1994) An Introduction to Homological Algebra Cambridge University Press Cambridge
A. Weinstein (1983) ArticleTitleThe local structure of Poisson manifolds J. Differential Geom. 18 523–557
A. Weinstein (1987) ArticleTitlePoisson geometry of the principal series and non-linearizable structures J. Differential Geom. 25 55–73
A. Weinstein (2000) ArticleTitleLinearization problem for Lie algebroids and Lie groupoids Lett. Math. Phys. 52 IssueID1 93–102
A. Weinstein (2002) ArticleTitleLinearization of regular proper groupoids J. Inst. Math. Jussieu 1 493–511
N.T. Zung (2003) ArticleTitleLevi decomposition of analytic Poisson structures and Lie algebroids Topology. 42 IssueID6 1403–1420
Zung N.T. A geometric proof of Conn’s linearization theorem for analytic Poisson structures, preprint SG/0207263.
Zung N.T. Linearization of proper groupoids, Preprint math.DG/0301297.
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Fernandes, R.L., Monnier, P. Linearization of Poisson Brackets. Lett Math Phys 69, 89–114 (2004). https://doi.org/10.1007/s11005-004-0340-4
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DOI: https://doi.org/10.1007/s11005-004-0340-4