Open Access
Oct 2004 Levi decomposition for smooth Poisson structures
Philippe Monnier, Nguyen Tien Zung
J. Differential Geom. 68(2): 347-395 (Oct 2004). DOI: 10.4310/jdg/1115669514


We prove the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids near a singular point. This Levi decomposition is a kind of normal form or partial linearization, which was established in the formal case by Wade [10] and in the analytic case by the second author [15]. In particular, in the case of smooth Poisson structures with a compact semisimple linear part, we recover Conn's smooth linearization theorem [5], and in the case of smooth Lie algebroids with a compact semisimple isotropy Lie algebra, our Levi decomposition result gives a positive answer to a conjecture of Weinstein [13] on the smooth linearization of such Lie algebroids. In the appendix of this paper, we show an abstract Nash-Moser normal form theorem, which generalizes our Levi decomposition result, and which may be helpful in the study of other smooth normal form problems.


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Philippe Monnier. Nguyen Tien Zung. "Levi decomposition for smooth Poisson structures." J. Differential Geom. 68 (2) 347 - 395, Oct 2004.


Published: Oct 2004
First available in Project Euclid: 9 May 2005

zbMATH: 1085.53074
MathSciNet: MR2144250
Digital Object Identifier: 10.4310/jdg/1115669514

Rights: Copyright © 2004 Lehigh University

Vol.68 • No. 2 • Oct 2004
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