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Levi decomposition for smooth Poisson structures

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 [Wad97] and in the analytic case by the second author [Zun03]. In particular, in the case of smooth Poisson structures with a compact semisimple linear part, we recover Conn’s smooth linearization theorem [Con85], 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 [Wei00] on the smooth linearization of such Lie algebroids. In the appendix of this paper, we show an abstract NashMoser normal form theorem, which generalizes our Levi decomposition result, and which may be helpful in the study of other smooth normal form problems.