# zbMATH — the first resource for mathematics

Levi decomposition of analytic Poisson structures and Lie algebroids. (English) Zbl 1033.53069
The classical Levi-Malcev theorem says that the exact sequence $$0\to r\to {\mathcal L}\to{\mathcal L}/r\to 0$$, where $${\mathcal L}$$ is an $$n$$-dimensional Lie algebra of linear functions on $$\mathbb{K}^n$$ under the Poisson bracket of the linear part of a given Poisson structure in a neighbourhood of 0 in $$\mathbb{K}^n$$, $$r$$ is the radical of $${\mathcal L}$$, admits a splitting (called the Levi decomposition), i.e. there is an injective section of $${\mathcal L}\to {\mathcal L}/r$$. And $${\mathcal L}$$ can be written as a semi-direct product of an image of this inclusion and a solvable Lie algebra $$r$$. The author proves the existence of such a local analytic Levi decomposition for analytic Poisson structures and Lie algebroids.

##### MSC:
 53D17 Poisson manifolds; Poisson groupoids and algebroids 32S65 Singularities of holomorphic vector fields and foliations
SUSYGEN
Full Text: