Levi decomposition of analytic Poisson structures and Lie algebroids. (Q1398219)

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.