The first jet of a Poisson tensor at a fixed-point encodes precisely the structure of a Lie algebra (i.e. the isotropy Lie algebra). The corresponding linear Poisson structure on the dual of the isotropy Lie algebra (i.e. the Kirillov-Kostant-Souriau Poisson structure) represents the first jet approximation of the Poisson tensor.

Much more intricate is the semi-global version of this construction due to Yuri Vorobiev, which provides a first order approximation for a Poisson structure around a symplectic leaf, depending only on the first order jet at the leaf (encoded by a transitive Lie algebroid).

In this talk I will explain a similar model for first order approximations of Poisson structures around Poisson submanifolds. This model generalizes Vorobiev's construction, it depends only on the first jet of the Poisson structure, it is unique up to isomorphisms, but does not always exist. I will also discuss an existence criterion.

This is joint work with Rui Loja Fernandes.