In a paper of A. Weinstein on the geometry of Poisson manifolds, he relates the formal linearization around a zero, p, of the Poisson bivector to extensions of the Lie algebra induced by the bivector on the tangent space over p. 


I wanted to know if this is part of a big picture, possibly relating deformations of Lie brackets to some extensions of Lie algebras. 



Is there any relation between deformation and extension of Lie algebras?
In fact the picture is extremely simple and works indeed for any type of algebra as follows :


Let $\mu$ be a Lie algebra on a vector space $V$, $c$ a two-cocycle $c\in CE^2(V;M)$ (Chevalley Eilenberg cohomology) where $M$ is a module over the Lie algebra. The extension of $\mu$ by $c$ is nothing else than a deformation of $\mu$, but in the space of Lie algebras on the vector space $V\oplus M$. The deformed Lie algebra has a bracket $\mu'$ given by $\mu'(v+m,v'+m')=(\mu(v,v'),v.m'-v'.m+c(v,v'))$. One can easily check that the Jacobi condition for the deformed algebra $\mu'$ is equivalent to the data of the Jacobi condition for $\mu$, the module structure of $M$ and the cocycle condition for $c$. From this point of view one can also view $\mu'$ as the semi-direct product of $\mu$ and $M$ when the cocycle $c$ is null.


