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.
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.
There are big pictures that I'll let others describe. Here's a little picture which cogeneralizes the Weinstein remark. (To "cogeneralize" is to make more specific, rather than less.)
Recall that a Lie bialgebra is a vector space $\mathfrak g$ with a "Lie bracket" $\mathfrak g^{\wedge 2} \to \mathfrak g$ satisfying Jacobi, a "Lie cobracket" $\mathfrak g \to \mathfrak g^{\wedge 2}$ satisfying Jacobi, and such that the two structures satisfy a compatibility condition which has lots of equivalent formulations: one of them is that the cobracket is a 1-cochain for the Chevalley-Eilenberg complex of $\mathfrak g$ with values in $\mathfrak g^{\wedge 2}$ (diagonal adjoint action).
Then the first result you prove about these things is: Any such structure defines (and is equivalent to) an "extension" (although it's not a short exact sequence), called the double of $\mathfrak g$. As a vector space, the double is the sum $\mathfrak g \oplus \mathfrak g^\ast$ (where $\mathfrak g^\ast$ is the dual vector space, and is a Lie algebra by turning around the Lie cobracket), and indeed each of the summands $\mathfrak g,\mathfrak g^\ast$ inside the double is a Lie subalgebra. The two terms do interact: they interact in the unique way making the canonical pairing $(\mathfrak g \oplus \mathfrak g^\ast)^{\otimes 2} \to \mathbb k$ ad-invariant. For various equivalent descriptions, and if you want to see this all in pictures, I have a short expository note on Lie bialgebras at http://math.berkeley.edu/~theojf/GraphicalLanguage.pdf .
Anyway, why is this a cogeneralization of what Alan's doing? There is a generalization of Lie algebra to Lie algebroid, which I can define if you like, but I would assume that it's in Alan's paper, and one example of a Lie algebroid is that the tangent bundle of a manifold has a canonical Lie algebroid structure. A Lie algebroid structure on the cotangent bundle is precisely the same as a Poisson bivector. So a Poisson manifold is (almost) an example of a "Lie bialgebroid", because the tangent bundle is both an algebroid and a coalgebroid. I say "almost", because a priori there is no cocycle condition. But the linearization of a Poisson structure near a zero thereof I think should satisfy a cocycle condition --- I haven't worked out the details, so take this paragraph with a grain of salt.
Anyway, having not read this paper, I'm not sure if I've answered the question you asked, or a related one.
Hinich says in "Deformation theory and lie algebra homology", that Grothendieck says that "to each deformation problem we can assign a sheaf of Lie algebras over X; the sheaf of infinitesimal automorphisms".
