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.