Could somebody provide some *simple* examples of Courant algebroids coming from Poisson geometry? 



Courant algebroids from Poisson geometry
I think most examples of Courant algebroids come from Poisson geometry. I don't have a particular favorite example: do you have a favorite Poisson manifold? Recall the motivating construction: let $X$ be a (finite-dimensional smooth) manifold, $A \to X$ a (finite-rank smooth) vector bundle, and suppose that both $A$ and $A^\ast$ are equipped with Lie algebroid structures. These structures are compatible iff you can give the "doubled" vector bundle $A \oplus A^\ast$ (with its canonical pairing) the structure of a Courant algebroid, for which $A,A^\ast$ are Lie subalgebroids thereof. (Note that not every Courant algebroid arises as the double of a Lie bialgebroid. A comparison is important: if you drop the "oid" part, then you get Lie bialgebras on the one hand, and Courant algebroids correspond to Lie algebras equipped with invariant metrics, e.g. any semisimple.)


So, whence arise vector bundles with dual Lie algebroid structures? Precisely in Poisson geometry. Recall that the tangent bundle $\mathrm T X \to X$ is always equipped with a canonical Lie algebroid structure. Recall also that a choice of Poisson structure $\varpi \in \Gamma(\mathrm T^{\wedge 2}X)$ on $X$ makes the cotangent bundle $\mathrm T^\ast X$ into a Lie algebroid (the anchor map is $\varpi: \mathrm T^\ast X \to \mathrm T X$, and the bracket is extended from the requirement that $[\mathrm d f, \mathrm d g] = \mathrm d \langle \varpi, \mathrm d f \wedge \mathrm d g\rangle$, where $\langle,\rangle$ is the pairing $\mathrm T^{\wedge 2} \otimes (\mathrm T^\ast)^{\wedge 2} \to \mathbb R$, and maybe I'm off by some signs and $\frac12$s). These two structures are compatible in the appropriate sense.


This is all better explained in the foundational articles on the subject:


* Zhang-Ju Liu, Alan Weinstein, Ping Xu. Manin Triples for Lie Bialgebroids. 1995. <http://arxiv.org/abs/dg-ga/9508013>
* Dmitry Roytenberg. Courant algebroids, derived brackets and even symplectic supermanifolds. 1999. <http://arxiv.org/abs/math/9910078>
* —. On the structure of graded symplectic supermanifolds and Courant algebroids. 2002. <http://arxiv.org/abs/math/0203110>
* Pavol Severa. Some title containing the words "homotopy" and "symplectic", e.g. this one. 2001. <http://arxiv.org/abs/math/0105080>


Note particularly the main insight in all but the first article, which is that Courant (and indeed Lie) algebroids should best be understood as special cases of something much more general in graded geometry. Namely, one can very generally consider $\mathbb Z$-graded ("super") manifolds which are equipped with some geometric structure. What ends up being very interesting is to equip the manifold with: (1) a grading=$1$ vector field that squares to zero (and hence makes the algebra of smooth functions into a dg algebra); (2) a "symplectic" structure in some interesting grading (which makes the algebra of smooth functions, shifted by said degree, into a dgla). In many cases, the vector field is necessarily hamiltonian, and so you can often replace (1) with (1') a function satisfying some conditions. The notions of Poisson manifold, Lie algebroid, and Courant algebroid are special cases of this set-up, for particular choices of the degree of the symplectic structure (2), and for restrictions on the gradings of coordinate functions (say to be all non-negative). One just as quickly recovers in this way some of the theory of "strongly homotopy Lie algebra/oids" and also some of the theory of Koszul duality.


