  Deformation theory refers to an apparatus in many parts of math and physics
for going from an infinitesimal (= first order) deformation to a full
deformation, either formal or convergent appropriately. If the algebra being
deformed is that of observables, the result is deformation quantization,
independent of any realization in terms of Hilbert space operators. There are
very important but rare cases in which a formula for a full deformation is
known. For physics, the most important is the Moyal-Weyl star product formula.
In this paper, we concentrate on deformations of Courant algebroid structures
via star products on graded manifolds. In particular, we construct a graded
version of the Moyal-Weyl star product. Recently, in Double Field Theory (DFT),
deformations of the C-bracket and O(d,d)-invariant bilinear form to first order
in the closed string sigma model coupling $\alpha '$ were derived by analyzing
the transformation properties of the Neveu-Schwarz B-field. By choosing a
particular Poisson structure on the Drinfel'd double corresponding to the
Courant algebroid structure of the generalized tangent bundle, we reproduce
these deformations for a specific solution of the strong constraint of DFT as
expansion of a graded version of the Moyal-Weyl star product.
