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Originally proposed as an $O(d,d)$-invariant formulation of classical closedstring theory, double field theory (DFT) offers a rich source of mathematicalstructures. Most prominently, its gauge algebra is determined by the so-calledC-bracket, a generalization of the Courant bracket of generalized geometry, inthe sense that it reduces to the latter by restricting the theory to solutionsof a "strong constraint". Recently, infinitesimal deformations of thesestructures in the string sigma model coupling $\alpha'$ were found. In thisshort contribution, we review constructing the Drinfel'd double of a Liebialgebroid and offer how this can be applied to reproduce the C-bracket of DFTin terms of Poisson brackets. As a consequence, we are able to explain the$\alpha'$-deformations via a graded version of the Moyal-Weyl product in aclass of examples. We conclude with comments on the relation between $B$- and$\beta$-transformations in generalized geometry and the Atiyah algebra on theDrinfel'd double.

Star products on graded manifolds and $\alpha'$-corrections to double field theory