On deformation quantization of quadratic Poisson structures

We study the deformation complex of the dg wheeled properad of $mathbb{Z}$-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck-Teichmuller group acts on the genus completion of that wheeled properad faithfully and essentially transitively. As a second application we classify all universal quantizations of $mathbb{Z}$-graded quadratic Poisson structures together with the underlying (so called) homogeneous formality maps.