On Deformation Quantization of Quadratic Poisson Structures

We study the deformation complex of the dg wheeled properad of 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 the universal quantizations of Z-graded quadratic Poisson structures (together with the underlying homogeneous formality maps). In particular we showthat two universal quantizations of Poisson structures are equivalent if the agree on generic quadratic Poisson structures.

Automated summary

We study the deformation complex of Z-graded quadratic Poisson structures and prove its similarity to the even M. Kontsevich graph complex. As a first application, we demonstrate the faithful and essentially transitive action of the Grothendieck-Teichmuller group on the genus completion of the wheeled properad. As a second application, we classify all universal quantizations of Z-graded quadratic Poisson structures.