On deformation quantization of quadratic Poisson structures

TL;DR

This paper explores the deformation complex of quadratic Poisson structures, establishing a quasi-isomorphism with the Kontsevich graph complex, and classifies universal quantizations with implications for the Grothendieck-Teichmüller group.

Contribution

It demonstrates a quasi-isomorphism between the deformation complex of quadratic Poisson structures and the Kontsevich graph complex, and classifies all universal quantizations of these structures.

Findings

Deformation complex is quasi-isomorphic to the Kontsevich graph complex.

Grothendieck-Teichmüller group acts faithfully on the wheeled properad.

Classification of all universal quantizations of quadratic Poisson structures.

Abstract

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-Teichm\"uller 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. In particular we show that two universal quantizations of Poisson structures are equivalent if the agree on generic quadratic Poisson structures.