From the Lie operad to the Grothendieck-Teichm\"uller group

We study the deformation complex of the standard morphism from the degree $d$ shifted Lie operad to its polydifferential version, and prove that it is quasi-isomorphic to the Kontsevich graph complex $\mathbf{GC}_d$. In particular, we show that in the case $d=2$ the Grothendieck-Teichm\"uller group $\mathbf{GRT_1}$ is a symmetry group (up to homotopy) of the aforementioned morphism. We also prove that in the case $d=1$ corresponding to the usual Lie algebras the standard morphism admits a unique homotopy non-trivial deformation which is described explicitly with the help of the universal enveloping construction. Finally we prove the rigidity of the strongly homotopy version of the universal enveloping functor in the Lie theory.