Formality theorem for Lie bialgebras and quantization of twists and coboundary r-matrices

Let (g,δℏ) be a Lie bialgebra. Let (Uℏ(g),Δℏ) a quantization of (g,δℏ) through Etingof–Kazhdan functor. We prove the existence of a L∞-morphism between the Lie algebra C(g)=Λ(g) and the tensor algebra (without unit) T+U=T+(Uℏ(g)[−1]) with Lie algebra structure given by the Gerstenhaber bracket. When s is a twist for (g,δ), we deduce from the formality morphism the existence of a quantum twist F. When (g,δ,r) is a coboundary Lie bialgebra, we get the existence of a quantization R of r.