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

Abstract

Let $(g,{\delta }_{\hslash })$ be a Lie bialgebra. Let $(U_{\hslash }(g),{\Delta }_{\hslash })$ a quantization of $(g,{\delta }_{\hslash })$ through Etingof-Kazhdan functor. We prove the existence of a $L_{\infty }$-morphism between the Lie algebra $C(\g)=Λ(g)$ and the tensor algebra $TU=T(U_{\hslash }(g)[-1])$ with Lie algebra structure given by the Gerstenhaber bracket. When $(g,{\delta }_{\hslash },r)$ is a coboundary Lie bialgebra, we deduce from the formality morphism the existence of a quantization $R$ of $r$.