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

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

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