Consider a Poisson-Lie group $G$, with whatever additional requirements (quasi-triangular, compact, simply connected).
We can consider $G$ as a Poisson Manifold and apply Kontsevich formality to obtain a star-product on $C^\infty(G)$, or equivalently a Maurer-Cartan element in $\star\in\Gamma (D_{\mathrm{poly}}^\bullet(G))$, the dg-LA of (formal) polydifferential operators.
Conversely we can consider the Lie algebra $\mathfrak g$, and then we can deform $U\mathfrak g$ using Etinghof-Kazhdan deformation quantization to a non-cocommutative Hopf algebra. Let us assume $\mathfrak g$ is quasi-triangular so that we only deform the coproduct $\Delta_\hbar \colon U_\hbar\mathfrak g\to U_\hbar\mathfrak g\otimes U_\hbar \mathfrak g$.
These two types of deformation quantization have to be related. The first thing to note is that $U{\mathfrak g}^{\otimes n}$ can be interpreted as invariant polydifferential operators on $C^\infty(G)$. Furthermore, since $\Delta_\hbar$ is an algebra morphism, it is determined by its restriction $\Delta_\hbar|_{\mathfrak g}\colon \mathfrak g\to U_\hbar\mathfrak g\otimes U_\hbar \mathfrak g$. The fact that $\Delta_\hbar$ is an algebra morphism means that this is a cocycle, and hence should integrate to a map $G\to U_\hbar\mathfrak g\otimes U_\hbar \mathfrak g$. Coassociativity of $\Delta_\hbar$ should translate to this being star product, modulo some details. We note that this star product is invariant (in the same sense of invariance as the Poisson structure on $G$.) Differentiating invariant star products should then also produce deformed coproducts on the universal enveloping algebra.
Does this give an equivalence between invariant star products and deformations of $U\mathfrak g$? Invariant star products are the Maurer-Cartan elements of some dgLA. Can we interpret deformations of $U\mathfrak g$ as Maurer-Cartan elements of some appropriate dgLA, and are the dgLA's quasi-isomorphic?
This seems like a fairly basic question, but I have been unable to find any literature discussing this.
