Universal deformation formula, formality and actions

Abstract

In this paper we provide a quantization via formality of Poisson actions of a triangular Lie algebra $(\mathfrak{g},r)$ on a smooth manifold $M$. Using the formality of $E$-polydifferential operators for Lie algebroids $E$ over $M$, we obtain a deformation quantization of $M$ together with a quantum group ${\mathcal{U}}_{{\rho }_{\hslash }}(\mathfrak{g})$ and a map of associated DGLA's. This motivates a definition of quantum action in terms of $L_{\infty }$-morphisms which generalizes the well-known definition given by Drinfeld.