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 polydifferential operators on Lie algebroids we obtain a deformation quantization of $M$ together with a quantum group ${\mathcal{U}}_{\hslash }(\mathfrak{g})$ and a map of associated DGLA's. This motivates a definition of quantum action in terms of $L_{\infty }$-morphisms which generalizes the one given by Drinfeld.

26 pages. Comments are welcome!