# Universal Deformation Formula, Formality and Actions

## TL;DR

This paper develops a method to quantize Poisson actions of triangular Lie algebras on manifolds using formality, leading to a deformation quantization framework with quantum groups and generalized quantum actions.

## Contribution

It introduces a new quantization approach for Poisson actions via formality, extending the concept of quantum actions through $L_$-morphisms.

## Key findings

- Deformation quantization of manifolds with Poisson actions.
- Construction of quantum groups $U_ ext{ extbackslash}hbar( g)$.
- Generalization of quantum actions using $L_$-morphisms.

## 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 $\mathscr{U}_\hbar(\mathfrak{g})$ 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.