# Group Actions in Deformation Quantisation

## TL;DR

This paper reviews the theory of deformation quantization with symmetries, discussing star products, group actions, Drinfeld twists, and quantum reduction, with a focus on existing results and conceptual frameworks.

## Contribution

It provides a comprehensive overview of deformation quantization with symmetries, linking classical and quantum concepts, and discusses various approaches and results without presenting new research.

## Key findings

- Connection between invariant star products and Drinfeld twists
- Description of quantum analogues of classical reduction
- Discussion on convergence issues of star products

## Abstract

This set of notes corresponds to a mini-course given in September 2018 in Bedlewo; it does not contain any new result; it complements -- with intersection -- the introduction to formal deformation quantization and group actions, corresponding to a course given in Villa de Leyva in July 2015. After an introduction to the concept of deformation quantization, we briefly recall existence, classification and representation results for formal star products. We come then to results concerning the notion of formal star products with symmetries; one has a Lie group action (or a Lie algebra action) compatible with the Poisson structure, and one wants to consider star products such that the Lie group acts by automorphisms (or the Lie algebra acts by derivations). We recall in particular the link between left invariant star products on Lie groups and Drinfeld twists, and the notion of universal deformation formulas. Classically, symmetries are particularly interesting when they are implemented by a moment map and we give indications to build a corresponding quantum moment map. Reduction is a construction in classical mechanics with symmetries which allows to reduce the dimension of the manifold; we describe one of the various quantum analogues which have been considered in the framework of formal deformation quantization. We end up by some considerations about convergence of star products.