  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.
