If a Lie algebra acts on a Poisson manifold via derivations, the problem of deformation quantization of the manifold can be shifted to the existence of a normalized 2-cocycle on the universal enveloping algebra. This process is called Drinfel'd twist deformation quantization. Unlike a general star product, a twist star product inhabits many additional properties which are of particular interest in physics. However, the twisting procedure is limited, i.e. not every deformation quantization is of this form. It is the task of the talk to introduce the notion of Drinfel'd twist deformation quantization and to discuss its strenght and disadvantages together with several examples and counterexamples.
