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This is the final version of my master thesis. I prove that there are notwist star products on the 2-sphere and on the higher genus pretzel surfacesdeforming a symplectic structure. One of the key arguments is that a connectedcompact symplectic manifold endowed with a twist star product has to be ahomogeneous space. Furthermore, there is a non-degenerate r-matrix on a Liealgebra whose Lie group is acting transitively and effectively on the manifoldin this case. I recap the theory of homogeneous spaces, in particular theclassification of transitive effective actions on the 2-sphere, and discuss thenotion of r-matrices and Drinfel'd twists. Some results of this thesis areavailable as a preprint with Pierre Bieliavsky, Chiara Esposito and StefanWaldmann on arxiv.org under the title "Obstructions for Twist Star Products".

Star Products that can not be induced by Drinfel'd Twists