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

TL;DR

This thesis demonstrates that certain symplectic manifolds, like the 2-sphere and higher genus pretzel surfaces, cannot admit twist star products deforming their symplectic structures, due to their geometric and algebraic properties.

Contribution

It proves the non-existence of twist star products on specific symplectic manifolds, linking geometric structures with algebraic obstructions.

Findings

No twist star products on the 2-sphere or higher genus pretzel surfaces.

Connected compact symplectic manifolds with twist star products are homogeneous spaces.

Existence of a non-degenerate r-matrix on the acting Lie algebra is necessary.

Abstract

This is the final version of my master thesis. I prove that there are no twist star products on the 2-sphere and on the higher genus pretzel surfaces deforming a symplectic structure. One of the key arguments is that a connected compact symplectic manifold endowed with a twist star product has to be a homogeneous space. Furthermore, there is a non-degenerate r-matrix on a Lie algebra whose Lie group is acting transitively and effectively on the manifold in this case. I recap the theory of homogeneous spaces, in particular the classification of transitive effective actions on the 2-sphere, and discuss the notion of r-matrices and Drinfel'd twists. Some results of this thesis are available as a preprint with Pierre Bieliavsky, Chiara Esposito and Stefan Waldmann on arxiv.org under the title "Obstructions for Twist Star Products".