Convergence of the Gutt Star Product: A strict Deformation Quantization

Abstract: This work is the final version of my master thesis. Many, but not all of its key results are already available as a preprint with Chiara Esposito and Stefan Waldmann on arxiv.org under the title "Convergence of the Gutt Star Product", which has been submitted to a journal. The main goal of this work is to understand a certain deformation of the commutative product on the symmetric tensor algebra over a given (locally convex) Lie algebra in a topological setting. Therefore, the tensor algebra over the Lie algebra is topologized in a locally convex manner. This finally provides a new example of a strict deformation quantization, i.e. a continuous star product. The related formal star product was found by Simone Gutt and is therefore called the Gutt star product. Together with this star product, the symmetric tensor algebra is isomorphic to the universal enveloping algebra of the Lie algebra. Hence, this work also presents an explicit locally convex topology on the universal enveloping algebra of certain locally convex Lie algebras. Since this construction has good functorial properties, it is also an interesting object of study not only in quantization and quantum group theory, but also in Lie theory.