Convergence of the Gutt Star Product: A strict Deformation Quantization

TL;DR

This paper establishes a strict deformation quantization of the symmetric tensor algebra over a locally convex Lie algebra by topologizing the tensor algebra, providing a new example of a continuous star product called the Gutt star product.

Contribution

It introduces a new locally convex topology on the universal enveloping algebra and demonstrates the convergence of the Gutt star product in this setting.

Findings

Provides a topological framework for the Gutt star product

Constructs a continuous star product as a strict deformation quantization

Offers an explicit topology on the universal enveloping algebra

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.