Topological Quantum Groups, Star Products and their relations

TL;DR

This paper reviews recent advances in quantum compact groups and star products, focusing on their algebraic deformations, duality, and connections with quantum double models.

Contribution

It introduces an abstract deformation framework for star products and explores their application to quantum groups and duality structures.

Findings

Reformulation of star products as algebraic deformations

Introduction of Montel topologies for quantum group duality

Analysis of deformations of Hopf algebras on compact groups

Abstract

This short summary of recent developments in quantum compact groups and star products is divided into 2 parts. In the first one we recast star products in a more abstract form as deformations and review its recent developments. The second part starts with a rapid presentation of standard quantum group theory and its problems, then moves to their completion by introduction of suitable Montel topologies well adapted to duality. Preferred deformations (by star products and unchanged coproducts) of Hopf algebras of functions on compact groups and their duals, are of special interest. Connection with the usual models of quantum groups and the quantum double is then presented.