A study of Galois objects for algebraic quantum groups

TL;DR

This paper advances the understanding of Galois objects in algebraic quantum groups by analyzing their structure, functionals, and symmetries, including reflection procedures and examples.

Contribution

It provides a detailed structural analysis of Galois objects, introduces new concepts like reflection across Galois objects, and extends the theory of algebraic quantum groups.

Findings

Galois objects have a rich structure similar to quantum groups

Existence of two distinguished weak K.M.S. functionals related by a modular element

Reflection across Galois objects can produce new algebraic quantum groups

Abstract

We supplement the study of Galois theory for algebraic quantum groups started in the paper 'Galois Theory for Multiplier Hopf Algebras with Integrals' by A. Van Daele and Y.H. Zhang. We examine the structure of the Galois objects: algebras equipped with a Galois coaction such that only the scalars are coinvariants. We show how their structure is as rich as the one of the quantum groups themselves: there are two distinguished weak K.M.S. functionals, related by a modular element, and there is an analogue of the antipode squared. We also show how to reflect the quantum group across the Galois object to obtain a (possibly) new algebraic quantum group. We end by considering an example.