Quantum groups with projection on von Neumann algebra level

TL;DR

This paper develops an axiomatic framework for semidirect products of locally compact quantum groups within the von Neumann algebra setting, enabling advanced analysis using crossed product techniques and introducing a braided comultiplication structure.

Contribution

It presents a new axiomatization of semidirect products of quantum groups on the von Neumann algebra level, differing from prior approaches and facilitating the use of crossed product methods.

Findings

Existence of a braided comultiplication on the algebra

Introduction of a von Neumann algebraic axiomatization

Application of crossed product techniques to quantum groups

Abstract

We introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study properties. Our approach is slightly different from the one introduced in the thesis of S.~Roy and, unlike the investigations of Roy, we work within the von Neumann algebraic picture. This allows the use of powerful techniques related to crossed products by actions of locally compact quantum groups. In particular we show existence of a "braided comultiplication" on the algebra spanned by slices of "braided multiplicative unitary".