Braided quantum groups and their bosonizations in the $C^*$-algebraic framework

TL;DR

This paper develops a comprehensive theory of braided quantum groups within the C*-algebraic framework, establishing their structure, classification, and examples, including the quantum plane and quantum E(2) group.

Contribution

It introduces a new construction of braided C*-quantum groups from multiplicative unitaries and generalizes bosonization to this setting.

Findings

Established a one-to-one correspondence between braided C*-quantum groups and C*-quantum groups with projection.

Constructed braided C*-quantum groups from manageable multiplicative unitaries.

Identified the quantum plane as a braided C*-quantum group over the circle group.

Abstract

We present a general theory of braided quantum groups in the C*-algebraic framework using the language of multiplicative unitaries. Starting with a manageable multiplicative unitary in the representation category of the quantum codouble of a regular quantum group $\mathbb{G}$ we construct a braided C*-quantum group over $\mathbb{G}$ as a C*-bialgebra in the monoidal category of the $\mathbb{G}$-Yetter-Drinfeld C*-algebras. Furthermore, we establish the one to one correspondence between braided C*-quantum groups and C*-quantum groups with projection. Consequently, we generalise the bosonization construction for braided Hopf-algebras of Radford and Majid to braided C*-quantum groups. Several examples are discussed. In particular, we show that the complex quantum plane admits a the braided C*-quantum group structure over the circle group $\mathbb{T}$ and identify its bosonization with the simplified quantum $E(2)$ group.