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

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.