Braided Quantum Groups and Their Bosonizations in the C*-Algebraic Framework

Abstract

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