Quantum E(2) groups for complex deformation parameters

We construct a family of $q$ deformations of $E(2)$ groups for nonzero complex parameters $|q|<1$ as locally compact braided quantum groups over the circle group $\mathbb{T}$ with respect to the unitary $R$-matrix $\chi\colon\mathbb{Z}\times\mathbb{Z}\to\mathbb{T}$ defined by $\chi(m,n):=(\zeta)^{mn}$, where $\zeta:= q/\bar{q}$. For real $0<|q|<1$, the deformation coincides with Woronowicz's $E_{q}(2)$ groups.