Quantum E(2) groups for complex deformation parameters

We construct a family of $q$ deformations $E(2)$ group for nonzero complex parameters $|q|<1$ as locally compact braided quantum groups over the circle $\mathbb{T}$ viewed quasitriangular with respect to unitary R-matrix $R(m,n):=(\zeta)^{mn}$ all $m,n\in\mathbb{Z}$. For real $0<|q|<1$, deformation coincides Woronowicz's $E_{q}(2)$ groups. As an application, we study analogue contraction procedure between $SU_{q}(2)$ and in spirit classic In\"on\"u-Wigner contraction. Consequently, obtain bosonisation by contracting $U_{q}(2)$