Narrow Results

We construct a family of $q$ deformations of $\mathrm{\text{E}}(2)$ group for nonzero complex parameters $\left|q\right|<1$ as locally compact braided quantum groups over the circle group $𝕋$ viewed as a quasitriangular quantum group with respect to the unitary $R$-matrix $\mathrm{\text{R}}(m,n):={(q/\bar{q})}^{mn}$ for all $m,n\in \mathbb{Z}$. For real $0<\left|q\right|<1$, the deformation coincides with Woronowicz’s ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups. As an application, we study the braided analogue of the contraction procedure between ${\mathrm{\text{SU}}}_{\mathrm{\text{q}}}(2)$ and ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups in the spirit of Woronowicz’s quantum analogue of the classic Inönü–Wigner group contraction. Consequently, we obtain the bosonization of braided ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups by contracting ${\mathrm{\text{U}}}_{\mathrm{\text{q}}}(2)$ groups.