Quantum $E(2)$ groups for complex deformation parameters

Abstract

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

22 pages. Some proofreading and minor corrections. The paper will appear in Reviews in Mathematical Physics

References in corpus (4)

• Quantum group-twisted tensor products of C*-algebras
• Quantum group-twisted tensor products of C*-algebras II
• Braided quantum SU(2) groups
• Semidirect products of C*-quantum groups: multiplicative unitaries approach

Cited by in corpus (3)

• Anyonic quantum symmetries of finite spaces
• Braided quantum groups and their bosonizations in the $C^{\ast }$-algebraic framework
• Homogeneous quantum symmetries of finite spaces over the circle group