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 𝕋 viewed as a quasitriangular quantum group with respect to the unitary R-matrix R(m,n) := (q/q̄)mn for all m,n ∈ ℤ. For real 0 < |q| < 1, the deformation coincides with Woronowicz’s Eq(2) groups. As an application, we study the braided analogue of the contraction procedure between SUq(2) and Eq(2) groups in the spirit of Woronowicz’s quantum analogue of the classic Inönü–Wigner group contraction. Consequently, we obtain the bosonization of braided Eq(2) groups by contracting Uq(2) groups.

中文翻译：

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用于复杂变形参数的 Quantum E(2) 组

我们构建了一个家庭q的变形乙(2)非零复杂参数的组|q| < 1作为圆群上的局部紧致编织量子群𝕋看成一个关于酉的准三角量子群R-矩阵R(米,n) ：= (q/q̄)米n对所有人米,n ∈ ℤ. 真的0 < |q| < 1, 变形与 Woronowicz 的一致乙q(2)团体。作为一个应用程序，我们研究了之间的收缩过程的编织类似物苏q(2)和乙q(2)Woronowicz 的经典 Inönü-Wigner 群收缩的量子类似物精神中的群。因此，我们得到了编织的玻色化乙q(2)通过承包分组üq(2)团体。