Jordanian twist quantization of D=4 Lorentz and Poincaré algebras and D=3 contraction limit

Abstract

We describe in detail the two-parameter nonstandard quantum deformation of the D=4 Lorentz algebra $\mathfrak{o}(3,1)$, linked with a Jordanian deformation of $\mathfrak{sl}(2;\mathbb{C})$. Using the twist quantization technique we obtain the explicit formulae for the deformed co-products and antipodes. Further extending the considered deformation to the D=4 Poincaré algebra we obtain a new Hopf-algebraic deformation of four-dimensional relativistic symmetries with a dimensionless deformation parameter. Finally, we interpret $\mathfrak{o}(3,1)$ as the D=3 de Sitter algebra and calculate the contraction limit $R\rightarrow\infty$ (R is the de Sitter radius) providing an explicit Hopf algebra structure for the quantum deformation of the D=3 Poincaré algebra (with mass-like deformation parameters), which is the two-parameter light-cone κ-deformation of the D=3 Poincaré symmetry.