Jordanian Twist Quantization of D=4 Lorentz and Poincare Algebras and D=3 Contraction Limit

TL;DR

This paper develops a new two-parameter quantum deformation of D=4 Lorentz and Poincaré algebras using twist quantization, and explores the contraction limit to D=3 Poincaré algebra, providing explicit Hopf algebra structures.

Contribution

It introduces a novel two-parameter nonstandard quantum deformation of D=4 Lorentz and Poincaré algebras via twist quantization, including a contraction to D=3 Poincaré algebra.

Findings

Explicit formulas for deformed coproducts and antipodes.

New Hopf algebraic deformation of D=4 relativistic symmetries.

Contraction limit yields a two-parameter light-cone κ-deformation of D=3 Poincaré algebra.

Abstract

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