Large Quantum Poincare Subgroup of q-Conformal Group and q-Minkowski Geometry

TL;DR

This paper constructs a quantum deformation of the Poincaré group derived from the conformal group and explores the conditions under which a Poincaré subgroup exists, revealing a unique nonstandard deformation with a commuting Minkowski space.

Contribution

It introduces a quantum deformation of the Poincaré group based on the conformal group and identifies conditions for the existence of a Poincaré subgroup in this quantum setting.

Findings

Quantum deformation of $SU(2,2)$ constructed from $GL(4,C)_{q_{ij},r}$

Poincaré subgroup exists only for a specific nonstandard deformation with $r=1$

Deformation leads to a commuting affine Minkowski space and a simplified Lie algebra structure.

Abstract

We construct quantum deformation of Poincar\'e group using as a starting point $SU(2,2)$ conformal group and twistor-like definition of the Minkowski space. We obtain quantum deformation of $SU(2,2)$ as a real form of multiparametric $GL(4,C)_{q_{ij},r}$. It is shown that Poincar\'e subgroup exists for special nonstandard one-parametric deformation only, the deformation parameter $r$ being equal to unity. This leads to commuting affine structure of the corresponding Minkowski space and simple structure of the corresponding Lie algebra, the deformation of the group being non-trivial.