Towards a topological (dual of) quantum $\kappa$-Poincar\'{e} group

TL;DR

This paper establishes a $C^*$-algebraic formulation of the $"kappa$-Poincaré group, revealing its relation to Lie group factorizations and extending the momentum manifold to resolve singularities, with implications for Doubly Special Relativity.

Contribution

It provides an explicit $C^*$-algebraic construction of the $"kappa$-Poincaré group and analyzes its orbit structure and momentum manifold extension.

Findings

Explicit factorization of the Lie group related to $"kappa$-deformation

Extension of the momentum manifold to eliminate singularities

Analysis of the global properties of the extended manifold

Abstract

We argue that the $\kappa$-deformation is related to a factorization of a Lie group, therefore {\em an approproate version of $\kappa$-Poincar\'{e} does exist on the $C^*$-algebraic level}. The explict form of this factorization is computed that leads to an ``action'' of the Lorentz group (with space reflections) considered in Doubly Special Relativity theory. The orbit structure is found and ``the momentum manifold'' is extended in a way that removes singularities of the ``action'' and results in a true action. Some global properties of this manifold are investigated