Polyhedron phase space using 2-groups: kappa-Poincare as a Poisson 2-group

TL;DR

This paper develops a phase space framework for 3D cellular complexes using Poisson 2-groups, specifically focusing on kappa-deformations of the Poincare group to model discrete geometries with curved edges.

Contribution

It introduces a novel phase space construction based on crossed modules with Poisson structures, extending classical models to include kappa-deformations of the Poincare 2-group.

Findings

Recovered deformations of the Poincare group as deformations of the Poincare 2-group

Provided a natural model for discrete geometries with curved edge decorations

Generalized classical phase space for 3D triangulations to include quantum group deformations

Abstract

We construct a phase space for a three dimensional cellular complex with decorations on edges and faces using crossed modules (strict 2-groups) equipped with a (non-trivial) Poisson structure. We do not use the most general crossed module, but only the ones where the target map (t-map) is trivial. As a particular case, we recover that deformations of the Poincare group can be exported to deformations of the Poincare 2-group. The kappa-deformation case provides a natural candidate for describing discrete geometries with curved edge decorations (but still flat face decorations). Our construction generalizes the classical phase space defined in [5] for a 3d triangulation in terms of an un-deformed Poincare 2-group.