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

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.