Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity

Understand

We present an in-depth investigation of the ${\rm SL}(2,\mathbb{R})$ momentum space describing point particles coupled to Einstein gravity in three space-time dimensions.

• We introduce different sets of coordinates on the group manifold and discuss their properties under Lorentz transformations.
• In particular we show how a certain set of coordinates exhibits an upper bound on the energy under deformed Lorentz boosts which saturate at the Planck energy.
• We discuss how this deformed symmetry framework is generally described by a quantum deformation of the Poincar\'e group: the quantum double of ${\rm SL}(2,\mathbb{R})$.