On free Lie algebras and particles in electro-magnetic fields

Understand

The Poincar\'e algebra can be extended (non-centrally) to the Maxwell algebra and beyond.

• These extensions are relevant for describing particle dynamics in electro-magnetic backgrounds and possibly including the backreaction due the presence of multipoles.
• We point out a relation of this construction to free Lie algebras that gives a unified description of all possible kinematic extensions, leading to a symmetry algebra that we call Maxwell${}_\infty$.
• A specific dynamical system with this infinite symmetry is constructed and analysed.