# On free Lie algebras and particles in electro-magnetic fields

## TL;DR

This paper explores the connection between free Lie algebras and extended symmetry algebras like Maxwell_infinity, providing a unified framework for describing particles in electromagnetic backgrounds and analyzing a related dynamical system.

## Contribution

It introduces a novel link between free Lie algebras and the Maxwell_infinity symmetry, offering a comprehensive description of kinematic extensions in electromagnetic fields.

## Key findings

- Unified description of all kinematic extensions via free Lie algebras
- Construction of a dynamical system with Maxwell_infinity symmetry
- Analysis of particle dynamics in electromagnetic backgrounds

## Abstract

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.