On free Lie algebras and particles in electro-magnetic fields
Joaquim Gomis, Axel Kleinschmidt
TL;DR
The paper identifies a maximal infinite extension of the Poincaré algebra, $Maxwell_{\infty}$, as the semi-direct product of $so(1,D-1)$ with a free Lie algebra generated by translations, tying this structure to Chevalley–Eilenberg cohomology and unfolding dynamics. It provides a concrete non-linear coset construction and a low-derivative point-particle model with universal equations of motion that couple an infinite tower of generalized momenta to space-time coordinates, capturing multipole backreactions. The authors construct and analyze level-by-level generators up to $\ell=4$, show how finite truncations reproduce known Maxwell algebras, and discuss quotients that yield unfolded electromagnetic backgrounds. This framework offers a general, symmetry-driven approach to particle dynamics in generic EM backgrounds, with potential links to unfolded dynamics and higher-spin/ gauge theories, and suggests several directions for Hamiltonian formulation and extensions to non-relativistic or supersymmetric settings.
Abstract
The Poincaré 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.