Action principle for $κ$-Minkowski noncommutative $U(1)$ gauge theory from Lie-Poisson electrodynamics

Lie-Poisson electrodynamics describes a semiclassical approximation of noncommutative $U(1)$ gauge theories with Lie-algebra-type noncommutativities. We obtain a gauge-invariant local classical action with the correct commutative limit for a generic Lie-Poisson gauge model, and present the corresponding deformed Maxwell equations. At the semiclassical level, our results provide a relatively simple solution to the old problem of constructing an admissible Lagrangian formulation for the $U(1)$ gauge theory on the four-dimensional $κ$-Minkowski space-time. On the one hand, we derive an explicit expression for the classical action which yields the deformed Maxwell equations previously proposed in JHEP 11 (2023) 200 for this noncommutativity on general grounds. On the other hand, according to our analysis, these Maxwell equations follow from the action proposed in the present paper as the Euler-Lagrange equations for any Lie-algebra-type noncommutativity.