Gauge theory on $\rho$-Minkowski space-time

We construct a gauge theory model on the 4-dimensional $\rho$-Minkowski space-time, a particular deformation of the Minkowski space-time recently considered. The corresponding star product results from a combination of Weyl quantization map and properties of the convolution algebra of the special Euclidean group. We use noncommutative differential calculi based on twisted derivations together with a twisted notion of noncommutative connection. The twisted derivations pertain to the Hopf algebra of $\rho$-deformed translations, a Hopf subalgebra of the $\rho$-deformed Poincar\'e algebra which can be viewed as defining the quantum symmetries of the $\rho$-Minkowski space-time. The gauge theory model is left invariant under the action of the $\rho$-deformed Poincar\'e algebra. The kinetic part of the action is found to coincide with the one of the usual (commutative) electrodynamics.