Conservation laws in Lie-Poisson classical field theories

Lie-Poisson classical field theory is a field-theoretical model embedded in a non-commutative structure related to the framework of Poisson electrodynamics. In this paper, we follow the recently developed action principle for Lie-Poisson electrodynamics to derive the conservation laws of the theory. The energy-momentum tensor is obtained, along with the conserved electric charge and the momentum operator. We consider non-interacting examples for real and complex scalar fields, as well as the Dirac field, within the $\kappa$-Minkowski spacetime framework. In the latter case, we show that the non-relativistic limit for the $\kappa$-Minkowski Dirac equation introduces an orbital Zeeman coupling term for the fermionic fields, and the energy shift in the first excited state depends exclusively on the $\kappa$-parameter.