Poisson gauge theory

The Poisson gauge algebra is a semi-classical limit of complete non-commutative algebra. In the present work we formulate theory which dynamical field theoretical model having as corresponding symmetries. proposed designed to investigate features full with coordinate dependent non-commutativity $\Theta^{ab}(x)$, especially whose non-constant rank. We derive expression for covariant derivative matter field. commutator relation derivatives defines strength under transformations and reproduces standard $U(1)$ in commutative limit. Bianchi identities. equations fields are obtained from invariant action. consider different examples linear coordinates structures well non-linear ones, obtain explicit expressions all constructions. Our unique up invertible redefinitions transformations.