New Seiberg-Witten Fields Maps Through Weyl Symmetrization and The Pure Geometric Extension of The Standard Model

A unified description of a symmetrized and anti-symmetrized Moyal star product of the noncommutative infinitesimal gauge transformations is presented and the corresponding Seiberg-Witten maps are derived. Moreover, the noncommutative covariant derivative, field strength tensor as well as gauge transformations are shown to be consistently constructed not on the enveloping but on the Lie and/or Poisson algebra. As an application, a pure geometric extension of the standard model is shown explicitly.