General non-commutative supersymmetric quantum mechanics models in two and three dimensions are constructed and some two and three dimensional examples are explicitly studied. The structure of the theory studied suggest other possible applications in
physical systems with potentials involving spin and non-local interactions.
The chiral anomaly in the context of an extended standard model with minimal Lorentz invariance violation is studied. Taking into account bounds from measurements of the speed of light, we argue that the chiral anomaly and its consequences are genera
l results valid even beyond the relativistic symmetry.
