Cubic supersymmetry and abelian gauge invariance
We consider a 4 dimensional space-time symmetry, which is a non-trivial extension of the Poincar\'e algebra. This new symmetry extract "cubic roots" of translations, hence being different from the supersymmetric extension and not contradicting the no-go theorems. Some field theoretical aspects are investigated. Boson multiplets and free Lagrangians are explicitely built. We also study in detail the interplay between this symmetry and abelian gauge symmetry.