The general solution of the real Z ⊗ N 2 graded contractions of so ( N + 1 )

The general solution of the graded contraction equations for a Z ⊗N 2 grading of the real compact simple Lie algebra so(N +1) is presented in an explicit way. It turns out to depend on 2 N −1 independent real parameters. The structure of the general graded contractions is displayed for the low dimensional cases, and kinematical algebras are shown to appear straightforwardly. The geometrical (or physical) meaning of the contraction parameters as curvatures is also analysed; in particular, for kinematical algebras these curvatures are directly linked to geometrical properties of possible homogeneous space-times.