An Analytic Method for S-Expansion involving Resonance and Reduction

In this work we describe an analytic method able to give the multiplication table(s) of the set(s) involved in an S-expansion process (with either resonance or 0-resonant-reduction) for reaching a target Lie (super)algebra from a starting one,after having properly chosen the partitions over subspaces of the considered (super)algebras. This analytic method gives us a simple set of expressions to find the subset decomposition of the set(s) involved in the process. Then, we use the information coming from both the initial (super)algebra and the target one for reaching the multiplication table(s) of the mentioned set(s). We also give some interesting examples of application which check and corroborate our analytic procedure andgeneralize some result already presented in the literature