An Analytic Method for $S$-Expansion involving Resonance and Reduction

TL;DR

This paper introduces an analytic method to determine the multiplication tables of sets involved in $S$-expansions of Lie (super)algebras, facilitating the construction of target algebras from initial ones with verification of associativity.

Contribution

The paper presents a novel analytic approach to derive set multiplication tables in $S$-expansions, simplifying the process and extending previous results.

Findings

The method successfully derives multiplication tables for $S$-expansions.

It verifies associativity to distinguish semigroup from abelian set structures.

Applications confirm the method's effectiveness and generalize existing results.

Abstract

In this paper 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 $0_S$-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 partitions over 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). Finally, we check associativity with an auxiliary computational algorithm, in order to understand whether the obtained set(s) can describe semigroup(s) or just abelian set(s) connecting two (super)algebras. We also give some interesting examples of application, which check and corroborate our analytic procedure and also generalize some result already presented in the literature.